## A positive mass theorem and Penrose type inequalities for the Gauss-Bonnet-Chern mass

In a recent preprint [GWW1], Ge, Wang and Wu proposed a family of new masses ${m_k(g)}$ (${k<\frac{n}{2}}$) for an asymptotically flat manifold ${(M^n,g)}$ and proved a positive mass theorem and some Penrose type inequalities for graphs, at least when ${k\le 2}$. They remarked that their results can be generalized for all ${k}$. While highly plausible, it does not seem obvious that their calculation can be extended to the case where ${k>2}$ (especially for the Penrose type inequalities). In this note, we present a more concise proof of the positive mass theorem and Penrose type inequalities for graphs.

1. Introduction

Let us first recall some definitions. We will adopt the following definition of an asymptotically flat manifold:

 Definition 1 A complete ${n}$-dimensional Riemannian manifold ${(M,g)}$ is said to be asymptotically flat of order ${\tau}$ (with one end) if there is a compact subset ${K}$ such that ${M\setminus K}$ is diffeomorphic to ${\mathbb R^n\setminus B_\rho(0)}$ for some ${\rho>0}$ and in the standard coordinates in ${\mathbb R^n}$, the metric ${g}$ satisfies: $\displaystyle g_{ij}=\delta_{ij}+\sigma_{ij}$ with $\displaystyle |\sigma_{ij}|+r|\partial \sigma_{ij}|+r^2|\partial ^2\sigma_{ij}|+r^3|\partial ^3 \sigma_{ij}|=O(r^{-\tau}),$ for some constant ${\tau>0}$. Here ${\partial }$ denotes the ordinary partial derivative and ${r}$ is the Euclidean distance from the origin.

A coordinate system of ${M}$ near infinity is said to be admissible if the metric tensor in this system satisfies the above decay conditions.

If we further assume that ${\tau>\frac{n-2}{2}}$ and the scalar curvature ${R(g)}$ is integrable in ${(M, g)}$, then in general relativity, the ADM mass ([ADM]) is defined by (under an admissible coordinate system):

$\displaystyle m_{ADM}(g)= \frac{1}{2(n-1)\omega_{n-1}}\lim_{r\rightarrow \infty}\int_{S_r}(\partial _j g_{ji}-\partial _i g_{jj})\nu_idS.$

Here ${S_r}$ is the coordinate sphere with Euclidean radius ${r}$, ${\nu}$ is the outward unit normal of ${S_r}$ w.r.t. ${\delta}$ and ${\omega_{n-1}}$ is the volume of the standard unit sphere in ${\mathbb{R}^n}$. (In this note, unless otherwise stated, we will sum over any index (except ${n}$) that appears more than once. )

In [GWW2], [GWW1], Ge, Wang and Wu proposed a new mass which generalizes the ADM mass on an asymptotically flat manifold ${(M^n,g)}$, which they called the Gauss-Bonnet-Chern mass, defined by

$\displaystyle m_k(g)= c_k(n)\lim_{r\rightarrow \infty}\int_{S_r} P_{(k)}^{ijlm}\partial _m g_{jl}\nu_i dS.$

Here

$\displaystyle c_k(n)= \frac{(n-2k)!}{2^{k-1}(n-1)!\omega_{n-1}},$

${P_{(k)}}$ ${(k<\frac{n}{2})}$ is a ${4}$-tensor whose definition will be given in Subsection 2. We remark that when ${k=1}$, then

$\displaystyle P_{(1)}^{ijlm}= \frac{1}{2}(g^{il}g^{jm}- g^{im}g^{jl})$

and ${m_1(g)}$ is just the ADM mass of ${(M,g)}$. It was proved in [GWW1] that if ${\tau>\frac{n-2k}{k+1}}$, then ${m_g(g)}$ is a geometric invariant (see also [Bartnik]), i.e. it is independent of the choice of the admissible coordinate system near infinity.

