text:
The idea of the proof is to create a grid of functions on $\Xc$ based on a covering of $\Fc$ to cover $\Ec_n = \{ \eps_{n,f} : f \in \Fc\}$. From this grid, we will deduce coverings of $\Gc_n$ and $\Gc_n^{(d)}$.

\paragraph{Step 1: covering of $\Ec_n$.}

By assumption, the class $\Fc$ is VC of parameters $(v,A)$ with respect to an envelope $F$. That means that for any $0 < \eta < 1$ and for any probability measure $Q$ on $\Xc$, we have
\[
	\Nc\left(\Fc, L^2(Q), \eta \|F\|_{L^2(Q)}\right) \le \left( \frac{A}{\eta} \right)^{v} .
\]
Moreover, a single ball centered at the constant function equal to zero and with radius $\| F \|_{L^2(Q)}$ is enough to cover $\Fc$. Thus, for any $\eta \in (0, \infty)$, the covering number is bounded from above by
\[
	\Nc\left(\Fc, L^2(Q), \eta\right) \le \left( \frac{A \|F\|_{L^2(Q)}}{\eta} \right)^{v} \vee 1.
\]
Fix $\eta > 0$, write $\eta_P = \eta/(4 \ninf{q_n}^{\half})$ and $\eta_Q = \eta/4$, and define the covering numbers
\begin{align}
\label{eq:NPNQ}
	N_P &= \Nc\left(\Fc, L^2(P), \eta_P / 2 \right), &
	N_Q &= \Nc\left(\Fc, L^2(Q), \eta_Q / 2 \right)
\end{align}
associated to the open balls
\begin{align*}
	B_P(f, \delta) &= \left\{g \in L^2(P) : \|g-f\|_{L^2(P)} < \delta \right\}, &
	B_Q(f, \delta) &= \left\{g \in L^2(Q) : \|g-f\|_{L^2(Q)} < \delta \right\},
\end{align*}
for $\delta > 0$. The balls in the definition of the covering numbers in \eqref{eq:NPNQ} have their centers in $L^2(P)$ and $L^2(Q)$ but not necessarily in $\Fc$. At the price of doubling the radii, the triangle inequality permits us to find functions $f_1^{(P)},\ldots, f_{N_P}^{(P)}$ and $f_1^{(Q)},\ldots, f_{N_Q}^{(Q)}$ in $\Fc$ such that
\begin{align*}
\label{eq:FcBPBQ}
\Fc &\subset \bigcup_{i=1}^{N_P} B_P( f_i^{(P)}, \eta_P ), &
\Fc &\subset \bigcup_{j=1}^{N_Q} B_Q( f_j^{(Q)}, \eta_Q ).
\end{align*}
Therefore, the class $\Fc$ is covered by the union of the intersections between the balls, that is to say
\begin{equation}\label{eq:inclusion}
	\Fc \subset \bigcup_{\substack{1 \le i\le N_P \\ 1\le j\le N_Q}}
	\left( B_P( f_i^{(P)}, \eta_P ) \cap B_Q( f_j^{(Q)}, \eta_Q ) \right).
\end{equation}
Define the support of this covering as
\[
\mathsf{S} = \left\{(i,j) \in  \left\{1,\ldots, N_P\right\} \times \left\{1,\ldots, N_Q\right\}: B_P( f_i^{(P)}, \eta_P ) \cap B_Q( f_j^{(Q)}, \eta_Q ) \neq \emptyset \right\}.
\]
For every $(i,j)\in\mathsf{S}$, we fix an arbitrary function $f_{i,j}\in B_P( f_i^{(P)}, \eta_P ) \cap B_Q( f_j^{(Q)}, \eta_Q )$.

Let $f\in \Fc$ and let $(i,j) \in \mathsf{S}$ be such that $f \in B_P( f_i^{(P)}, \eta_P ) \cap B_Q( f_j^{(Q)}, \eta_Q )$. We will show that $\norm{\eps_{n,f} - \eps_{n,f_{i,j}}}_{L^2(Q)} \le \eta$. Since $f$ and $f_{i,j}$ belong to the same intersection in \eqref{eq:inclusion}, we have
\begin{align}
\label{eq:ffijL2PL2Q}
	\norm{f - f_{i,j}}_{L^2(P)} &< 2 \eta_P, &
	\norm{f - f_{i,j}}_{L^2(Q)} &< 2 \eta_Q.
\end{align}
The residual functions can be expressed in terms of the whitened feature map $\hbar_n$ via
\begin{align*}
	\eps_{n,f} &= f - \hbar_n^\T P(\hbar_n f), &
	\eps_{n,f_{i,j}} &= f_{i,j} - \hbar_n^\T P(\hbar_n f_{i,j}).
\end{align*}
By the triangle inequality, we find
\begin{align}
	\label{eq:residuals}
	\norm{ \eps_{n,f} - \eps_{n, f_{i,j}} }_{L^2(Q)}
	\le \norm{ f - f_{i,j} }_{L^2(Q)} + \norm{ \hbar_n^\T P[\hbar_n(f - f_{i,j})] }_{L^2(Q)}.
\end{align}
Recall $M_n \ge \sup_{f \in \Fc} \ninf{\eps_{n,f}}$, a constant envelope for the class $\Ec_n$, and recall the leverage function $q_n = |\hbar_n|_2^2$. The Cauchy--Schwarz inequality and the orthonormality of $\prn{\hbar_{n,1},\ldots,\hbar_{n,d_n}}$ give
\begin{align}
\nonumber
\norm{ \hbar_n^\T P[\hbar_n(f - f_{i,j})] }_{L^2(Q)}^2
&= \int_{y\in\Xc} \left\{ \hbar_n(y)^\T \int_{x\in \Xc} \hbar_n(x) (f-f_{i,j})(x) \, \diff P (x) \right\}^2 \,\diff Q(y) \\
\nonumber
&\le \int_{y\in\Xc} |\hbar_n(y)|_2^2 \left| \int_{x\in \Xc} \hbar_n(x) (f-f_{i,j}) (x) \, \diff P (x) \right|_2^2 \,\diff Q(y) \\
\label{eq:hnPhnfL2Q}
&\le \ninf{q_n} \norm{f - f_{i,j}}^2_{L^2(P)}.
\end{align}
The combination of \eqref{eq:ffijL2PL2Q}, \eqref{eq:residuals} and \eqref{eq:hnPhnfL2Q} yields
\[
	\norm{\eps_{n,f} - \eps_{n, f_{i,j}}}_{L^2(Q)}
	< 2 \eta_Q + 2 \ninf{q_n}^{\half} \eta_P
	= \eta/2 + \eta/2
	= \eta.
\]
We have thus constructed the covering
\begin{equation*}
\Ec_n \subset \bigcup_{(i,j)\in\mathsf{S}} B_Q( \eps_{n,f_{i,j}}, \eta )
\end{equation*}
of $\Ec_n$ with $L^2(Q)$ balls of radius at most $\eta$. The covering number of $\Ec_n$ is thus bounded by
\[
\Nc\left(\Ec_n, L^2(Q), \eta\right)
\le \# \mathsf{S}
\le \Nc\left(\Fc, L^2(P), \eta_P / 2 \right) \cdot \Nc\left(\Fc, L^2(Q), \eta_Q / 2 \right).
\]
Using the definition of a VC-class and since $\norm{q_n}_\infty \ge P(q_n) =  P(|\hbar_n|_2^2) = d_n \ge 1$, a case-by-case analysis reveals that
\begin{align}
\nonumber
	\Nc\left(\Ec_n, L^2(Q), \eta\right)
	&\le
	\left(\frac{2 A \norm{F}_{L^2(P)}}{\eta_P}\vee 1 \right)^v \cdot
	\left(\frac{2 A \norm{F}_{L^2(Q)}}{\eta_Q} \vee 1 \right)^v \\
\nonumber
	&\le
	\left(\frac{64 A^2 \norm{F}_{\infty}^2\ninf{q_n}^{\half}}{\eta^2}\right)^v
	\vee \left(\frac{8 A \norm{F}_{\infty} \ninf{q_n}^{\half}}{\eta}\right)^v \vee 1 \\

\label{eq:NE}
	&\le \left(\frac{8 A \norm{F}_{\infty} \ninf{q_n}^{\half}}{\eta}\right)^{2v} \vee 1.

\end{align}

Lemma~\ref{lemma:maj M_n} implies that the sequence $A_n$ defined in~\eqref{eq:def A_n} satisfies $A_n\ge 1$.

From~\eqref{eq:NE} we deduce
\[
	\forall \eta\in(0,1],\qquad
	\Nc\prr{\Ec_{n},L^2(Q),\eta M_n} \le (A_n/\eta)^{2v}.
\]
Therefore, the residual class $\Ec_n$ is VC of parameters $(2v, A_n)$ with respect to the envelope $M_n$.

\paragraph{Step 2: covering of $\Gc_n$.}

Consider two functions $f, \tilde{f}\in\Fc$ and a probability measure $Q$ on $\Xc^2$ with marginals $Q_1, Q_2$ on $\Xc$, that is, $Q_1(B) = Q(B \times \Xc)$ and $Q_2(B) = Q(\Xc \times B)$ for measurable $B \subset \Xc$. By definition, every function in $\Gc_n$ is written as $(x,y)\mapsto \hbar_n(x)^\T \hbar_n(y) \eps_{n,f}(x)\eps_{n,f}(y)$. The Cauchy--Schwarz inequality gives
\begin{equation}
\label{eq:hxhybound}
	\forall (x, y) \in \Xc^2, \qquad
	|\hbar_n(x)^\T \hbar_n(y)|^2
	\le |\hbar_n(x)|_2^2 \, |\hbar_n(y)|_2^2
	= q_n(x) q_n(y)
	\le \ninf{q_n}^2.
\end{equation}

For $f, \tilde{f} \in \Fc$ and $(x, y) \in \Xc^2$, define
\[
	g_{n,f,\tilde{f}}(x, y)
	= \eps_{n,f}(x) \hbar_n(x)^\T \hbar_n(y) \eps_{n,\tilde{f}}(y).
\]
Note that $g_{n,f} = g_{n,f,\tilde{f}}$. By the Minkowski inequality,
\[
	\norm{ g_{n, f} - g_{n, \tilde{f}} }_{L^2(Q)}
	\le
	\norm{ g_{n, f} - g_{n, f, \tilde{f}} }_{L^2(Q)}
	+
	\norm{ g_{n, f, \tilde{f}} - g_{n, \tilde{f}} }_{L^2(Q)}.
\]
Let us look at square of the first term on the right-hand side: by~\eqref{eq:hxhybound},
\begin{align*}
	\norm{ g_{n, f} - g_{n, f, \tilde{f}} }_{L^2(Q)}^2
	&= \int_{\Xc^2}
		\eps_{n,f}(x)^2
		\left|\hbar_n(x)^\T \hbar_n(y)\right|^2
		\left( \eps_{n,f}(y) - \eps_{n,\tilde{f}}(y) \right)^2
		\diff Q(x, y) \\
	&\le \int_{\Xc^2}
		\eps_{n,f}(x)^2 q_n(x) q_n(y)
		\left( \eps_{n,f}(y) - \eps_{n,\tilde{f}}(y) \right)^2
		\diff Q(x, y) \\
	&\le \ninf{q_n \eps_{n,f}^2}
	\int_{\Xc}
		q_n(y)
		\left( \eps_{n,f}(y) - \eps_{n,\tilde{f}}(y) \right)^2
	\diff Q_2(y) \\
	&\le \ninf{q_n \eps_{n,f}^2} \ninf{q_n} \norm{\eps_{n,f} - \eps_{n,\tilde{f}}}_{L^2(Q_2)}^2.
\end{align*}
The term $\norm{g_{n,f,\tilde{f}} - g_{n,\tilde{f}}}_{L^2(Q)}$ can be treated similarly, yielding
\begin{align}
	\lefteqn{
	\norm{ g_{n, f} - g_{n, \tilde{f}} }_{L^2(Q)}
	} \nonumber \\
	\nonumber
	&\le
	\ninf{q_n \eps_{n,f}^2}^{1/2} \ninf{q_n}^{1/2} \norm{\eps_{n,f} - \eps_{n,\tilde{f}}}_{L^2(Q_2)}
	+
	\ninf{q_n \eps_{n,\tilde{f}}^2}^{1/2} \ninf{q_n}^{1/2} \norm{\eps_{n,f} - \eps_{n,\tilde{f}}}_{L^2(Q_1)} \\
	\label{eq:gnfft}
	&\le M_n \ninf{q_n}
	\left(
	\norm{\eps_{n,f} - \eps_{n,\tilde{f}}}_{L^2(Q_1)}
	+
	\norm{\eps_{n,f} - \eps_{n,\tilde{f}}}_{L^2(Q_2)}
	\right).
\end{align}

Fix $\eta > 0$. Following the approach in Step~1, we can for $\ell \in \{1, 2\}$ construct a covering of $\Ec_n$ by $L^2(Q_\ell)$-balls of at most radius $\eta$. The centers of the balls are of the form
\[
	\forall \ell = 1, 2, \;
	\forall k = 1, \ldots, m_\ell, \qquad
	\eps_{n,f^{(\ell)}_{k}} \in \Ec_n,
\]
where $f^{(\ell)}_{k}$ belongs to $\Fc$ and where $m_\ell$ is the number of such balls needed, a number which is bounded by $(A_n M_n / \eta )^{2v}\vee 1$ for $A_n$ defined in~\eqref{eq:def A_n} Consider the intersections
\[
	\forall i = 1, \ldots, m_1, \; \forall j = 1, \ldots, m_2, \qquad
	B(i, j) = B_{Q_1} \bigl(\eps_{n,f^{(1)}_{i}}, \eta \bigr) \cap B_{Q_2}\bigl(\eps_{n,f^{(2)}_{j}}, \eta\bigr).
\]

The set $\Ec_n$ is covered by the union of all those intersections $B(i,j)$. For each $(i, j) \in \{1,\ldots,m_1\} \times \{1,\ldots,m_2\}$ such that $\Ec_n$ intersects $B(i,j)$, pick an arbitrary $f_{i,j} \in \Fc$ such that $\eps_{n,f_{i,j}} \in \Ec_n \cap B(i,j)$. Note that the functions $f_{i,j}$ are different from the ones denoted in the same way in Step~1.

