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\begin{document}

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\title{Gauss-Bonnet on certain open manifolds (Joint work with Tian)}
%\author{÷��ǿ}
\author{Jiaqiang Mei}
%\institution{�Ͼ���ѧ��ѧϵ\\�Ͼ���ѧ�ִ���ѧ�о���}
\institution{Department of Mathematics \\ Institute of Mathematical Science \\
Nanjing University }
\email{meijq@nju.edu.cn}
\slideCaption{Talk at Ningbo 2006}

\maketitle

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% slide 1
% the number 6 in next line means you have 6 items in this slide.
\overlays{6}{%
\begin{slide}[R]{Content}
\bis
\item Introduction

\item Gauss-Bonnet in 19th Century 

\item Gauss-Bonnet in 20th Century

\item Gauss-Bonnet on open manifolds 

\item Gauss-Bonnet Renormalized 
\item Conformally compact 4-manifolds 
\eis
\end{slide}
}

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% slide 2

\overlays{2}{%
\begin{slide}{Introduction}
\bis
\item
In this talk I will consider the Gauss-Bonnet-Chern formula on
 some open Riemannian manifolds.
\item
The question is as follows: \\
\vskip 1em
What is the Gauss-Bonnet-Chern formula on conformally compact four
manifolds?
\eis
\end{slide}
}
\overlays{2}{%
\begin{slide}{Introduction}
\bis
\item
Let $M$ be the interior of a compact manifold with boundary. According
 to Penrose, a complete metric $g$ on $M$ is {\it  conformally
compact}  if there is a smooth defining function $\rho $ on $\bar M =
M\cup\partial M,$ i.e. $\rho (\partial M) =$ 0, $d\rho  \neq $ 0 on
$\partial M$ and $\rho  > $ 0 on $M$, such that the metric
\begin{equation}
\bar g = \rho^{2}\cdot  g,
\end{equation}
extends to a smooth metric on $\bar M.$
\item
$\rho$ is called special if $|d\rho|_{\bar{g}}^2 =1$ on a neighborhood of the boundary.
\eis
\end{slide}
}
\overlays{2}{%
\begin{slide}{Introduction}
\bis
%\item
\item
Under mild conditions, the Gauss-Bonnet-Chern formula for a conformally
compact manifolds has the following form:
$$
\frac{1}{8\pi^{2}}\int_{M}(|W|^{2} -  \frac{1}{2}|z|^{2} +
\frac{1}{24}(s+12)^{2})dV
$$
$$
= \chi (M) -\frac{3}{4\pi^2}\hat{V},
$$
\item $W$: Weyl tensor, $z$: trace-free Ricci tensor, $s$: scalar curvature.
\eis
\end{slide}
}

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% 3
\overlays{3}{%
\begin{slide}{Gauss-Bonnet in 19th Century}
\bis
\item
Let's review briefly the history of the Gauss-Bonnet-Chern formula.
\item
Gauss, 1828: For a geodesic triangle $ABC$ in a surface in $R^3$, one has
$$\alpha + \beta +\gamma -\pi = \int_{ABC}k\mbox{d}s.$$
\item
Bonnet, 1848: extended the formula to smooth curves on surfaces.

\eis
\end{slide}
}
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% 4
\overlays{2}{%
\begin{slide}{Gauss-Bonnet in 20th Century}
\bis
\item
(Gauss-Bonnet)Let $\Sigma$ be a smooth closed oriented surface in $R^3$, then
$$\int_{\Sigma}k\mbox{d}s = 2\pi\chi (\Sigma).$$
\item
Hopf, 1925: For a hypersurface $M^n$ in $R^{n+1}$($n$ even), one has
$$\int_{M}k\mbox{d}v =\frac{1}{2}\mbox{vol}(S^n)\chi (M),$$
where $k$ is the Gauss-Kronecker curvature.