Inspired by an interesting preprint of Lam ([Lam]), Ge, Wang and Wu ([GWW1]) proved a positive mass theorem ([GWW1] Theorem 1.4) and some Penrose type inequalities ([GWW1] Theorem 1.6) regarding their new mass for graphs, at least when ${k=2}$. They remarked that their results can be generalized for all ${k<\frac{n}{2}}$. While this is very reasonable, it is not obvious to us that their calculation can be extended to include the case where ${k>2}$ (especially for the Penrose inequalities, see [GWW1] Proposition 5.2). In this note, following the ideas in [GWW1], we will present a somewhat more concise proof of the positive mass theorem and Penrose type inequalities for graphs. Indeed, most of the ingredients are already present in a classical paper of Reilly [Reilly1]. More precisely, we prove the following:

 Theorem 2 (Theorem 6) Let ${(M^n,g)}$ be the graph of an asymptotically flat function ${f: \mathbb{R}^n\rightarrow \mathbb{R}}$. Suppose ${L_k}$ is integrable on ${(M,g)}$, then $\displaystyle m_k(g)= \frac{c_k(n)}{2}\int_M \frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g.$ In particular, if ${L_k\ge 0}$, then ${m_k(g)\ge 0}$.

The Gauss-Bonnet curvature ${L_k}$ is defined in Subsection 2. Indeed, (up to a constant) it is the integrand for the general Gauss-Bonnet theorem for a Riemannian manifold, and is an intrinsic geometric quantity.

As a special case of Theorem 10, we also prove the following Penrose type inequalities (cf. also [GWW1], [Lam], [Schwartz]):

 Theorem 3 (Theorem 10) Let ${\Omega}$ be a bounded open set in ${\mathbb{R}^n}$ and ${\Sigma= \partial \Omega}$ (not necessarily connected). If ${f:\mathbb{R}^n\setminus \Omega\rightarrow \mathbb{R}}$ is a smooth asymptotically flat function such that each connected component of ${\Sigma}$ is in a level set of ${f}$ and ${|\nabla f(x)|\rightarrow \infty}$ as ${x\rightarrow \Sigma}$. Furthermore, assume that each connected component of ${\Sigma}$ is ${(2k-1)}$-convex and star-shaped, and ${L_k(g)\ge 0}$, then $\displaystyle \begin{array}{rl} m_k (g) \ge\frac{1}{2^k}\left(\frac{\int_\Sigma L_{k-1}^\Sigma dS}{P(n-1, 2k-2)\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+1}} \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma L_{k-2}^\Sigma dS}{P(n-1, 2k-4)\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+3}}\\ \ge&\displaystyle \cdots\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma L_1^\Sigma dS}{P(n-1, 2)\omega_{n-1}}\right)^{\frac{n-2k}{n-3}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\mathrm{Area}(\Sigma)}{\omega_{n-1}}\right)^{\frac{n-2k}{n-1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{n \mathrm{Vol(\Omega)}}{\omega_{n-1}}\right)^{\frac{n-2k}{n}}, \end{array}$ where ${P(n-1,l)= \frac{(n-1)!}{(n-1-l)!}}$. Note that ${L_m^\Sigma}$ is a quantity which is intrinsic to ${\Sigma}$.

2. Preliminaries

2.1. Higher order mean curvatures and the Newton’s tensors

Let us fix some notations first. Let ${M}$ be a hypersurface in an ${(n+1)}$-dimensional Riemannian manifold ${N}$. Let ${\eta}$ be a unit normal vector field (chosen to be outward pointing whenever this makes sense) of ${M}$ in ${N}$. The shape operator of ${M}$ in ${N}$ (w.r.t. ${\eta}$) is then defined by ${A=\overline \nabla \eta}$. Here ${\overline \nabla }$ is the connection on ${N}$. Let ${A(e_i)=A_i^j e_j}$, where ${\lbrace e_i\rbrace_{i=1}^n}$ is a local frame on ${M}$. In this and the following subsection, any index that appears more than once will be summed over ${\{1,\cdots, n\}}$.

Following [Grosjean] and [Reilly2], we define the ${k}$-th mean curvature of ${M}$ in ${N}$ (w.r.t. ${\eta}$) by

$\displaystyle H _k = \frac 1{k!} \varepsilon_{j_1\cdots j_k}^{i_1\cdots i_k}A_{i_1}^{j_1}\cdots A_{i_k}^{j_k}.$

Here ${\varepsilon_{i_1 \cdots i_k}^{j_1\cdots j_k}}$ is zero if ${i_m=i_l}$ or ${j_m=j_l}$ for some ${m\ne l}$, or if ${\lbrace i_1, \cdots, i_k\rbrace \ne \lbrace j_1, \cdots, j_k\rbrace}$ as sets, otherwise it is defined as the sign of the permutation ${(i_1, \cdots, i_k)\mapsto (j_1, \cdots, j_k)}$.