Let $f \in \Fc$ and let $(i,j)$ be such that $\eps_{n,f}$ and $\eps_{n,f_{i,j}}$ belong to the same intersection $B(i, j)$. Since the diameters of the two balls in the definition of $B(i,j)$ are bounded by $2\eta$ in view of the triangle inequality, we find
\[
	\forall \ell = 1, 2, \qquad
	\norm{\eps_{n,f} - \eps_{n,f_{i,j}}}_{L^2(Q_\ell)}
	< 2\eta.
\]
By \eqref{eq:gnfft}, it follows that
\[
	\norm{g_{n,f} - g_{n,f_{i,j}}}_{L^2(Q)}
	< M_n \ninf{q_n} [(2\eta) + (2\eta)]
	= 4 M_n \ninf{q_n} \eta.
\]

We find that $\Gc_n$ is covered by the union of the balls $B_Q(g_{n,f_{i,j}}, 4 M_n \ninf{q_n} \eta)$. Its covering number is thus bounded by
\begin{equation*}

	\Nc\left( \Gc_n, L^2(Q), 4 M_n \ninf{q_n} \eta \right)
	\le m_1 m_2 \\

	\le \left( \frac{A_n M_n}{\eta} \right)^{4v} \vee 1.
\end{equation*}
Rescaling $M_n \eta' = 4 \eta$, we get
\begin{equation}
\label{eq:NGM2q}
	\Nc\left( \Gc_n, L^2(Q), M_n^2 \ninf{q_n} \eta' \right)

	\le \left( \frac{4 A_n}{\eta'} \right)^{4v} \vee 1.
\end{equation}
In view of \eqref{eq:hxhybound}, the functions in $\Gc_n$ are uniformly bounded by $M_n^2 \ninf{q_n}$. Since $Q$ was an arbitrary probability measure on $\Xc^2$, we conclude that $\Gc_n$ is a VC-class with parameters $(4v, 4A_n)$ with respect to the constant envelope $M_n^2 \ninf{q_n}$.

\paragraph{Step 3: covering of $\Gc_n^{(d)}$.}

Let $Q$ be a probability measure on $\Xc$ and let $\eta > 0$. In Step~1, we found functions $f_{i,j} \in \Fc$ such that $\Ec_n$ is covered by the balls $B_Q(\eps_{n,f_{i,j}}, \eta)$, and we needed at most $(A_n M_n / \eta)^{2v} \vee 1$ of such functions. For $f \in \Fc$ we can thus find $(i, j)$ such that
\[
	\norm{\eps_{n,f} - \eps_{n,f_{i,j}}}_{L^2(Q)} < \eta
\]
and thus
\begin{align*}
	\norm{ \sqrt {q_n} \eps_{n,f} - \sqrt{q_n} \eps_{n,f_{i,j}} }_{L^2(Q)}^{2}

	&\le  \ninf{q_n}  \left\| \eps_{n,f} - \eps_{n,f_{i,j}} \right\|^2 _{L^2(Q)} \\
	&< \ninf{q_n}  \eta^2.
\end{align*}
It follows that the number of $L^2(Q)$ balls of radius $  \ninf{q_n} ^{1/2} \eta$ needed to cover $\Gc_n^{(d)}$ is bounded by the number of functions $f_{i,j}$ in the construction in Step~1, and so
\begin{align*}

	\Nc\left( \Gc_n^{(d)}, L^2(Q),    \ninf{q_n} ^{1/2} \eta \right)
	&\le
	\Nc\left( \Ec_n, L^2(Q), \eta \right) \\

	&\le
	\left( \frac{A_n M_n}{\eta} \right)^{2v} \vee 1.
\end{align*}
Upon rescaling $M_n \eta' =  \eta$, we find
\begin{equation}
\label{eq:NGdM2q}
\Nc\left( \Gc_n^{(d)}, L^2(Q), M_n     \ninf{q_n} ^{1/2}  \eta' \right)

\le \left( \frac{ A_n}{\eta'} \right)^{2v}\vee 1.
\end{equation}
We conclude that $\Gc_n^{(d)}$ is a VC-class with parameters $(2v, A_n)$ with respect to the constant envelope $M_n  \ninf{q_n}^{1/2}$. The proof of Proposition~\ref{prop:cover} is complete. \qed


TITLE: Use of □ in Modal Logic, Translated into English
QUESTION [0 upvotes]: For $\Box$, do you read it as “for all states. . .”, that which one, maybe, cannot withdraw from?
Does this, $\Box$, not request proving of itself?
$ \Box P \leftrightarrow  \neg \Diamond  \neg P$
How do you use it in English?
If, for all states $P$, if and only if, not in some states, not $P$?

REPLY [1 votes]: You shouldn't quantify over all states, merely the accessible ones. The most precise translation of $\Box$ is "For all accessible states," and of $\Diamond$ is "For some accessible state." So "$\neg \Diamond \neg P$" means "There is no accessible state such that $\neg P$ holds," which is equivalent to saying "In every accessible state, $P$ holds" - which is "$\Box P$."
More colloquially, "$\Box$" and "$\Diamond$" are usually translated as "Necessarily" and "Possibly," respectively. So "$\neg\Diamond\neg P$" means "It is not possible that $P$ fails," or equivalently "$P$ is necessarily true" - i.e. "$\Box P$."
Keep in mind that the accessibility relation between states may have nonintuitive properties - e.g. maybe $\alpha$ is accessible from $\beta$ and $\beta$ is accessible from $\gamma$, but $\alpha$ is not accessible from $\gamma$.

REPLY [0 votes]: There are several ways to interpret modal symbols in the normal langage. The common one is the following : $\Box$ means "it is necessary" and $\Diamond$ means "it is possible". Thus in english the formula $$\Box P \iff \neg \Diamond\neg P$$ can be read "It is necessary to have $P$ if and only if it is not possible to not have $P$." I guess you will agree that makes sense.


text:
\beforepreface

\prefacesection{Preface}

As the twin movements of open science and open source bring an ever greater
share of the scientific process into the digital realm, new opportunities arise
for the meta-scientific study of science itself, including of data science and
statistics. Future science will likely see machines play an active role in
processing, organizing, and perhaps even creating scientific knowledge. To make
this possible, large engineering efforts must be undertaken to transform
scientific artifacts into useful computational resources, and conceptual
advances must be made in the organization of scientific theories, models,
experiments, and data.

This dissertation takes steps toward digitizing and systematizing two major
artifacts of data science, statistical models and data analyses. Using tools
from algebra, particularly categorical logic, a precise analogy is drawn between
models in statistics and logic, enabling statistical models to be seen as models
of theories, in the logical sense. Statistical theories, being algebraic
structures, are amenable to machine representation and are equipped with
morphisms that formalize the relations between different statistical methods.
Turning from mathematics to engineering, a software system for creating machine
representations of data analyses, in the form of Python or R programs, is
designed and implemented. The representations aim to capture the semantics of
data analyses, independent of the programming language and libraries in which
they are implemented.

\subsection*{Guide to reading}

Data science and statistics, category theory and categorical logic, program
analysis and programming language theory all interact in this dissertation.
Individually these are large fields, but the group of people conversant in them
all cannot be very large. In the hope of expanding the readership, I have
exposited much that is already known in addition to my own contributions. So, if
the expert in some area is surprised to find me explaining something they know
very well, I hope that they will forgive me on the grounds that it is part of
the design.

After a general introduction in the first chapter, the dissertation proceeds in
two fairly independent directions. In
\cref{ch:algebra-statistics,ch:zoo-statistics}, I develop the algebra of
statistical theories, statistical models, and their morphisms. Knowledge of
probability theory and exposure to abstract algebra are assumed. Measure theory
is used, but more as an unifying language for probability than as a substantive
mathematical theory. The necessary background in category theory is presented in
\cref{ch:category-theory}. Readers familiar with category theory may omit this
chapter, referring to it as needed for the less standard definitions and
examples. An impressionistic view can be formed by reading the Introduction and
then perusing the many examples of \cref{ch:zoo-statistics}, with the
understanding that the mathematical underpinnings are supplied by
\cref{ch:category-theory,ch:algebra-statistics}.

In the second major part, I describe the design and implementation of a software
system for creating semantic representations of data science workflows.
\cref{ch:program-analysis}, on the analysis of data science code, assumes a
working knowledge of the programming languages Python and R. In
\cref{ch:semantic-enrichment}, I present the Data Science Ontology and a
procedure for the semantic enrichment of idealized computer programs. The formal
description of the ontology and procedure is category-theoretic, for which the
necessary background is again contained in \cref{ch:category-theory}. Although
they describe components of a unified software system,
\cref{ch:program-analysis,ch:semantic-enrichment} can be read independently of
each other. The concluding \cref{ch:conclusion} describes limitations of the
work, suggests directions for future work, and offers a general outlook on how
the structuralist approach to data analysis might transform the scientific
process.

Pictorially, the dependencies between the chapters are:

\begin{center}
  \begin{tikzcd}[math mode=false]
    \cref{ch:introduction} \arrow[r] \arrow[dr] \arrow[ddr]
      & \cref{ch:algebra-statistics} \arrow[r]
      & \cref{ch:zoo-statistics} \\
    \cref{ch:category-theory} \arrow[ur] \arrow[dr]
      & \cref{ch:program-analysis} \\
      & \cref{ch:semantic-enrichment}
  \end{tikzcd}
\end{center}

\subsection*{Acknowledgments}

Over the course of this project I have benefited from the advice and support of
many people. I am deeply grateful to my advisor, Emmanuel Cand{\`e}s, for
sharing his broad knowledge of statistics and applied mathematics and for
consistently supporting my research explorations, no matter where they led me.
Through his commitment to careful, intellectually serious science, he has been
and remains a great role model to me . I also thank the other members of my
thesis committee, namely John Chambers, Trevor Hastie, Mykel Kochenderfer, and
Mark Musen, for their advice. I am especially thankful to John Chambers for his
enthusiastic support of both parts of this project and for his helpful comments
on the manuscript.

The semantic modeling of data science code began during my time at IBM Research,
where I was fortunate to collaborate with Ioana Baldini, Aleksandra Mojsilovic,
and Kush Varshney. I thank them for their mentorship and for introducing me to a
line of research that I likely would not have otherwise pursued. This part of my
thesis has also benefited from conversations with Roxana Danger, Julian Dolby,
Yaoli Mao, Robert McBurney, Holie Schmidt, and Gustavo Stolovitzky.

The algebra of statistical theories and models grew out of interactions with the
field of applied category theory. I am grateful to John Baez and David Spivak
for encouraging me to explore this area at a time when I knew very little about
it. I thank James Fairbanks, Brendan Fong, Jade Master, and David Spivak for
helpful conversations related to this part of the thesis. I also thank
Arquimedes Canedo for the opportunity to work on applied category theory at
Siemens.

As a student in the Statistics Department at Stanford I benefited from
interactions with many excellent peers and colleagues. From my original student
cohort, I am especially grateful for time spent with my friends Paulo Orenstein
and Feng Ruan. From the Cand{\`e}s research group, I thank Stephen Bates, Qijia
Jiang, and Yaniv Romano for conversations and collaboration. I also thank the
department administrative staff, particularly Susie Ementon, for their patience
and willingness to help.

Finally, I wish to express my deep appreciation for the friends and family who
have supported me over a period of time that has not always been simple or easy.
Especially to my mother, Morgan; my brother, Wes; my uncle, Brian; and my
significant other, Julia, I am ever grateful for your unconditional love and
encouragement.

\prefacesection{List of Notation}

\subsection*{Commonly used symbols}

\printunsrtglossary[type=symbols,style=long]

\subsection*{Named categories}

\printunsrtglossary[type=abbreviations,style=long]

\afterpreface

config: arxiv


TITLE: To show that $\sum_{a=1}^{p-2}(a(a+1)/p)=-1$
QUESTION [3 upvotes]: Let $p$ be an odd prime and for any integer $a$ , relatively prime to $p$ , let $(a/p)$ denote the Legendre symbol . Then how to show that $\sum_{a=1}^{p-2}(a(a+1)/p)=-1$ ?
I know that $\sum_{a=1}^{p-1}(a/p)=0$ , so it is enough to show that modulo $p$ , the integers $a(a+1) ; a=1,...,p-2$ runs through a complete set of non-zero residues modulo $p$ except exactly one quadratic residue mod $p$ . But I am unable to show this . Please help . Thanks in advance

REPLY [2 votes]: Ah , I think I got it .
For every $1\le a \le p-1 , \exists $ unique $1\le a' \le p-1$ such that $aa'\equiv 1(mod \space p) $ . Now for $1 \le a \le p-2$ , gcd$(p.a(a+1))=1$
So for $1\le a \le p-2$ ,
$(a(a+1)/p))=(a(a+aa')/p)=(a^2(1+a')/p)=(a^2/p)((1+a')/p)=((1+a')/p)$ ,
and since $p-1$ is the inverse of itself modulo $p$ , so for $1\le a \le p-2$ , the distinct inverses $a'$ run through the residue system $\{1,...,p-2\}$ . Hence
$\sum_{a=1}^{p-2}(a(a+1)/p)=\sum_{a'=1}^{p-2}((a'+1)/p)=\sum_{m=2}^{p-1}(m/p)=0-(1/p)=-1$


TITLE: The two equations $F (x, y, u, v) = 0$ and $G (x, y, u, v) = 0$ determine $x$ and $y$ implicitly as functions of $u$ and $v$
QUESTION [1 upvotes]: The two equations  $F (x, y, u, v) = 0$ and  $G (x, y, u, v) = 0$  determine $x$  and $y$  implicitly as functions of u and v, say  $ x = X (u, v) $ and $ y = Y (u, v) $.   Show that
$$ {\partial X \over \partial u} = {{\dfrac{\partial (F, G)}{\partial (y, u)}} \over \dfrac{\partial (F, G)}{\partial (x, y )}} $$
at the points where the Jacobian $ {\partial (F, G) \over \partial (x, y)} \neq 0 $ and finds similar formulas for the partial derivatives $ {\partial X \over \partial v} , {\partial Y \over \partial u}, {\partial Y \over \partial v} $.
I need help with this exercise, I don't know how to take the function, should I define a composition or another function? Like
$h(u,v)=F(x,y,u,v)=F(X(u,v),Y(u,v),u,v)$?