\eis
\end{slide}
}
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% 5
\overlays{3}{%
\begin{slide}{Gauss-Bonnet in 20th Century}
\bis
\item
Allendoerfer and Weil(independently), 1940: Extended the formula to submanifolds
of any co-dimensions.
\item
Allendoerfer and Weil, 1943: For any abstract oriented riemannian manifolds, one has
$$\int_{M}\Theta = \chi (M^n)$$
\item
Remarks: For odd $n$, $\Theta = 0$; They use the local isometric embedding theorem to
obtain the global formula.
\eis
\end{slide}
}

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% 6
\overlays{4}{%
\begin{slide}[Blinds]{Gauss-Bonnet in 20th Century}
\bis
\item
Chern, 1944: "A simple intrinsic proof of the Generalized
Gauss-Bonnet theorem".
\item
Results for open manifolds:
\item
Cohn-Vossen, 1935: For complete surface $M$, if $\dim H_1(M, R)$ is finite, then
$$\int_{M}\Theta \leq \chi (M).$$
\item
Huber, 1957: Extended the above result to general 2-manifolds.
\eis
\end{slide}
}

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% 7
\overlays{3}{%
\begin{slide}[Blinds]{Gauss-Bonnet on open manifolds}
\bis
\item
Walter, 1975: For complete 4-manifolds with non-negative sectional
curvature,
$$\int_{M}\Theta \leq \chi (M).$$
\item
Greene and Wu, 1976: The above formula holds for 4-manifolds with positive
 sectional curvature outside some compact set.
\item
Cheeger and Gromov, 1985: They considered complete manifolds with bounded
curvature and finite volume.
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 8
\overlays{3}{%
\begin{slide}[Blinds]{Gauss-Bonnet on open manifolds}
\bis
\item
In this case, if the manifold is of finite
topological type, then
$$\int_{M}\Theta = \chi (M).$$
\item
Chang, Qing, Yang, 2000: For certain complete metric on $R^4$, one has
$$\int_{R^4} Qe^{4w}dx \le 4\pi^2 \chi(R^4) = 4\pi^2.$$
\item
Fang, 2005: Considered a class of complete locally conformally
flat manifolds.
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 9
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact manifolds}
\bis
\item
Let's now return back to conformally compact manifolds. When $(M,g)$
is a complete conformally compact Einstein metric with $Ric_g=-(n-1)g$,
then the sectional curvatures of $g$ necessarily approach $-1$ uniformly at
infinity at an exponential rate, i.e, the manifolds are asymptotically hyperbolic.
\item
The study of this kind of manifolds has become very active
recently due to the so called AdS/CFT correspondence in string theory.

\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 10
\overlays{2}{%
\begin{slide}[Blinds]{Renormalized volume}
\bis
\item
Let $\rho$ be a special defining function. Graham observed that, in even dimensions,
$$ \int_{\rho>\varepsilon} dvol_g =
 C_0 \varepsilon^{1-n} + C_2 \varepsilon^{3-n} + \ldots (\text{odd powers})$$
 $$ \ldots
 +C_{n-2} \varepsilon^{-1} + \widehat{V} + o(1), $$
\item
$\widehat{V}$ is known as the renormalized volume, it does not depend on the choice
 of special defining functions.
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 11
\overlays{2}{%
\begin{slide}[Blinds]{Gauss-Bonnet Renormalized}
\bis
\item Anderson (2001) showed that, for 4-dim conformally compact Einstein manifolds,
$$    \frac{1}{8(2\pi)^2}\int_{M} |W|^2 + \frac{3}{(2\pi)^2}\hat{V} = \chi(M) ,$$
where $W$ is the Weyl curvature tensor.
\item
This formula can be thought as a Renormalized Gauss-Bonnet formula. From it one can
also see that the renormalized volume $\hat{V}$ is only depend on $(M, g)$.