REPLY [0 votes]: Let $h(u,v):=F(X(u,v),Y(u,v),u,v)$ and $k(u,v):=G(X(u,v),Y(u,v),u,v)$. We know that $h,k$ are identically $0$, so their derivatives are also $0$
Then by the chain rule
$$
\begin{align}
0&=&h_u &=& F_x X_u +F_y Y_u + F_u+0\\
%0&=&h_v &=& F_x X_v +F_y Y_v + 0+ F_v\\
\text{and}\\
0&=&k_u &=& G_x X_u +G_y Y_u + G_u+0\\
%0&=&k_v &=& G_x X_v +G_y Y_v + 0+ G_v\\
\end{align}
$$
which we can write as
$$
\begin{pmatrix}
F_x & F_y\\
G_x & G_y
\end{pmatrix}
\begin{pmatrix}
X_u\\
Y_u
\end{pmatrix}
=-
\begin{pmatrix}
F_u\\
G_u
\end{pmatrix}.
$$
Inverting the $2\times 2$ matrix we have
$$
\begin{pmatrix}
X_u\\
Y_u
\end{pmatrix}
=
\frac{1}{\dfrac{\partial (F,G)}{\partial (x,y)}}
\begin{pmatrix}
-G_y & F_y\\
G_x & -F_x
\end{pmatrix}
\begin{pmatrix}
F_u\\
G_u
\end{pmatrix},
$$
and we can then see that
$$
X_u
=
{{\dfrac{\partial (F, G)}{\partial (y, u)}} \over \dfrac{\partial (F, G)}{\partial (x, y )}},
$$
with a similar expression for $Y_u$.
Differentiating $h,k$ with respect to $v$ will give the formulae for $X_v, Y_v$.


TITLE: What is the formal definition of a variable?
QUESTION [9 upvotes]: What is a variable?
I know that a ($n$-ary) connective can be thought of as a function from $\{ 0,1 \}^n$ to $\{ 0,1 \}$.
And a quantifier over $M$ can be thought of as a set of subsets of $M$.
What is the corresponding way to think of a variable ranging over $M$ ?

REPLY [3 votes]: You can see Categorial grammar and, in detail : Sara Negri & Jan von Plato, Structural Proof Theory (2001), Appendix A.2 : CATEGORIAL GRAMMAR FOR LOGICAL LANGUAGES [page 221].

In propositional logic, no structure at all is given to atomic propositions, but these are introduced just as pure parameters $P, Q, R$,..., with the categorizations
$P : Prop, Q : Prop, R : Prop$, ...
Connectives are functions for forming new propositions out of given ones. We have the constant function $\bot$, called falsity, for which we simply write $\bot : Prop$.
Next we have negation, with the categorization :
$Not: (Prop)Prop$.

And so on ...

We can put predicate logic into [this] framework [...] if we assume for simplicity that we deal with just one domain $\mathcal D$. The objects, individual constants, are denoted by $a, b, c$,.... Instead of the propositional constants $P, Q, R$,..., atomic propositions can be values of propositional functions over $\mathcal D$, thus categorized as $P : (\mathcal D)Prop, Q : (\mathcal D)(\mathcal D)Prop$, and so on. Next we have individual variables $x, y, z$ , . . . taking values in $\mathcal D$. Following the usual custom,
  we permit free variables in propositions. Propositions with free variables are understood as propositions under assumptions, say, if $A : (\mathcal D)Prop$, then $A(x) : Prop$ under the assumption $x : \mathcal D$. Terms $t,u,v$,... are either individual parameters or variables. The language of predicate logic is obtained by adding to propositional logic the quantifiers Every and Some, with the categorizations
$Every : ((\mathcal D)Prop)Prop$
$Some : ((\mathcal D)Prop)Prop$.
These are relative to a given domain $\mathcal D$. Thus, for each domain $\mathcal D$, a quantifier over that domain is a function that takes as argument a one-place propositional
  function $A : (\mathcal D)Prop$ and gives as value a proposition, here either $Every(A)$ or
  $Some(A)$.


text:
\section{Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring}
Tags: Completion of Normed Division Ring

\begin{theorem}
Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Complete Normed Division Ring|complete normed division ring]].
Let $\struct {S, \norm {\, \cdot \,}}$ be a [[Definition:Everywhere Dense|dense]] [[Definition:Normed Division Subring|normed division subring]] of $\struct {R, \norm {\, \cdot \,}}$.
Then:
:$\struct {R, \norm {\, \cdot \,} }$ is a [[Definition:Completion (Normed Division Ring)|completion]] of $\struct {S, \norm {\, \cdot \,}}$ where the [[Definition:Inclusion Mapping|inclusion mapping]] $i : S \to R$ is the required [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphism]].
\end{theorem}

\begin{proof}
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by the [[Definition:Norm on Division Ring|norm]] $\norm {\, \cdot \,}$.
By definition of a [[Definition:Complete Normed Division Ring|complete normed division ring]], the [[Definition:Metric Space|metric space]] $\struct{R, d}$ is [[Definition:Complete Metric Space|complete]].
From [[Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism]], $i : S \to R$ is a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphism]].
From [[Image under Inclusion Mapping]]:
:$\map {i^\to} S = S$
Thus $\map {i^\to} S$ is [[Definition:Everywhere Dense|dense]] in $\struct {R, \norm {\, \cdot \,} }$.
It follows that $\struct {R, \norm {\, \cdot \,} }$ is a [[Definition:Completion (Normed Division Ring)|completion]] of $\struct {S, \norm {\, \cdot \,}}$ by definition.
{{qed}}
[[Category:Completion of Normed Division Ring]]
aat9tvf411affe5v08pi40bw0ltscgi
\end{proof}

config: wiki


TITLE: How to isolate vector X in a vectorial equation with vectorial product?
QUESTION [1 upvotes]: I have no idea how to start solving this equation... It seems that the way of solving it is very different of non-vectorial equations... does anybody knows how to do it?
$$(\vec x \times \vec a)\times(\vec x \times \vec b)=\vec c$$

REPLY [1 votes]: You start with the LHS and use $\vec{a}\times(\vec{b}\times\vec{c})=\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})$
$$
{(\vec x \times \vec a)}\times(\vec x \times \vec b)=\vec x((\vec x \times \vec a)\cdot\vec b)-\vec b((\vec x \times \vec a)\cdot x)
$$
The second term is zero as $(\vec x \times \vec a)$ and $\vec x$ are perpendicular. Then use that $\vec a\cdot(\vec b\times\vec c)=\vec b\cdot(\vec c\times\vec a)$ i.e. the triple product is cyclic.
$$
\vec x((\vec x \times \vec a)\cdot\vec b)=\vec x(\vec x\cdot(\vec a\times \vec b))=\vec c
$$
Now we see that $\vec x$ and $\vec c$ must be parallel as the second term on the LHS is just a scalar thus $\vec x=\lambda\vec c$. Inserting this we find
$$
\lambda^2\vec c(\vec c\cdot(\vec a\times\vec b))=\vec c\quad\Rightarrow\quad \lambda^2(\vec c\cdot(\vec a\times\vec b))=1\quad\Rightarrow\quad \lambda=\frac{\pm 1}{\sqrt{\vec a\cdot(\vec b\times\vec c)}}\quad\Rightarrow\quad \vec x=\frac{\pm \vec c}{\sqrt{\vec a\cdot(\vec b\times\vec c)}}
$$
Which holds when $\vec a\cdot(\vec b\times\vec c)\neq 0$.


TITLE: Solutions of a periodic non-autonomous system
QUESTION [2 upvotes]: I must find solutions for the system
  $$
\left(
\begin{matrix}
\dot{x_1}\\
\dot{x_2}
\end{matrix}
\right)
=
\left(
\begin{matrix}
\cos(t) & -\sin(t)\\
\sin(t) & \cos(t)
\end{matrix}
\right)
\left(
\begin{matrix}
x_1\\
x_2
\end{matrix}
\right)
$$
  and then, obtain the period advance map $\varphi(2\pi,0)=P$.

My problem is how to find the solutions of the non-autonomous system above.
Could someone help me or refer me to somewhere where I can find information about it?

REPLY [3 votes]: Rewrite this as $\dot z(t)=\mathrm e^{\mathrm it}z(t)$ where $z(t)=x_1(t)+\mathrm ix_2(t)$, hence $$z(t)=\mathrm e^{\mathrm i(1-\exp(\mathrm it))}z(0),$$
in particular, $z(t+2\pi)=z(t)$ for every $t$, that is, $$(x_1(t+2\pi),x_2(t+2\pi))=(x_1(t),x_2(t)).$$ To get the solutions $(x_1(t),x_2(t))$, one determines the real and imaginary parts of the expression of $z(t)$ as a complex number above. Thus, one starts from $$x_1(t)+\mathrm ix_2(t)=\mathrm e^{\sin(t)}\mathrm e^{\mathrm i(1-\cos(t))}(x_1(0)+\mathrm ix_2(0)),$$ which yields $$x_1(t)=\mathrm e^{\sin(t)}(\cos(1-\cos(t))x_1(0)-\sin(1-\cos(t))x_2(0),$$ and $$x_2(t)=\mathrm e^{\sin(t)}(\cos(1-\cos(t))x_2(0)+\sin(1-\cos(t))x_1(0),$$


text:
\begin{document}
\title{ On the behavior of quasi-local mass at the infinity along nearly round surfaces }
\author{Yuguang Shi$^1$}
\address{Key Laboratory of Pure and Applied mathematics, School of
 Mathematics Science, Peking University,
Beijing, 100871, P. R. China.}\email{ygshi@math.pku.edu.cn}

\author{Guofang Wang}

\address{Faculty of Mathematics, University Magdeburg, D-39016, Magdebrug, Germany}
\email{gwang@math.uni-magdeburg.de}
\thanks{$^1$Research partially supported by Grant of NSFC (10725101) and by   973 Program (2006CB805905)
 of China }

\author{Jie Wu$^1$}

\address{Key Laboratory of Pure and Applied mathematics, School of
 Mathematics Science, Peking University,
Beijing, 100871, P. R. China.} \email{wujie@math.pku.edu.cn}
\renewcommand{\subjclassname}{
  \textup{2000} Mathematics Subject Classification}
\subjclass[2000]{Primary 53C20; Secondary 83C99}
\date{May 2008}
\begin{abstract}
In this paper,  we  study the limiting behavior  of the Brown-York
mass  and Hawking mass along nearly round surfaces at infinity of an
asymptotically flat manifold.  Nearly round surfaces can be defined
in an intrinsic way. Our results show that the ADM mass of an
asymptotically flat $3$-manifold can be approximated by some
geometric invariants of a family of nearly round surfaces which
approach to infinity of the manifold.
\end{abstract}

\maketitle\markboth{ Yuguang Shi, Guofang Wang and Jie Wu}{Infinity
limits of Quasilocal mass}

\section{Introduction}
The ADM mass  of an asymptotically flat (AF) manifold is a basic
conserved quantity in General relativity. To state its explicit
definition, we need the following:

\begin{defi}\label{defaf}
A complete three manifold $(M,g)$ is said to be {\rm asymptotically flat}
(AF) of order $\tau$ (with one end) if there is a compact subset $K$
such that $M\setminus K$ is diffeomorphic to $\R^3\setminus B_R(0)$
for some $R>0$ and in the standard coordinates in $\R^3$, the metric
$g$ satisfies:
\begin{equation} \label{daf1}
g_{ij}=\delta_{ij}+\sigma_{ij}
\end{equation}
with
\begin{equation} \label{daf2}
|\sigma_{ij}|+r|\p \sigma_{ij}|+r^2|\p\p\sigma_{ij}|=O(r^{-\tau}),
\end{equation}
for some constant $1\ge\tau>\frac{1}{2}$, where $r$ and $\p$ denote
the Euclidean distance and standard derivative operator on $\R^3$
respectively.
\end{defi}

A coordinate system of $M$ near infinity so that the metric tensor
in these coordinates satisfies the decay conditions in the
definition is said to be {\it admissible}, in such a coordinate
system, we have

\begin{defi}
The Arnowitt-Deser-Misner (ADM) mass (see \cite{ADM}) of an
asymptotically flat manifold $M$ is defined as:
\begin{equation} \label{defadm1}
m_{ADM}(M)=\lim_{r\to\infty}\frac{1}{16\pi}\int_{S_r}
\lf(g_{ij,i}-g_{ii,j}\ri)\nu^jd\sigma^0,
\end{equation}
where $S_r$ is the Euclidean sphere, $d\sigma^0$ is the volume
element induced by the Euclidean metric, $\nu$ is the outward unit
normal of $S_r$ in $\R^3$ and the derivative is the ordinary
partial derivative.
\end{defi}
We  always assume that the scalar curvature is in $L^1(M)$ so that
the limit exists in the definition. Under the decay conditions in
the definition of AF manifold, the definition of the ADM mass  is
independent of the choice of admissible coordinates  by the result
of Bartnik \cite{BTK86}. Indeed $S_r$ in the above definition does
not need to be the Euclidean sphere in some admissible coordinates,
it could be   a connected boundary of an exhausting domain with its
area growth like $r^2$. Here $r= \min_{x\in \Si} r(x)$, $r(x)$ is
the  distance function to some fixed point(see Proposition 4.1,
\cite{BTK86}). Hence the ADM mass of an AF manifold is actually a
geometric quantity. With these facts in mind and in the view point
of geometry, one may intend to ask: {\it Whether or not one can
define certain geometric invariants on  a family of surfaces defined
in an intrinsic way, i.e. are independent of the choice of
admissible coordinates,  that tends to the ADM mass as the surfaces
approach to the infinity of the manifold ?} In this paper, we will
investigate this problem and give an affirmative answer to it.