\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 12
\overlays{2}{%
\begin{slide}[Blinds]{Gauss-Bonnet Renormalized}
\bis
\item
Albin (2005) then proved a Renormalized Gauss-Bonnet formula for any even dimensional
conformally compact Einstein manifolds:
$$\sideset{^{R}}{}\int_M \Theta = \chi (M).$$
\item
A particular case was also obtained by Epstein (2001) for convex cocompact hyperbolic manifold:
$$\frac{(-1)^{m/2}}{2^{m/2}(2\pi)^{m/2}} \frac{m!}{(m/2)!} \hat{V} = \chi(M).$$
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 13
\overlays{2}{%
\begin{slide}[Blinds]{Gauss-Bonnet Renormalized}
\bis
\item
Also, Chang, Qing, and Yang (2004) obtained the following general formula:
$$\int_M \widetilde{W} dvol_g
       + (-1)^{\frac{m}{2}} \frac{\Gamma\frac{m+1}{2}}{\pi^{\frac{m+1}{2}}} \hat{V}
       = \chi (M),$$
where $\widetilde{W}$ is a full contraction of the Weyl tensor and its covariant derivatives.
\item
Question 1. Both formulas are the generalizations of the Gauss-Bonnet-Chern formula. What's
the relation between them?
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 14
\overlays{3}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
 Question 2. What happens if the manifolds are not Einstein?
\item
To our knowledge, the answer to question 1 is unclear up to now. We consider
question 2 for the case of dimension 4.
\item
Let $M$ be a 4-dimensional open manifold with a complete metric $g$.
Suppose $\rho$ is a positive function on $M$ such that $\rho^2\cdot g$ can be
extended to a metric $\bar{g}$ on $\bar{M}=M\cup\partial M$. So $\rho |_{\partial M}=0$.
\eis
\end{slide}
}
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% 15
\overlays{3}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
Let $K_{ij}$, $\bar{K_{ij}}$ be the sectional curvatures on $M$ and 
$\bar{M}$
respectively. We have
\item
$$\bar{K_{ij}} = \rho^{-2}(K_{ij} + |\bar{\nabla}\rho|^2)$$
$$-\rho^{-1}[\bar{D}^2\rho(\bar{e_i}, \bar{e_i}) + 
\bar{D}^2\rho(\bar{e_j}, \bar{e_j})]$$
\item
Assume that
$$ $$
$ i). |\bar{\nabla}\rho| = 1$ near $\partial M$, $ ii). \bar{D}^2\rho = 
O({\rho}).$
\eis
\end{slide}
}
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% 16
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
Then we have
$$K_{ij} +1 = O(\rho^2)$$
i.e, $(M, g)$ is asymtotically hyperbolic. Also
\item
$$Ric +3 = \rho^2\cdot\bar{Ric} + 2\rho\cdot\bar{\nabla}^2\rho + 
\rho\cdot\bar{\Delta}\rho ,$$
$$ s + 12 = \rho^2\cdot\bar{s} + 6\rho\cdot\bar{\Delta}\rho.$$
%\item

%\item

\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 17
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
Let $\rho = e^{-r}$, and $\lambda_i, \bar{\lambda}_i$ be the eigenvalues
of $D^2 r$ and $\bar{D}^2 \rho$ respectively. We have
$$\lambda_i = 1 - \rho\cdot\bar{\lambda}_i$$
\item
Since $|\bar{\nabla}\rho| = 1$ near $\partial M$, the integral curves of 
$\bar{\nabla}\rho$ are geodesics. So along these geodesics, we have the 
Ricatti
equation:
$$ \bar{H}' + |\bar{A}|^2 + \bar{Ric}(\bar{\nabla}\rho, 
\bar{\nabla}\rho) =0.$$
Where $\bar{H}$ is the mean curvature of $\partial M$.
%\item

%\item

\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 18
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
In particular, Since
$$Ric(4,4) +3 = \rho^2\cdot\bar{Ric}(4,4) + \rho\cdot\bar{H},$$
we have the following estimate
\item
$$Ric(4,4) + 3 = -\frac{1}{3}\rho^3\cdot\bar{H}''(0) + O(\rho^4)$$

which means $Ricci$ along normal direction decays at rate of order at 
least 3.
%\item