Intuitively, the ADM mass is a kind of total mass of $(M, g)$. In many
cases, we want to measure how much mass is contained in a bounded
domain. For this purpose, the notion of quasi-local energy (mass) is
needed. The Brwon-York mass and the Hawking mass are two of them which are
used frequently in literature and both of them are geometric
invariants of the surfaces (see the definitions below). Physically,
one natural property of quasi-local mass need to have is : {\it the
limit of quasi-local mass of the boundary of exhausting domains of
an AF manifold should approach the ADM mass }(see \cite{CY}). Many
people have studied this problem, they verified that for boundary of
certain exhausting domains this property is true for the Brown-York mass
and the Hawking mass, see \cite{BLP, BBWYY,BY2, HKGH1, Ma}, see also
\cite{FST, ST02}. However, the definitions of these boundaries
considered in above mentioned  papers depend on some special coordinates.
So, these surfaces are not intrinsic.

In this paper, we will discuss the problem  mentioned above in the
case that surfaces are {\it nearly round } at infinity of  an AF
manifold $(M, g)$. Let us begin with the following definition

\begin{defi}
Let $\{\Si_r\}$ be a family of surfaces which are topological
sphere in $(M, g)$, $r= \min_{x\in \Si} r(x)$,  then we call
$\Si_r$ as nearly round  when $r$ tends to infinity if it
satisfies:
\begin{enumerate}
    \item $|\overset\circ A| +r |\nabla \overset\circ A| \leq C r^{-1-\tau}$,
    \item $\max_{x\in \Si_r} r(x) \leq C\min_{x\in \Si_r} r(x)+C$,
    \item $\diam (\Si_r) \leq C r$,
    \item $\Area (\Si_r)\leq C r^2$.
\end{enumerate}
Here  $C$  is a constant independent of $r$. $\diam (\Si_r)$,
$\nabla$, and $|\cdot|$, denote  diameter of the surface, covariant
derivatives and the norm with respect to the induced metric of $g$
respectively, $r(x)$ is the distance of $x$ to some fixed point in
$(M,g)$, $A$ is the second fundamental forms of $\Si_r$ in $(M,g)$ and
$\overset\circ A$ is the trace free part of $A$.
\end{defi}

\begin{rema}
\begin{enumerate}
\item It is easy to see that the above definition of nearly round
surface is intrinsic, i.e. it does not depend on any coordinates.
  \item We suspect that the third and the fourth assumptions are
superfluous, since both of them can be derived from the first and
second assumptions in the Euclidean space case.
  \item It is not difficult to see that the third assumption
implies the second one.
\end{enumerate}
\end{rema}

One of very important and also quite natural surfaces in AF
manifolds are those with constant mean curvature and approach to
infinity of the manifolds, the existence of these surfaces was
proved by \cite{Ye}, and \cite{HY} many years ago, and later the
uniqueness was obtained by \cite{QT}(see also \cite{HY}). It is not
so difficult to see that these constant mean curvature surfaces are
nearly round (see the discussion at the beginning of Section 2).
Besides this, all the surfaces considered in \cite{BLP, BBWYY,BY2,
HKGH1, Ma,FST, ST02} are nearly round.

Now, let us move to the definition of the Brown-York mass and the Hawking
mass.

Let $\lf(\Omega, g\ri)$ be a compact three manifold with smooth
boundary $\p \Omega$. Suppose the Gauss curvature of $\p\Omega$ is
positive, then the Brown-York quasi local mass of $\p\Omega$ is
defined as  (see \cite{BY1,BY2}):
\begin{defi}
\begin{equation} \label{defbym1}
m_{BY}\lf(\p \Omega\ri)=\frac{1}{8\pi}\int_{\p \Omega}(H_0-H)d\sigma
\end{equation}
 where  $H$ is the mean curvature of $\p\Omega$ with respect to the
 outward unit normal and
  the metric $g$, $d\sigma$ is the volume element induced on $\p\Omega$ by
 $g$ and $H_0$ is the mean curvature  of $\p \Omega$ when
  embedded in  $\R^3$.
\end{defi}
The Brown-York mass is well-defined because by the result of
Nirenberg \cite{Nr}, $\p\Omega$ can be isometrically embedded in
$\R^3$ and the embedding is unique by
\cite{Herglotz43,Sacksteder62,PAV}. In particular, $H_0$ is
completely determined by the metric on $\p\Omega$. However, this is
a global property.  In contrast, the norm of the mean curvature
vector of an embedding of $\p\Omega$ into the light cone in the
Minkowski space can be expressed explicitly in terms of the Gauss
curvature, see \cite{BLY99}. Hence in the the study of Brown-York
mass,   one of the difficulties  is to estimate
$\int_{\p\Omega}H_0d\sigma$. We will use the Minkowski formulae
\cite{KLBG} and the estimates of Nirenberg \cite{Nr} to deal with
this problem.

The Hawking quasi local mass is  defined as (see \cite{HKG}):
\begin{defi}
\begin{equation} \label{defhkgm1}
m_H(\p \Omega)=\frac{|\p
\Omega|^{1/2}}{(16\pi)^{3/2}}\lf(16\pi-\int_{\p
\Omega}H^2d\sigma\ri)
\end{equation}
 where $d\Sigma$ is the volume element induced on $\p\Omega$ by
 $g$
 and $|\p \Omega|$ is the area of $\p \Omega$.
\end{defi}

Our main results in this paper are:

\begin{theo}\label{BYmass}
Let $(M,g)$ be an AF manifold, $\{\Si_r\}$ be  nearly  round
surfaces of $(M, g)$ when $r$ tends to infinity, then
$$
\lim_{r\rightarrow \infty}m_{BY}(\Si_r)=m_{ADM}(M).
$$
\end{theo}

\begin{theo}\label{Hawkingmass}
Let $(M,g)$ be an AF manifold, $\{\Si_r\}$ be  nearly  round
surfaces of $(M, g)$ when $r$ tends to infinity then
$$
\lim_{r\rightarrow \infty}m_{H}(\Si_r)=m_{ADM}(M).
$$
\end{theo}

The remaining of this paper is organized  in the following way. In  Section 2 we
will discuss the  geometry of nearly round surfaces,
show that many interesting surfaces are nearly round and  present
some useful formulae. In Section 3 we will show some estimates of
isometric embedding and  in Section 4 we will prove the main theorems.

\section{Geometry of nearly round  surfaces }

In this section, we want to give some examples of nearly round
surfaces and to investigate their geometric properties,  and  also
we will derive some basic formulae which will be used later. Let us
begin with some interesting examples.

\begin{exam}
Constant mean curvature (CMC) surfaces constructed  in \cite{HY}
are nearly round at the infinity of the manifold .
\end{exam}

Note that the CMC surfaces constructed in \cite{HY} are convex at
infinity, and then by Propositions 3.5, 3.9 and 3.12 in \cite{HY} we
see that they are nearly round at infinity of the manifolds.

Also, by a direct computations, it is not difficult to see that

\begin{exam}
Let $(M,g, x^i)$ , $1\leq i\leq 3$, be an AF manifold with
admissible coordinates $x^i$, then the coordinate sphere
$S_r=\{(x^1, x^2, x^3)|$\quad$ (x^1)^2+(x^2)^2+(x^3)^2=r^2\}$ is
nearly round when $r$ tends to infinity.
\end{exam}

Our next example relates to the surfaces in  Kerr solution to vacuum
Einstein equations.

\begin{exam}
The well-known  Kerr
metric is given by
\begin{equation}
\begin{split}
ds^2 =&-(1-\frac{2mr}{r^2+ a^2 \cos^2 \theta})dt^2 -\frac{2ma r \sin
^2
\theta}{r^2+ a^2 \cos^2 \theta}(dt d\phi +d\phi dt)\\
&+\frac{r^2+ a^2 \cos^2 \theta}{r^2 -2m r +a^2}dr^2 + (r^2+ a^2
\cos^2)d\theta^2 \\
&+\frac{\sin^2 \theta}{(r^2+ a^2 \cos^2 \theta)^2}[(r^2 +a^2)^2 -
a^2 (r^2-2mr +a^2) \sin^2 \theta]d\phi^2.
\end{split}
\end{equation}
For instance, see page 261 in \cite{Ca}.
Let $(M, g)$ be the slice  with $t= const.$, then it can be shown
directly that $(M, g)$ is an AF manifold. Let $\Si_\tau$ be the
surface in $(M,g)$ with $r=\tau$, then one can verify that $\Si_\tau$
is nearly round in $(M, g)$  when $\tau$ goes to infinity. Indeed,
if we let $e_1 = (\tau^2+ a^2 \cos^2)^{-\frac12} \frac{\p}{\p
\theta}$, $e_2= \frac{\tau^2+ a^2 \cos^2 \theta}{\sin
\theta}[(\tau^2 +a^2)^2 - a^2 (\tau^2-2m\tau +a^2) \sin^2
\theta]^{-\frac12}\frac{\p}{\p \phi}$, then it is easy to see that
$e_1$, $e_2$ is an  orthonormall frame of $\Si_\tau$, and in  this
frame the second fundamental form of $\Si_\tau$
with respect to outward unit normal vector in $(M,g)$ is
$$
A_{11}=\frac{2}{\tau}
\lf(\frac{1-\frac{2m}{\tau}+\frac{a^2}{\tau^2}}{1+\frac{a^2}{\tau^2}\cos^2
\theta}\ri)^\frac12 (1+\frac{a^2}{\tau^2}\cos^2 \theta)^{-1},
\quad
A_{12}=0,
$$
\begin{equation}
\begin{split}
A_{22}=& \frac{2}{\tau^2}
 \lf(\frac{1-\frac{2m}{\tau}+\frac{a^2}{\tau^2}}{1+\frac{a^2}{\tau^2}\cos^2
\theta}\ri)^\frac12 \lf(\frac{ 1+\frac{a^2}{\tau^2}\cos^2 \theta
}{(1+\frac{a^2}{\tau^2})^2 -\tau^{-2}
(1-\frac{2m}{\tau}+\frac{a^2}{\tau^2})a^2\sin^2 \theta}\ri)\\
&\cdot \frac{1}{(\tau^2 +a^2 \cos^2 \theta)^2} \left\{\tau^5 +2a^2
\tau^3 \cos^2 \theta -a^2 \tau^2 m \sin^2 \theta\right.
\\ & \left. + a^4 \tau (\cos
2\theta +\sin^4
\theta)
+m a^4 \sin^2 \theta \cos^2 \theta \right\}.
\end{split}\nonumber
\end{equation}
When $\tau$ tends to infinity, we have
$$
A_{11}=\frac{2}{\tau}-\frac{2m}{\tau^2} +O(\tau^{-3}),
\quad
A_{12}=0,
$$
$$
A_{22}=\frac{2}{\tau}-\frac{2m}{\tau^2} +O(\tau^{-3}).
$$
Hence
$$
|\overset\circ A|\leq C \tau^{-3}.
$$
Similarly  we  have
$$
|\nabla \overset \circ A|\leq C \tau^{-4}.
$$
Here $C$ is a constant  independent of $\tau$. Hence, $\Si_\tau$
is a nearly round surface when $\tau$  tends to infinity.

\end{exam}

\begin{lemm}\label{estimate of area}
Let $\Si_r$ be  nearly round  surfaces in $(M, g)$  when $r$ goes to
infinity, then there is a positive constant $\Lambda$ which is
independent of $r$, and with $\Lambda^{-1} r^2 \leq \Area
(\Si_r)\leq \Lambda r^2$.
\end{lemm}
\begin{proof}

It suffices to show the lower bound of the area. Since $g$ is an AF
metric, without loss of generality we may assume $g$ and $\hat g$
are equivalent on $M\setminus K$. Here and in the sequel, $\hat g$ is
background Euclidean metric on $M\setminus K$. Thus, we only need to
show $\Area (\Si_r, \hat g)\geq C r^2$, where $\Area (\Si_r, \hat g)$
is the area of $\Si_r$ with respect to metric $\hat g$. From the
second assumption of nearly round sphere surfaces, we know that the
standard sphere with radius $\frac{r}{2}$, denoted by
$\mathbb{S}^2 _{\frac{r}{2}}$, is in the domain enclosed by
$\Si_r$. Let $\Omega$ be the domain enclosed by $\Si_r$ and
$\mathbb{S}^2 _{\frac{r}{2}}$, $X=(X^1, X^2, X^3)$ be the Euclidean
coordinates. In view of
$$
\div_{\hat g} (\frac{X}{|X|})=\frac{2}{|X|}>0,
$$
we have
\begin{equation}
\begin{split}
\int_{\Si_r}\frac{X}{|X|}\cdot n -\int_{\mathbb{S}^2
_{\frac{r}{2}}}\frac{X}{|X|}\cdot n &= \int_\Omega \div_{\hat g}
(\frac{X}{|X|})\\
&\geq 0,
\end{split}\nonumber
\end{equation}
where $n$ is the outward unit normal vector of on one part  of the boundary of $\Omega$, $\Si_r$
and $-n$ is the  outward unit normal vector of on another part of the boundary of $\Omega$, $\mathbb{S}^2 _{\frac{r}{2}}$.
Therefore, we see that
$$
\Area (\Si_r, \hat g)\geq \pi r^2.
$$
This finishes the proof of the Lemma.
\end{proof}

Our next lemma is on the estimation of the decay of the second
fundamental forms of nearly round sphere surfaces.

\begin{lemm}\label{decay2ndforms}
Let $\Si_r$ be  nearly round  surfaces in $(M, g)$ when $r$ goes to
infinity, then we have
$$
|A|\leq C r^{-1},
$$
where $C$ is  a positive  constant independent of $r$.
\end{lemm}

\begin{proof}

Let $e_0$, $e_1$ and $e_2$ be the orthonormall frame of $(M,g)$ at
any fixed point of $\Si_r$, $e_1$ and $e_2$ be the tangential
vectors of $\Si_r$. Let $\nabla_k A_{ij}$ be the components of
$\nabla A$, then by a direct computation for $1\leq$ $i$, $j$, $k$
$\leq 2$,  we have
$$
\nabla_i A_{jk} =E_{ijk}+F_{ijk},
$$
where
$$
E_{ijk}=\frac14(\nabla_i H \bar g_{jk}+\nabla_j H \bar
g_{ik}+\nabla_k H \bar g_{ij})-\frac12 w_i \bar g_{jk}+\frac12 (w_j
\bar g_{ik}+w_k \bar g_{ij}).
$$
Here $w_i = R_{0kil}\bar g^{kl}$ and $\bar g_{ij}$ is the induced
metric on $\Si_r$. By the Codazzi equations, we have

$$
\langle E_{ijk}, F_{ijk}    \rangle=0,
$$
and
$$
|E_{ijk}|^2 =\frac35 |\nabla H|^2 +|w|^2 -\langle w_i, \nabla_i H
\rangle.
$$
thus combining these equalities  we have
\begin{equation}
\begin{split}
|\nabla \overset {\circ}A|^2 &=|\nabla A|^2 -\frac12 |\nabla H|^2
 \geq |E|^2-\frac12 |\nabla H|^2 \\
&\geq \frac{1}{20}|\nabla H|^2 -C |w|^2
\geq \frac{1}{200}(|\nabla A|^2- |\nabla \overset {\circ}A|^2)
-C|w|^2,
\end{split}
\end{equation}
where $C$ is a  constant. Thus, we see that
$$
|\nabla A| \leq C (|\nabla \overset\circ A|+  r^{-2-\tau}).
$$
Combining this  with the first assumption of nearly round sphere
surfaces we have
$$
|\nabla H |\leq |\nabla A| \leq  C r^{-2-\tau}.
$$
Due to the assumption $\diam (\Si_r) \leq C r$, we see that for any
$x\in \Si_r$ we have

\begin{equation}
\begin{split}
|H(x)-H(x_0)|&\leq |\nabla H| \cdot \diam (\Si_r)\\
&\leq C r^{-1-\tau}.\nonumber
\end{split}
\end{equation}
Here $x_0$ is a fixed point on $\Si_r$.
Setting
$$
r_1 =\frac{2}{H(x_0)},
$$
we have $$
H=\frac{2}{r_1}+O(r^{-1-\tau}).
$$
We now claim that there is a    constant $C> 1$ independent of $r$ with
$$
C^{-1} r \leq r_1 \leq C r.
$$
Indeed, due to the assumption $|\overset\circ A| \leq C
r^{-1-\tau}$ and the Gauss equations, we have
$$
K=\frac{1}{r^2 _1}+r^{-1} _{1} O(r^{-1-\tau}) +O(r^{-2-\tau}) ,
$$
where $K$ is the Gauss curvature of $\Si_r$ with  respect to the metric induced
from $g$. Then by the Gauss-Bonnet formula we get

$$
\frac{\Area (\Si_r)}{r^{2} _{1}}+r^{-1} _{1} O(r^{1-\tau})
+O(r^{-\tau})=4\pi.
$$
Together with Lemma \ref{estimate of area},  it follows that the claim
is true. Note that $A_{ij}=\overset\circ A_{ij}
+\frac{H}{2}\bar{g}_{ij}$, where $\bar{g}_{ij}$ is the  metric on
$\Si_r$ induced from $g$. The first assumption and the claim imply
$$
|A|\leq C r^{-1},
$$
which finishes the proof of  the lemma.
\end{proof}

As mentioned before, we may regard $\Si_r$ as a surface in
$M\setminus K$ with the Euclidean  metric $\hat g$. Our next lemma
is about the relationship between $A$ and $\hat A$, here and in the
sequel, $\hat A$ is the second fundamental forms of $\Si_r$ with
respect to outward unit normal vector and metric $\hat g$.

\begin{lemm}\label{relation2ndform}
Let $\rho$ be the Euclidean distance function to $\Si_r$,
$\frac{\p}{\p x^i}$, $1\leq i \leq 3$, be the standard coordinate
frame in $\mathbb{R}^3$, $\Gamma^k _{ij}$ is the Christoffel
symbols of metric $g$ with respect to $\frac{\p}{\p x^i}$, then
$$
\hat A(X,Y)=|\nabla_g \rho|\cdot A(X,Y)+X^i Y^j\Gamma^k
_{ij}\frac{\p \rho}{\p x^k},
$$
 where $X=X^i\cdot \frac{\p}{\p x^i}$ and $Y=Y^i\cdot \frac{\p}{\p x^i}$ are tangential vectors of
 $\Si_r$ and $\nabla_g$ is the Livi-Civita connection with respect to $g$.
\end{lemm}

\begin{proof}
By the definition of $\rho$,  on $\Si_r$ we have
\begin{equation}
\begin{array}{rcl}
\nabla^2 \rho (X, Y)&=&XY(\rho)-\nabla_X Y (\rho)\\
&=&-\nabla_X Y (\rho)\\
&=& -\langle\nabla_X Y , v  \rangle v(\rho)\\
&=& A(X,Y)v(\rho),
\end{array}\nonumber
\end{equation}
where $v$ is the outward unit normal vector of $\Si_r$. Again,  a
direct computations gives
$$
v=|\nabla_g \rho|^{-1} g^{ij}\frac{\p \rho}{\p x^i}\frac{\p}{\p
x^j},
$$
Hence, we have
$$
v(\rho)=|\nabla_g \rho|,
$$
$$
A(X,Y)=\frac{1}{|\nabla_g \rho|}\nabla^2 \rho (X, Y)
$$
and
$$
\nabla^2 \rho (X, Y)=X^i Y^j \frac{\p^2 \rho}{\p x^i \p x^j}-X^i
Y^j\Gamma^k _{ij}\frac{\p \rho}{\p x^k}.
$$
Similarly  we have

$$
\hat A(X,Y)=X^i Y^j \frac{\p^2 \rho}{\p x^i \p x^j },
$$
Combining these formulas together, we have
$$
\hat A(X,Y)=|\nabla_g \rho|\cdot A(X,Y)+X^i Y^j\Gamma^k
_{ij}\frac{\p \rho}{\p x^k}.
$$
\end{proof}

Let $v=\sum_{i=1}^3 v^i \frac{\p}{\p x^i} $, then $h_{ij}=g_{ij}-v_i
v_j$, $1\leq i$, $j\leq 3$. Then $h_{ij}dx_idx_j$ is the induced
metric on $\Si_r$ in $(M, g)$. Define $h^{ij}:=g^{is}g^{jt}h_{st}$.
We have
\begin{lemm}\label{2ndfundamentalforms}
Let $\rho$ be the  Euclidean distance function to $\Si_r$, then on
$\Si_r$, we have
$$
\frac{\p^2 \rho}{\p x^i \p x^j}= B_{ij} +\frac{\hat
H}{2}h_{ij},
$$
where $B_{ij}=\overset{\circ}{\hat A}((\frac{\p}{\p x^i})^T,
(\frac{\p}{\p x^j})^T )$, and $(\frac{\p}{\p x^i})^T$ is the
tangential part of $\frac{\p}{\p x^i}$.
\end{lemm}

\begin{proof} Let $\hat \nabla$ be the covariant derivatives with
respect to $\hat g$, then by a direct computation gives
\begin{equation}
\begin{split}
\frac{\p^2 \rho}{\p x^i \p x^j}& =\hat \nabla^2 \rho
((\frac{\p}{\p
x^i})^T, (\frac{\p}{\p x^j})^T )\\
&=\hat A ((\frac{\p}{\p x^i})^T, (\frac{\p}{\p x^j})^T )\\
&=\overset{\circ}{\hat A}((\frac{\p}{\p x^i})^T, (\frac{\p}{\p
x^j})^T ) +\frac{\hat H}{2}h_{ij}.
\end{split}
\end{equation}

\end{proof}

Combining Lemma (\ref{decay2ndforms}), Lemma (\ref{relation2ndform})
and Lemma (\ref{2ndfundamentalforms}), we obtain

\begin{coro}\label{estimate of 2ndderivative}

Let $\Si_r$ be   nearly round  surfaces in $(M, g)$ as $r$ goes to
infinity, then on $\Si_r$, we have:

$$
|\frac{\p^2 \rho}{\p x^i \p x^j}|\leq C r^{-1},
$$
where $C$ is a  constant independent of $r$.
\end{coro}

As a corollary, we have

\begin{coro}\label{decay2ndformsinR3}
Let $\overset{\circ}{\hat A}$ be the trace free part of $\hat A$,
$\bar D$ be the covariant derivatives of $\Si_r$ with respect to induced metric
from $(M\setminus K, \hat g)$, then we have
$$
|\overset{\circ}{\hat A}|+r |\bar D \overset{\circ}{\hat A}|\leq
Cr^{-1-\tau},
$$
where $C$ is a positive constant  independent of $r$.
\end{coro}
\begin{proof}
By Lemma \ref{relation2ndform}  and direct computations, we have
$$
|H-\hat H|\leq Cr^{-1-\tau},
$$
where $C$ is a positive constant  independent of $r$, and hence
$$
|\overset{\circ}{\hat A}|\leq Cr^{-1-\tau}.
$$
Hence it suffices to show the second part estimate of the corollary is
true. Let $p$  be any point of $\Si_r$, $e_1$ and $e_2$ be the
orthonormal frame at $p$   with $\bar D_{e_i} e_j =0$, $1\leq
i$, $j\leq 2$. Let $X_i$, $1\leq i\leq 3$, be one of $e_k$.  Then,
at $p$, we have
\begin{equation}
\begin{split}
(\bar D_{X_3} \hat A)(X_1, X_2)=&X_3 (\hat A(X_1, X_2))\\
=&X_3 (|\nabla_g \rho|)\cdot A(X_1,X_2)+|\nabla_g \rho|X_3
(A(X_1,X_2))\\
&+(X_3 (X^i _1)X^j _2 + X^i _1 X_3 (X^j _2))\Gamma^k _{ij}\frac{\p
\rho}{\p
x^k}+ X^i _1 X^j _2 X_3 (\Gamma^k _{ij})\frac{\p \rho}{\p x^k}\\
&+X^i _1 X^j _2 \Gamma^k _{ij}X_3 (\frac{\p \rho}{\p x^k}).
\end{split}
\end{equation}
Here we assume $X_i =X_i ^k \frac{\p}{\p x^k}$. Since
$\rho$ is Euclidean distance to $\Si_r$,  we have
$$
\sum_{i=1}^3 (\frac{\p \rho}{\p x^i})^2 =1.
$$
Thus,
\begin{equation}
\begin{split}
|\sum_{i=1} ^3 g^{ij} X_3 (\frac{\p \rho}{\p x^i})\frac{\p \rho}{\p
x^j}|&\leq |\sum_{i=1}^3 \sigma^{ij} X_3 (\frac{\p \rho}{\p
x^i})\frac{\p
\rho}{\p x^j}|+C r^{-1-\tau}\\
&\leq  Cr^{-1-\tau},
\end{split}\nonumber
\end{equation}
where  $C$ is a constant  independent of $r$ and $p$ and the
orthonormal frames that we choose. Note that   Corollary
\ref{estimate of 2ndderivative} was used  in the last equality.