%\item

\eis
\end{slide}
}
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% 19
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
The idea of proof of the renormalized Gauss-bonnet-Chern formula is to 
apply
the above computations to  manifolds with boundary.
\item
$$
\frac{1}{8\pi^{2}}\int_{D}(|R|^{2} -  4|z|^{2}) = \chi (D) -
$$
$$
\frac{1}{2\pi^2}\int_{\partial D}\prod_{i=1}^3\lambda_i - 
\frac{1}{8\pi^{2}}\int_{\partial D}
\sum_{\sigma\in S_3}K_{\sigma_1\sigma_2}\cdot\lambda_{\sigma_3}
$$
%\item

%\item

\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 20
\overlays{1}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
Take $D=B(r)=\{ \log\rho^{-1}\leq r \}\subset M$, $\partial D = S(r)$. 
It follows that  
$$ \frac{1}{8\pi^2}\int_{B(r)}[|W|^2 - \frac{1}{2}|z|^2 + 
\frac{1}{24}(s+12)^2] $$
$$= \chi (B(r)) - \frac{3}{4\pi^2}[I + II +III] + O(\rho),$$
$$I = volB(r) - \frac{1}{3}volS(r) = 
\frac{1}{3}\rho^{-1}\cdot\int_{\bar{S}(0)}\bar{H}'$$
$$-\frac{1}{6}\log\rho\cdot\int_{\bar{S}(0)}\bar{H}'' + C_1 + o(1)$$

%\item

%\item

\eis
\end{slide}
}
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% 21
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis
\item
$$II = \frac{1}{6}\int_{B(r)}(s+12) - \frac{1}{6}\int_{S(r)}(s+12)$$
$$= -\frac{1}{6}\log\rho\cdot\int_{\bar{S}(0)}[2\bar{\tau}'(0) + 
\bar{H}''(0)]
+C_2 + o(1)$$

\item

$$III = \frac{1}{3}\int_{S(r)}(\rho^2\bar{H}' -2\rho\bar{H})$$
$$=-\frac{1}{3}\rho^{-1}\int_{\bar{S}(0)}\bar{H}'(0) + O(\rho)$$
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 22
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis

\item
Thus we have
$$
\frac{1}{8\pi^2}\int_{B(r)}[|W|^2 - \frac{1}{2}|z|^2 + 
\frac{1}{24}(s+12)^2]$$
$$
=\chi (B(r)) + C_3\cdot\log\rho +C_4 + o(1)
$$
This implies that the constants $C_3$ is $0$. 
\item
The final formula:
$$\frac{1}{8\pi^2}\int_{M}[|W|^2 - \frac{1}{2}|z|^2 + 
\frac{1}{24}(s+12)^2]$$
$$=\chi (M) - \frac{3}{4\pi^2}\hat{V}$$
\eis
\end{slide}
}
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% 23
\overlays{2}{%
\begin{slide}[Blinds]{Conformally compact 4-manifolds}
\bis

\item
where $\hat{V}$ is the following limit:

$$\hat{V}=\lim_{r\rightarrow +\infty}[volB(r) - \frac{1}{3}volS(r) + 
\frac{1}{6}\int_{B(r)}(s+12) $$
$$ - \frac{1}{6}\int_{S(r)}(s+12) 
+ \frac{1}{3}\int_{S(r)}(\rho^2\bar{H}' -2\rho\bar{H})] $$

\item
$\hat{V}$ is called the renormalized volume.
\eis
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 24
\overlays{4}{%
\begin{slide}[Blinds]{Remarks}
\bis

\item
What's the meaning of the renormalized volume $\hat{V}$ ?
\item
Which metric $g$ can be conformally compactified ?
\item 
How about the Gauss-Bonnet-Chern formula on higher dimensional 
manifolds?

\item

Acknowledgement: THANKS FOR YOUR PATIENCE!

\eis
\end{slide}
}
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