Due to the asymptotically flatness of  manifold $(M,g)$, by Lemma (\ref{decay2ndforms}) and Corollary (\ref{estimate of
2ndderivative}) we have
$$
|X_3 (|\nabla_g \rho|)\cdot A(X_1,X_2)|+|X^i _1 X^j _2 X_3 (\Gamma^k
_{ij})\frac{\p \rho}{\p x^k}|+|X^i _1 X^j _2 \Gamma^k _{ij}X_3
(\frac{\p \rho}{\p x^k})|\leq C r^{-2-\tau},
$$
where   $C$ is a positive constant independent of $r$ and $p$ and
the orthonormal frames that we choose. In order to get the estimates
of the remaining part, we need to estimate covariant derivatives of
$X_i$ at $p$ first. Note that by Lemma \ref{relation2ndform}, we see
that there is a  constant $C$ which is independent of $r$  and with
$$
|A-\hat A|\leq C r^{-1-\tau}.
$$
On the other hand, by the fundamental equations of the surface, for
$i=1$, $2$,  we have
\begin{equation}
\begin{split}
 \bar{\nabla}_{X_3} X_i -\bar {D}_{X_3} X_i &=(\nabla_{X_3} X_i
-D_{X_3} X_i)+ (\hat A -A)(X_3, X_i)v +\hat A (X_3 , X_i)(\hat v -v)\\
&=X^i _3 X^k _i \Gamma^l _{ik} \frac{\p}{\p x^l}+(\hat A -A)(X_3,
X_i)v +\hat A(X_3, X_i)(\hat v-v),
\end{split}\nonumber
\end{equation}
where $\nabla$, $\bar \nabla$ is covariant derivatives with respect
to metric $g$ and its induced metric on $\Si_r$ respectively. By a
direct computations, it is not difficult to see that
$$
|\hat v- v|\leq C r^{-\tau},
$$
where $C$ is a positive universal constant independent of $r$.
Hence, by the choice of $X_i$ and decay of $|A-\hat A|$,  we
get
\begin{equation}\label{estimateof covariant derivative}
|\bar \nabla_{X_3} X_i|\leq C r^{-1-\tau},
\end{equation}
at $p$, where $C$ is a  constant independent of $r$  and $p$ and the
orthonormal frames that we choose. Together this with   decay of
$|\bar\nabla A|$ and the equality
$$
X_3 (A(X_1, X_2))=(\bar \nabla_{X_3} A)(X_i, X_j) + A(\bar
\nabla_{X_3} X_i, X_j)+A(\bar \nabla_{X_3} X_j, X_i),
$$
$i$, $j=$ $1$, $2$,  we get
$$
|X_3 (A(X_i, X_j))|\leq C r^{-2-\tau},
$$
where $C$ is a  constant independent of $r$  and $p$ and the
orthonormal frames that we choose. By the choice of $X_i$, we have
for $i=1$, $2$,
$$
\bar D_{X_3}X_i =D_{X_3}X_i -\hat A (X_3, X_i)\hat v=0,
$$
 at $p$,
 and hence
 $$
X_3 (X_i ^k)=\hat g (\hat A (X_3, X_i)\hat v, \frac{\p}{\p x^k})
 $$
at $p$
 which implies
 $$
|X_3 (X_i ^k)|\leq C r^{-1}
 $$
for a  constant $C$ independent of $r$  and $p$ and the orthonormal
frames that we choose. Combining the above estimates, we have
$$
|\bar D \hat A| \leq C r^{-2-\tau}
$$
at $p$ for some universal constant independent of $r$  and $p$ and
the orthonormal frames that we choose. Since $p$ is arbitrary, we
complete to prove the corollary.
\end{proof}

Let $H$ and $\hat H$ be the mean curvature of $\Si_r$ in $(\R^3
\setminus K, g)$ and  $(\R^3 \setminus K, \hat g)$ respectively.
Both of them are  with respect to outward unit normal vector.

\begin{lemm}\label{difference of mean curvature}
Let $\rho$ be the Euclidean distance to $\Si_r$ in $\R^3$, then
\begin{equation}\label{meancuravture}
\begin{split}
H=&\hat H +\frac{H}{2}\sigma_{ij}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^j}+\frac12 \sigma_{st,i}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t}\\
&-\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p x^j}-g_{ij,i}\frac{\p
\rho}{\p x^j}
+\frac12 g_{jj,i}\frac{\p \rho}{\p x^i}+O(r^{-1-2\tau}).\\
\end{split}
\end{equation}
Here and in the sequel, $\sigma_{ij,k}=\frac{\p \sigma_{ij}}{\p
x^k}$ and $g_{ij,k}=\frac{\p g_{ij}}{\p x^k}$.

\end{lemm}

\begin{proof}
 We first note that on $\Si$ we have
$$
\Delta_0 \rho =\hat H,
$$
where $\Delta_0$ is Laplacian with respect to $\R^3$. The unit
normal vector of $\Si_r$ in $\R^3$ is $\nabla_0 \rho =\frac{\p
\rho}{\p x^i}\frac{\p}{\p x^i}$,  denoted by $\hat v$. Let $\{e_1,
e_2, v\}$ be the orthonormal frame in $(M\setminus K, g)$ and $\{e_1,
e_2\}$ the tangential vector of $\Si$. We have
$$
\Delta \rho = \nabla^2 \rho (e_1 , e_1) + \nabla^2 \rho (e_2 ,
e_2)+ \nabla^2 \rho (v , v)
$$
and
\begin{equation}
\begin{split}
 \nabla^2 \rho (e_1 , e_1) + \nabla^2 \rho (e_2 , e_2)&=e_1 e_1 \rho
 -\nabla_{e_1}e_1 \rho +e_2 e_2 \rho
 -\nabla_{e_2}e_2 \rho\\
 &=-(\nabla_{e_1}e_1 +\nabla_{e_2}e_2)\rho\\
 &=\langle\nabla_{e_1}e_1 +\nabla_{e_2}e_2, v\rangle v(\rho)\\
 &=H\cdot v(\rho).
\end{split}
\end{equation}
On the other hand, it is clear that
$$
\hat v = \langle \hat v, v\rangle_g v + T,
$$
where, $T$ is the tangential part of $\hat v$ on $\Si_r$ and
$\langle, \rangle_g$ is the inner product with respect to the metric
$g$. Thus we have
$$
\hat v(\rho)=\langle \hat v, v\rangle_g v(\rho).
$$
Since $\hat v (\rho)=1$ on $\Si_r$, we get
$$
v(\rho)=\langle \hat v, v\rangle_g ^{-1}.
$$
It is easy to see that
$$
v=\frac{\nabla \rho}{|\nabla \rho|}=|\nabla \rho|^{-1} g^{ij}
\frac{\p \rho}{\p x^i}\frac{\p}{\p x^j}
$$
and
\begin{equation}
\begin{split}
\langle \hat v, v\rangle_g&=|\nabla \rho|^{-1}g^{ij} \frac{\p
\rho}{\p x^i}\langle \frac{\p}{\p x^j},  \frac{\p}{\p
x^k}\rangle_g
\frac{\p \rho}{\p x^k}\\
&=|\nabla \rho|^{-1} g^{ij}\frac{\p \rho}{\p x^i}g_{jk}\frac{\p
\rho}{\p x^k}\\
&=|\nabla \rho|^{-1}.
\end{split}
\end{equation}
Hence, we have
$$
v(\rho)=|\nabla \rho|,\quad
\langle \hat v, v\rangle_g = |\nabla \rho|^{-1}.
$$
Since $v$ is the unit norm vector along $\Si_r$, it is clear that
$\langle \nabla_v v, v   \rangle_g =0$, which implies $\nabla_v v$
is a tangential vector on $\Si_r$. Therefore, we get
$$\nabla_v v (\rho)=0.$$
Combining these facts we obtain
$$
\nabla^2 \rho (v, v) = v(|\nabla \rho|)
$$
and
\begin{equation}\label{meancurvatureinafmfd}
\Delta \rho =H |\nabla \rho| + v(|\nabla \rho|).
\end{equation}

Now we compute  the second term in (\ref{meancurvatureinafmfd}).
\begin{equation}
\begin{split}
v(|\nabla \rho|^2)=&v(g^{st}\frac{\p \rho}{\p x^s}\frac{\p
\rho}{\p
x^t})\\
=&|\nabla \rho|^{-1}g^{ij}\frac{\p \rho}{\p x^i}\frac{\p}{\p
x^j}(g^{st}\frac{\p \rho}{\p x^s}\frac{\p \rho}{\p x^t})\\
=&|\nabla \rho|^{-1}g^{ij}\frac{\p \rho}{\p x^i}g^{st}_{,j}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t} +2|\nabla
\rho|^{-1}g^{ij}\frac{\p \rho}{\p x^i}g^{st}\frac{\p^2 \rho}{\p x^s
\p x^j}\frac{\p \rho}{\p x^t},
\end{split}
\end{equation}
where
$$
g^{st}_{, j}=\frac{\p g^{st}}{\p x^j}
$$
and
$$
g^{ij}=\delta^{ij}-\sigma_{ij}+O(r^{-2\tau}).
$$
Putting these things together, we get
\begin{equation}
\begin{split}
v(|\nabla \rho|^2)=&\frac{\p \rho}{\p x^i}g^{st}_{,i}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t}+2|\nabla \rho|^{-1}\frac{\p
\rho}{\p x^i}\frac{\p^2 \rho}{\p x^i \p x^j}\frac{\p \rho}{\p
x^j}\\
&-2|\nabla \rho|^{-1}\sigma_{ij}\frac{\p \rho}{\p x^i}\frac{\p^2
\rho}{\p x^s \p x^j}\frac{\p \rho}{\p x^s}-2|\nabla
\rho|^{-1}\frac{\p \rho}{\p x^i}\frac{\p^2 \rho}{\p x^i \p
x^s}\frac{\p \rho}{\p x^t}\sigma_{st}\\
&+ O(r^{-1-2\tau}).
\end{split}
\end{equation}
 From
$$
\sum_{i} \frac{\p \rho}{\p x^i}\frac{\p^2 \rho}{\p x^i \p
x^s}=\frac12 \sum_{i} \frac{\p}{\p x^s}((\frac{\p \rho}{\p
x^i})^2)=0
$$
and
$$
v(|\nabla \rho|^2)= 2 |\nabla \rho|v(|\nabla \rho|),
$$
 we get
 $$
v(|\nabla \rho|)=\frac12 g^{st}_{,i}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t}+O(r^{-1-2\tau}).
 $$
Thus, we have
$$
\Delta \rho =H |\nabla \rho| +\frac12 g^{st}_{,i}\frac{\p \rho}{\p
x^i}\frac{\p \rho}{\p x^s}\frac{\p \rho}{\p x^t}+O(r^{-1-2\tau}).
$$
On the other hand, by the definition of Laplacian operator we
have
\begin{equation}
\begin{split}
\Delta \rho &=g^{ij} \frac{\p^2 \rho}{\p x^i \p
j}+\frac{1}{\sqrt
g}\frac{\p}{\p x^i}(\sqrt {g} g^{ij})\frac{\p \rho}{\p x^j}\\
&=\hat H -\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p x^j}+
g^{ij}_{,i}\frac{\p \rho}{\p x^j}+g^{ij}\frac{1}{\sqrt
g}\frac{\p}{\p x^i}(\sqrt g)\frac{\p \rho}{\p x^j}.
\end{split}
\end{equation}
Noticing that
$$
\frac{\p}{\p x^i}(\sqrt g)= \frac12 g_{jj,i} + O(r^{-1-2\tau})
$$
and
$$
g^{ij}_{,i}=-g_{ij,i},
$$
we get
\begin{equation}\label{meancuravture}
\begin{split}
H=&\hat H +\frac{H}{2}\sigma_{ij}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^j}+\frac12 \sigma_{st,i}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t}\\
&-\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p x^j}-g_{ij,i}\frac{\p
\rho}{\p x^j}
+\frac12 g_{jj,i}\frac{\p \rho}{\p x^i}+O(r^{-1-2\tau})\\
=&\hat H+O(r^{-1-\tau})+O(r^{-1-2\tau})
\end{split}
\end{equation}
\end{proof}

In the sequel, we want to calculate the integral of $ H-\hat H$ on
$\Si_r$. Let us begin with
\begin{equation}\label{3rdterminmeancurv}
\begin{split}
&\int_{\Si}\sigma_{st,i}\frac{\p \rho}{\p x^i}\frac{\p \rho}{\p
x^s}\frac{\p \rho}{\p x^t}d\sigma^0 =\int_{\Si}\frac{\p}{\p
x^i}(\sigma_{st}\frac{\p \rho}{\p x^s})\frac{\p \rho}{\p
x^i}\frac{\p \rho}{\p x^t}d\sigma^0
-\int_{\Sigma}\sigma_{st}\frac{\p^2 \rho}{\p x^i \p x^s}\frac{\p
\rho}{\p x^i}\frac{\p \rho}{\p x^t}d\sigma^0\\
&=-\int_{\Si}(\delta_{it}-\frac{\p \rho}{\p x^i}\frac{\p \rho}{\p
x^t})\frac{\p}{\p x^i}(\sigma_{st}\frac{\p \rho}{\p x^s})d\sigma^0
+\int_{\Si}\frac{\p}{\p x^t}(\sigma_{st}\frac{\p \rho}{\p
x^s})d\sigma^0\\
&=-\int_{\Si}\hat H \sigma_{st}\frac{\p \rho}{\p x^s}\frac{\p
\rho}{\p x^t}d\sigma^0 +\int_{\Si}\sigma_{st,t}\frac{\p \rho}{\p
x^s}d\sigma ^0 +\int_{\Si}\sigma_{st}\frac{\p^2 \rho}{\p x^s \p
x^t}d\sigma^0,
\end{split}
\end{equation}
where $d\sigma^0$ is the area element with respect to the Euclidean
induced metric, and we have used the divergence theorem in the
last equality.  By Lemma (\ref{estimate of area}), we have
\begin{equation}
\begin{split}
\int_{\Si}(H-\hat H)d \sigma =&\int_{\Si}(H-\hat H)d\sigma^0
+O(r^{1-2\tau})\\
=&\frac12 \int_{\Si}(H-\hat H)\sigma_{st}\frac{\p \rho}{\p
x^s}\frac{\p \rho} {\p x^t}d\sigma^0 +\frac12
\int_{\Si}(g_{ii,j}-g_{ij,i})\frac{\p
\rho}{\p x^j}d\sigma^0 \\
&-\frac12 \int_{\Si}\sigma_{st}\frac{\p^2 \rho}{\p x^s \p
x^t}d\sigma^0 +O(r^{1-2\tau}).
\end{split}
\end{equation}
Noticing that
$$H-\hat H = O(r^{-1-\tau})$$
we have
\begin{equation}\label{integral of diff meancurv}
\int_{\Si}(H-\hat H)d \sigma =\frac12
\int_{\Si}(g_{ii,j}-g_{ij,i})\frac{\p \rho}{\p x^j}d\sigma^0
-\frac12 \int_{\Si}\sigma_{st}\frac{\p^2 \rho}{\p x^s \p
x^t}d\sigma^0 +O(r^{1-2\tau}).
\end{equation}

\medskip

\section{Estimations of isometric embedding of nearly round   surfaces}

In this section, we study the isometric embedding of nearly round
surfaces, and the main purpose is to get the expansion of the mean
curvature nearly round   surfaces at the infinity of $(M,g)$.

 Let $(\mathbf{S}^2, g_0)$ be the
standard unit sphere and $i_0$ an isometric embedding of
$(\mathbf{S}^2, g_0)$ into $\mathbb{R}^3$. Let $K_g$ denote the Gauss curvature
of the metric $g$.
 We want to show the
following

\begin{theo}\label{estimateof isometric emb} There exists a positive constant
$\e_0>0$ such that for any metric $g$ on  $\mathbf{S}^2$ with
$$
\| K_g-1\|_{C^1} \leq \e_0,
$$
 there exists an isometric embedding
$$
i:(\mathbf{S}^2, g)\rightarrow \mathbb{R}^3
$$
and a conformal transformation $\Psi_1$ of $(\mathbf{S}^2, g_0)$
with
$$\|i\circ \Psi^{-1} _1 -i_0\|_{C^{2,\alpha}}\leq C_0 \| K-1\|_{C^\alpha}, $$
  for some $0<\alpha<1$. Here  $C_0$ is a positive  constant only depending on $\alpha$, and
$\|\cdot\|_{C^{2,\alpha}}$, $\|\cdot\|_{C^{\alpha}}$ are taken with respect to
$g_0$.
\end{theo}
\begin{proof}
The key point of the proof of above theorem is to show that there is a
conformal transformation $\Psi_1$ of the standard unit sphere with
\begin{equation}\label{perturbmetric}
\|\Psi_1 ^* (g)-g_0 \|_{C^{2,\alpha}} \leq C_1\| K-1\|_{C^\alpha},
\end{equation}
where $C_1$ is a  constant only depending on $\alpha$. Once
(\ref{perturbmetric}) is verified then by the arguments in P353 of
\cite{Nr}, we see that the conclusion of Theorem is true. Due to the
Uniformization Theorem, we see that there is conformal
differmorphism $\Phi$: $(\mathbf{S}^2, g_0)\mapsto (\mathbf{S}^2,
g)$ with $\Phi^* (g) = e^{2u} g_0$ and

\begin{equation}\label{equation1}
\Delta_{\mathbf{S}^2} u + K e^{2u}=1.
\end{equation}
Without loss of generality, we may assume for $1\leq i \leq 3$
\begin{equation}\label{masscenter}
\int_{\mathbf{S}^2}e^{2u} x_i =0,
\end{equation}
where $x_i$, $1\leq i \leq 3$, is the coordinate function of the
standard sphere in $\mathbb{R}^3$, integral is taken on the standard
sphere $(\mathbf{S}^2 , g_0)$. Otherwise, by Lemma 2, part 3 in
\cite{Ch} we can find a conformal transformation $\Phi_1$:
$(\mathbf{S}^2, g_0)\mapsto(\mathbf{S}^2, g_0)$ so that
$u\circ\Phi_1$ satisfying (\ref{equation1}), and (\ref{masscenter}).
Due to $a'$ in the proof of Theorem 1, Part 7 in \cite{Ch}, we know
that there a   constant $C$ only depending on $w$ and $W$ (here
$0<w\leq K \leq W$)  and with
\begin{equation}\label{compactness}
|u|_{C^0(\mathbf{S}^2)} \leq C.
\end{equation}
We now claim that for any $\eta>0$, there is $\delta=\delta(\eta)$
so that
\begin{equation}\label{small}
|u|_{C^0 (\mathbf{S}^2)}\leq \eta,
\end{equation}
provided $|K-1|_{C^0 (\mathbf{S}^2)}\leq \delta$.

Suppose the claim  fails, then we may find a constant $\eta_0
>0$, and a  sequence of $K_j$ and $u_j$ satisfying
(\ref{equation1}) and (\ref{masscenter}), and $|K_j -1|$ tends to
zero while $|u_j|_{C^0 (\mathbf{S}^2)}\geq \eta_0$. By
(\ref{compactness}) and the standard estimates in elliptic PDE,  we
may take a subsequence still denoted by $u_j$ which converges to
$u_\infty$ in the sense of $C^1(\mathbf{S}^2)$, and $u_\infty$
satisfying (\ref{masscenter}) and
$$
\Delta_{\mathbf{S}^2} u_\infty + e^{2u_\infty}=1.
$$
By this we see that $u_\infty =0$ which is contradiction to the
choice of $u_j$. Thus,  claim is true.

Let
$$u=u_0 + u_1 +u_2$$
with
$$
u_0=\frac{1}{4\pi}\int_{\mathbf{S}^2} u, \quad
u_1 =a_1 x_1 +a_2 x_2 +a_3 x_3, $$
where
$$
a_i =\frac{3}{4\pi} \int_{\mathbf{S}^2} u \cdot x_i
$$
for $1\leq 1\leq 3$.
Taking integral on (\ref{equation1}) on $(\mathbf{S}^2 , g_0)$, we
get
$$
\int_{\mathbf{S}^2}  K e^{2u} =4\pi.
$$
Together this with the assumption of $K$, we get
\begin{equation}\label{u0}
|u_0|\leq C (\|K-1\|_{C^0 (\mathbf{S}^2)}+\|u\|^2 _{L^2
(\mathbf{S}^2)}).
\end{equation}
Here  and in the sequel, $C$ is  always a  constant
independent of $u$.
Let
$$
L(u)
:=\Delta_{\mathbf{S}^2} u+ 2u,
$$
By equation (\ref{equation1}) and the definition of $u_2$,
we get
$$
L(u_2) = (1-K)e^{2u}+(1+2u -e^{2u})-2u_0,
$$
from which we have
\begin{equation}\label{estimate1}
\begin{split}
\int_{\mathbf{S}^2}( |\nabla u_2|^2 -2|u_2|^2 )\leq & C
(\|K-1\|_{C^0
(\mathbf{S}^2)}\|u_2\| _{L^2 (\mathbf{S}^2)}\\
& +\|1+2u -e^{2u}\|_{L^2 (\mathbf{S}^2)}\|u_2\| _{L^2
(\mathbf{S}^2)}\\
&+ |u_0|\|u_2\| _{L^2 (\mathbf{S}^2)}),
\end{split}
\end{equation}

From the
definition of $u_2$ we see that
\begin{equation}\label{estimate2}
\int_{\mathbf{S}^2}( |\nabla u_2|^2 -2|u_2|^2 )\geq C \|u_2 \|^2
_{L^2 (\mathbf{S}^2)}
\end{equation}
where $C$ is a positive universal constant  independent of $u$.
By (\ref{small}) we get
\begin{equation}\label{estimate3}
\|1+2u -e^{2u}\|_{L^2 (\mathbf{S}^2)} \leq C \|u\|^2 _{L^2
(\mathbf{S}^2)}
\end{equation}
Combining (\ref{estimate1}), (\ref{estimate2}), (\ref{estimate3})
and (\ref{u0}) we obtain
\begin{equation}\label{u2}
\|u_2 \| _{L^2 (\mathbf{S}^2)}\leq C (\|K-1\|_{C^0
(\mathbf{S}^2)}+\|u\|^2 _{L^2 (\mathbf{S}^2)})
\end{equation}
On the other hand, by the (\ref{masscenter}) and direct
computations we obtain, for each $i$,
$$
|a_i| \leq C \|u\|^2 _{L^2 (\mathbf{S}^2)},
$$
 which  implies
$$
\|u_1\| _{L^2 (\mathbf{S}^2)} \leq C  \|u\|^2 _{L^2
(\mathbf{S}^2)}.
$$

All the constants $C$ above are independent of $u$. Putting this
estimate with (\ref{u0}), (\ref{u2}), and (\ref{small}), we get
\begin{equation}\label{u}
\|u\| _{L^2 (\mathbf{S}^2)} \leq C \|K-1\|_{C^0 (\mathbf{S}^2)}.
\end{equation}
Together this with (\ref{equation1}), we get
$$
\|u \|_{W^{2,2}(\mathbf{S}^2)}\leq C \|K-1\|_{C^0 (\mathbf{S}^2)},
$$
which implies
$$
\|u \|_{C^\alpha(\mathbf{S}^2)}\leq C \|K-1\|_{C^0
(\mathbf{S}^2)},
$$
for some $\alpha>0$, here $C$ is a constant that only depends on
$\alpha$ and is  independent of $u$. Then by the Schauder theory in
partial differential equations, we get
$$
\|u \|_{C^{2,\alpha}(\mathbf{S}^2)}\leq C \|K-1\|_{C^0
(\mathbf{S}^2)},
$$
Thus, we see that
$$
\|\Psi^* _1 (g)-g_0 \|_{C^{2,\alpha}} \leq C_1\| K-1\|_{C^\alpha},
$$
where $C_1$ is a constant  depending only on $\alpha$. It
implies the conclusion of the Theorem is true. Thus, we finish to
prove the Theorem.
\end{proof}

Let $X$,  $n$ be the position vector and  the outward unit normal
vector of $i (\mathbf{S}^2)$ in $\mathbb{R}^3$ respectively,
$H_0$ be its mean curvature with respect to $n$, then  as a
corollary, we have
\begin{coro}\label{estimate2of isometric emb}
Let $(\mathbf{S}^2, g)$ be a two dimensional Riemannian manifold,
$K$ be its Gauss curvature. Then there exist a positive constant
$\e_0$ which is independent of $g$ such that if
$$
\| K-1\|_{C^1} \leq \e_0,
$$
then
$$|X\cdot n-1|\leq C \|K-1\|_{C^0
(\mathbf{S}^2)}, $$
and
$$
|H_0 -2|\leq C \|K-1\|_{C^0 (\mathbf{S}^2)},
$$
where $C$ is a constant that independent of $g$.
\end{coro}

\begin{proof}

By Theorem \ref{estimateof isometric emb}, we see that the statement
of the theorem is true for $i\circ \Psi^{-1}_1 (\mathbf{S}^2)$ in
$\mathbb{R}^3$. It is also true for the case of $i (\mathbf{S}^2)$,
for the support function $X\cdot n$ and the mean curvature $H_0$ are
independent of a parametrization of the domain manifold.
\end{proof}

On the other hand, by Corollary \ref{decay2ndformsinR3} and the
similar arguments as in Prop 2.1 in \cite{HY}, we may prove the
following

\begin{prop}\label{nearround}
Let $\hat \lambda_i$ be the principal curvature of $\Si$ in
$(M\setminus K, \hat g)$, If $|\AA|=O(r^{-1-\tau})$, $|\bar\nabla
\AA|=O(r^{-2-\tau})$, then there is a number $r_0 \in \R$ and a
vector $\overrightarrow{a} \in \R^3$ such that
$$
\hat \lambda_i -r^{-1}_0 =O(r^{-1-\tau}),
$$
$$
|(y-\overrightarrow{a})-r_0 \hat n|=O(r^{1-\tau}).
$$
Here $y$ is the position vector of $\Si$ in $(M\setminus K,\hat
g)$ (which is regarded as a subdomain  of $\R ^3$), $\hat n$ is
the outward unit normal vector of $\Si_r$.
\end{prop}

By the Gauss-Bonnet formula, Lemma \ref{estimate of area} and the
arguments we used before, we know that there is a  constant $C>0$
which is independent of $r$  and $r_0$ and  with $C^{-1} r \leq r_0
\leq C r$.

Now, we are in the position to  study the isometric embedding of
nearly round surfaces. Let $(M,g)$ be the AF manifolds and $\Si_r$
the nearly round surface in $(M,g)$ as $r$ goes to infinity. Then we
have

\begin{theo}\label{estimate H0}
Let $\Si_r\subset (M, g)$ be a nearly round  surfaces in $(M,g)$ as
$r$ goes to infinity,  and $r_0$  defined as in Proposition
\ref{nearround}. Then there is isometrically embedding $X_{r}$ of
$\Si_r$ into $\mathbb{R}^3$ such that
\begin{equation}\label{suppfun}
X_{r}\cdot n_0 =r_0 +O(r_0 ^{1-\tau})
\end{equation}
and
\begin{equation}\label{meancurvexp}
\|H_0  -\frac{2}{r_0}\|_{C^0}\leq C_3 r_0 ^{-1-\tau },
\end{equation}
provided that $r_0$ is  large enough. Here $n_0$ and $H_0 $ are the
unit outward normal vector and mean curvature of $X_r (\Si_r)$ in
$\mathbb{R}^3$ respectively. $C_3$ is a constant that is independent
of $r$.
\end{theo}
\begin{proof}
By the assumption on $A$ and $\bar\nabla A$, we see that
$$K=\frac{1}{r_0 ^2}+ O(r ^{-2-\tau})$$
and
$$
\|\bar\nabla K\|_{C^0}\leq C r ^{-3-\tau}.
$$
Then by a rescaling, we see that the resulting surface satisfying
the assumptions of Theorem \ref{estimateof isometric emb}. Then
combining Theorem \ref{estimateof isometric emb} with   direct
computations we get the conclusion.
\end{proof}

\section{Brown-York mass and Hawking  mass of nearly  round surfaces at infinity}

In this section, we prove our main results.
\begin{theo}\label{BYmass}
Let $(M,g)$ be an AF manifold, $\Si_r$ be a nearly  round surface of
$(M, g)$  as $r$ goes to infinity, then
$$
\lim_{r\rightarrow \infty}m_{BY}(\Si_r)=m_{ADM}(M).
$$
\end{theo}

\begin{theo}\label{Hawkingmass}
Let $(M,g)$ be an AF manifold, $\Si_r$ be a nearly  round surface of
$(M, g)$ as $r$ goes to infinity, then
$$
\lim_{r\rightarrow \infty}m_{H}(\Si_r)=m_{ADM}(M).
$$
\end{theo}

\begin{proof}[Proof of Theorem \ref{BYmass}]
Let $H_0$ be the mean curvature of the  isometric embedding image of
$(\Si_r, g)$ in $\R^3$. Then by Theorem \ref{estimate H0}, we have
$$
H_0 =\frac{2}{r_0}+O(r^{-1-\tau}), \quad
X_r \cdot n_0 =r_0 +O(r^{1-\tau}), \quad
K=\frac{1}{r^2_0}+O(r^{-2-\tau}).
$$
Then, by the same arguments as that at page 11 in \cite{FST}, we claim
that
$$
\int_{\Si_r}H_0 d \sigma = 4\pi r_0
+\frac{\Area(\Si_r)}{r_0}+O(r^{1-2\tau}).
$$
In fact, let $K=\frac{1}{r^2 _0}+\bar{K}$, by one of the Minkowski
integral
 formulae  \cite[Lemma 6.2.9]{KLBG},   we have
\begin{equation}\label{BY-ADMH01}
\begin{split}
\int_{\Si_r}H_0d\sigma &=2\int_{\Si_r}KX_r\cdot n_0d\sigma\\
&=2\int_{\Si_r}\lf(\frac1{r_0 ^2}+\bar{K}\ri)X_r\cdot n_0d\sigma\\
&=\frac{2}{r^2 _0}\int_{\Si_r}X_r\cdot n_0d\sigma
+2\int_{\Si_r}\bar{K}
X_r\cdot n_0d\sigma\\
&=\frac{6V(r)}{r_0 ^2}+2\int_{\Si_r}\bar{K}\lf(r_0
+O\lf(r^{1-\tau}\ri)\ri)
d\sigma\\
&=\frac{6V(r)}{r_0 ^2}+2r_0\int_{\Si_r}\bar{K}+O(r^{1-2\tau})\\
&=\frac{6V(r)}{r_0 ^2}+2r_0\int_{\Si_r}\lf(K-\frac1{r_0 ^2}\ri)+O(r^{1-2\tau})\\
&=\frac{6V(r)}{r_0 ^2}+8\pi r_0
-\frac{2\Area(\Si_r)}{r_0}+O(r^{1-2\tau}),
\end{split}
\end{equation}
where $V(r)$ is the volume of the interior of the surface
$X_r(\Sigma_r)$ in $\R^3$. Here and in the sequel, $\Area(\Si_r)$ is
the area of $(\Si, g)$. On the other hand, by the above estimate we
know that  $H_0=\frac2r+H_1$ with $H_1=O\lf(r^{-1-\tau}\ri)$ and by
another Minkowski integral formula we have
\begin{equation}\label{BY-ADMH02}
\begin{split}
2\Area(\Si_r)&= \int_{\Si_r}H_0X_r\cdot n_0d\sigma\\
&=\frac{6V(r)}{r_0}+\int_{\Si_r}H_1X_r\cdot n_0d\sigma\\
&=\frac{6V(r)}{r_0}+r\int_{\Si_r}H_1d\sigma+O\lf(r^{2-2\tau}\ri)\\
&=\frac{6V(r)}{r_0}-2\Area(\Si_r)+r_0 \int_{\Si_r}H_0d\sigma
+O\lf(r^{2-2\tau}\ri),
\end{split}
\end{equation}
which implies
\begin{equation}\label{BY-ADMH03}
\int_{\Si_r}H_0d\sigma=-\frac{6V(r)}{r_0 ^2}+
\frac{4\Area(\Si_r)}{r_0} +O(r^{1-2\tau}).
\end{equation}
 The claim  follows from \eqref{BY-ADMH01} and \eqref{BY-ADMH03}.
 From the claim and (\ref{integral of diff meancurv}) we get
\begin{equation}
\begin{split}
&\int_{\Si_r}(H_0-H) d \sigma=4\pi r_0
+\frac{\Area(\Si_r)}{r_0}-\int_{\Si_r}\hat H d\sigma\\
&+\frac12 \int_{\Si_r}(g_{ij,i}-g_{ii,j})\frac{\p \rho}{\p
x^j}d\sigma_0 + \frac12 \int_{\Si_r}\sigma_{st} \frac{\p^2 \rho}{\p
x^s \p x^t}d\sigma^0 +O(r^{1-2\tau}).
\end{split}
\end{equation}

By the definition, we see that
$$
\frac{\Area(\Si_r)}{r_0}=\frac{\Area_0(\Si_r)}{r_0}+\frac12 r^{-1}_0
\int_{\Si_r}h^{st}\sigma_{st} d\sigma_0 +O(r^{1-2\tau}),
$$
where $\Area_0(\Si_r)$ is the area of $\Si_r$ with respect to induce
metric from $\hat g$.
We also have
$$
\int_{\Si_r}\hat H d\sigma =\int_{\Si_r}\hat H d\sigma_0 +\frac12
\int_{\Si_r}\hat H h^{st}\sigma_{st} d\sigma_0 +O(r^{1-2\tau}).
$$
By Lemma \ref{2ndfundamentalforms} we have
$$
\frac12 \int_{\Si_r}\sigma_{st}\frac{\p^2 \rho}{\p x^s \p
x^t}d\sigma=\frac12\int_{\Si_r}\sigma_{st}\overset{\circ}{\hat
A}_{st} +\frac14 \int_{\Si_r} \hat H h^{st}\sigma_{st}d\sigma_0
+O(r^{1-2\tau})
$$
Combining these things together we get

\begin{equation}
\begin{split}
&\int_{\Si_r}(H_0 -H)d \sigma =4\pi r_0
+\frac{\Area_0(\Si_r)}{r_0}-\int_{\Si_r}\hat H d\sigma_0
+\int_{\Si_r}(\frac{1}{2r_0}-\frac{\hat
H}{4})h^{st}\sigma_{st}d\sigma_0\\
&+ \frac12 \int_{\Si_r}(g_{ij,i}-g_{ii,j})\frac{\p \rho}{\p
x^j}d\sigma_0 +\frac12 \int_{\Si_r}\sigma_{st}B_{st}d\sigma.
\end{split}
\end{equation}

Set $\hat X =y-\overrightarrow{a} $. It is an isometric
embedding of $(\Si_r, \hat g)$ into $\R ^3$. By Proposition
\ref{nearround}, we see that
$$
\hat X\cdot \hat n =r_0 +O(r^{1-\tau}), \quad
\hat H =\frac{2}{r_0}+O(r^{-1-\tau}),\quad
\hat K=\frac{1}{r^2_0}+O(r^{-2-\tau}),
$$
where $\hat K$ is the Gauss curvature of $\Si_r$ in $(M \setminus K,
\hat g)$. Using the same arguments as that at page 11 in \cite{FST} to
$(\Si, \hat g)$ we get
$$
4\pi r_0 +\frac{\Area_0(\Si_r)}{r_0}-\int_{\Si}\hat H d\sigma_0
=O(r^{1-2\tau}).
$$
By Corollary \ref{decay2ndformsinR3} and Proposition \ref{nearround}
we see that
$$
\int_{\Si_r}(\frac{1}{2r_0}-\frac{\hat
H}{4})h^{st}\sigma_{st}d\sigma_0+\frac12
\int_{\Si_r}\sigma_{st}B_{st}d\sigma =O(r^{1-2\tau}).
$$
Thus we have
$$
\int_{\Si_r}(H_0 -H)d \sigma =\frac12
\int_{\Si_r}(g_{ij,i}-g_{ii,j})\frac{\p \rho}{\p x^j}d\sigma_0
+O(r^{1-2\tau}).
$$
The first term in the right hand side of the above equality is the ADM mass
of $(M,g)$. Thus, we finish to prove  Theorem \ref{BYmass}.\end{proof}

\begin{proof}[Proof of Theorem \ref{Hawkingmass}] By Lemma
\ref{difference of mean curvature} and (\ref{3rdterminmeancurv})
we have
\begin{equation}
\begin{split}
\int_{\Si_r}H^2 d \sigma=&\int_{\Si_r}{\hat H}^2 d \sigma
+\int_{\Si_r}H \cdot \hat H \frac{\p \rho}{\p x^i} \frac{\p \rho}{\p
x^j}\sigma_{ij} d \sigma\\
&+\hat{H}\int_{\Si_r} \sigma_{st,i}\frac{\p \rho}{\p x^i}\frac{\p
\rho}{\p x^s}\frac{\p \rho}{\p x^t}d\sigma^0 -2 \hat
{H}\int_{\Si_r}\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p x^j}d\sigma^0
\\
&-2\hat{H}\int_{\Si_r} g_{ij,i}\frac{\p \rho}{\p x^j}d\sigma^0
+\hat{H}\int_{\Si_r} g_{jj,i}\frac{\p \rho}{\p x^i}d\sigma^0 +
O(r^{-2\tau})\\
=&\int_{\Si_r}{\hat H}^2 d \sigma+\int_{\Si_r}H \cdot \hat H
\frac{\p \rho}{\p x^i} \frac{\p \rho}{\p x^j}\sigma_{ij} d
\sigma^0 \\
&-{\hat H}^2 \int_{\Si_r}\sigma_{st}\frac{\p \rho}{\p x^s} \frac{\p
\rho}{\p x^t}d \sigma^0 +\hat{H} \int_{\Si_r}\sigma_{st,t}\frac{\p
\rho}{\p x^s}d \sigma^0 \\
&+\hat {H}\int_{\Si_r}\sigma_{st}\frac{\p^2 \rho}{\p x^s \p
x^t}d\sigma^0
-2\hat {H}\int_{\Si_r}\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p
x^j}d\sigma^0 \\
&-2\hat{H}\int_{\Si_r} g_{ij,i}\frac{\p \rho}{\p x^i}d\sigma^0
+\hat{H}\int_{\Si_r} g_{jj,i}\frac{\p \rho}{\p x^i}d\sigma^0 +
O(r^{-2\tau})\\
=&\int_{\Si_r}{\hat H}^2 d \sigma +\hat{H}\int_{\Si_r}
(g_{jj,i}-g_{ij,j})\frac{\p \rho}{\p x^i}d\sigma^0 \\
&-\hat {H}\int_{\Si_r}\sigma_{ij}\frac{\p^2 \rho}{\p x^i \p
x^j}d\sigma^0 + O(r^{-2\tau}).
\end{split}
\end{equation}
Note that
\begin{equation}
\begin{split}
d\sigma &=(1+h^{ij}\sigma_{ij}+O(r^{-2\tau}))^{\frac12}d\sigma_0
=d\sigma^0 +\frac12 h^{ij}\sigma_{ij} d\sigma_0 +O(r^{-2\tau}).
\end{split}
\end{equation}
Combining above equalities with Lemma \ref{2ndfundamentalforms}, we
get
$$
\int_{\Si_r}H^2 d \sigma =\int_{\Si_r}{\hat H}^2 d \sigma_0
+\hat{H}\int_{\Si_r} (g_{jj,i}-g_{ij,j})\frac{\p \rho}{\p
x^i}d\sigma^0+ O(r^{-2\tau}).
$$
Hence,
\begin{equation}
\begin{split}
m_H (\Si_r)=&\frac{\Area(\Si_r)^{\frac12}}{{(16 \pi)}^\frac32}(16\pi
-\int_{\Si_r}{\hat H}^2 d
\sigma_0)\\
&-\frac{\Area(\Si_r)^{\frac12}}{{(16 \pi)}^\frac32}\cdot
\hat{H}\int_{\Si_r} (g_{jj,i}-g_{ij,j})\frac{\p \rho}{\p
x^i}d\sigma^0
+O(r^{1-2\tau})\\
=&-2\frac{\Area(\Si_r)^{\frac12}}{(16
\pi)^\frac32}\int_{\Si_r}|\overset{\circ}{\hat A}|^2 d\sigma^0
-\frac{\Area(\Si_r)^{\frac12}}{{(16 \pi)}^\frac32}\cdot
\hat{H}\int_{\Si_r} (g_{jj,i}-g_{ij,j})\frac{\p \rho}{\p
x^i}d\sigma^0 \\
&+O(r^{1-2\tau})\\
=&\frac{\Area(\Si_r)^{\frac12}}{{(16 \pi)}^\frac32}\cdot
\hat{H}\int_{\Si_r} (g_{ij,j}-g_{jj,i})\frac{\p \rho}{\p
x^i}d\sigma^0 +O(r^{1-2\tau}),
\end{split}
\end{equation}
where we have used estimate
$$
|\overset{\circ}{\hat A}|\leq C r^{-1-\tau}
$$
in the last equality. On the other hand, we  have

$$
K=\frac{1}{r^2 _0}+O(r^{-2-\tau})
$$
By the Gauss-Bonnet formula, we get
$$
\Area(\Si_r)=4\pi r^2 _0 + O(r^{2-\tau}),
$$
From  Proposition \ref{nearround} we see that
$$
\hat H =\frac{2}{r_0}+O(r^{-1-\tau}),
$$
Combining these formulas we obtain
$$
m_H (\Si_r)=\frac{1}{16 \pi}\int_{\Si_r} (g_{ij,j}-g_{jj,i})\frac{\p
\rho}{\p x^i}d\sigma^0 +O(r^{1-2\tau}).
$$
Thus, we finish to prove the Theorem.
\end{proof}


TITLE: The converse of "nilpotent elements are zero-divisors"
QUESTION [2 upvotes]: For commutative rings $A$ with identity $1\ne0$, nilpotent elements are zero-divisors. The converse is false, i.e. there is a commutative ring $A$ with identity $1\ne0$ and a zero-divisor $x$ in $A$ which is not nilpotent. Where is such an example?
Is it meaningful to ask the "percentage" of commutative rings $A$ with identity $1\ne0$ for which the converse holds, i.e. zero-divisors are nilpotent?

REPLY [0 votes]: Another class of examples are matrix rings over a field. A $n\times n$ matrix $A$ is a zero-divisor if and only if $A$ is not invertible.

Certainly if $A$ is a zero divisor then $A$ cannot be invertible. If $A$ is not invertible then $\ker A \ne 0$ and taking any non-zero matrix $B$ with $\operatorname{col}B \subseteq \ker A$ you have $AB = 0$. For example, let $v$ be a non-zero vector with $Av = 0$ and let $B$ be the matrix whose columns are all $v$.

The nilpotent matrices are rare among zero-divisors. Specifically, nilpotent matrices are solutions to the equation $A^n = 0$. As a rule, there are always more non-solutions to a polynomial equation than there are solutions.
In precise terms, we look at the zero set $\{A : \det A = 0\}$ (the set of zero-divisors). This zero-set is a manifold if the underlying field is $\mathbf{C}$ and is a variety in the general case. The non-nilpotents are dense in the Zariski topology looking over a general ring, and dense in the classical topology looking over $\mathbf C$.