Equivalently, ${H_k}$ is given by ${\displaystyle H_k =\sum_{1\le i_1<\cdots< i_k\le n}\lambda_{i_1}\cdots \lambda_{i_k}}$ where ${\{\lambda_i\}_{i=1}^n}$ are the principal curvatures. We also define ${H_0=1}$. The normalized ${k}$-th mean curvature is defined as

$\displaystyle \sigma_k = \frac{H_k}{{{n-1}\choose k}}.$

For convenience, we define ${\sigma_{-1}= X\cdot \eta}$ where ${X}$ is the position vector of ${\Sigma}$. The reason for this choice is that the corresponding Minkowski formula holds (cf. [Grosjean], [Reilly2]).

We define the ${k}$-th Newton tensor ${T_k}$ of ${A}$ (w.r.t. ${\eta}$) as follows:

$\displaystyle {(T_k)}_j^{\,i}= \frac 1 {k!} \varepsilon^{i i_1 \ldots i_k}_{j j_1 \ldots j_k} A_{i_1}^{j_1}\cdots A_{i_k}^{j_k}.$

2.2. Gauss-Bonnet curvatures

In the last subsection, we introduce some notions related to the extrinsic geometry of ${M}$. In this subsections, we instead introduce some notions which are intrinsic in nature.

 Definition 4 Let ${(M,g)}$ be a Riemannian manifold, then the Gauss-Bonnet curvature ${L_k(g)}$ is defined as $\displaystyle L_{k}(g)=\frac{1}{2^{k}}\varepsilon_{j_1\cdots j_{2k}}^{i_1\cdots i_{2k}}{R_{i_1i_2}}^{j_1j_2}\cdots {R_{i_{2k-1}i_{2k}}}^{j_{2k-1}j_{2k}}.$ Here we use the convention that ${R_{ijij}}$ is the sectional curvature. We also define $\displaystyle P_{(k)}^{ijpq}= g^{pl}g^{qm}(P_{(k)})^{ij}_{lm}=\frac{1}{2^k}g^{pl}g^{qm} \varepsilon_{ i_1 \ldots i_{2k}lm}^{j_1 \ldots j_{2k}ij}{R_{j_1 j_2}}^{i_1i_2}\cdots {R_{j_{2k-1} j_{2k}}}^{i_{2k-1} i_{2k}}. \ \ \ \ \ (1)$ As mentioned before, we define (under an admissible coordinate system): $\displaystyle m_k(g)= c_k(n)\lim_{r\rightarrow \infty}\int_{S_r} P_{(k)}^{ijlm}\partial _m g_{jl}\nu_i dS \ \ \ \ \ (2)$ where ${c_k(n)= \frac{(n-2k)!}{2^{k-1}(n-1)!\omega_{n-1}}.}$

From now on, we assume ${(M,g)}$ is a hypersurface in ${\mathbb{R}^{n+1}}$ given by the graph of a smooth function ${f:\mathbb{R}^n\setminus \Omega\rightarrow \mathbb{R}}$, with the induced metric from ${\mathbb{R}^{n+1}}$, where ${\Omega}$ is a (possibly empty) bounded open subset in ${\mathbb{R}^n}$. We say that such ${M}$ is graphical. Let ${\delta}$ be the Euclidean metric on ${\mathbb{R}^n}$ and ${\nabla }$ be the standard connection on ${\mathbb{R}^n}$. Let ${\{x^i\}_{i=1}^n}$ be the standard coordinates of ${\mathbb{R}^n}$, ${f_i=\partial _i f}$, and ${f_{ij}= \partial _i\partial _j f=\nabla ^2_{ij}f}$. Then the metric ${g}$ on ${M}$ can be expressed as

$\displaystyle g= \delta+ df\otimes df\textrm{ and }g^{ij}= \delta_{ij}- \frac{f_i f_j}{1+|\nabla f|^2}.$

Following [Lam], we say that ${f}$ is asymptotically flat (of order ${\tau}$) if

$\displaystyle f(x)\rightarrow 0\textrm{ as }x\rightarrow \infty \textrm{\quad and \quad} |\nabla f|+|x||\nabla^2f (x)|+ |x|^2 |\nabla^3 f (x)| =O(|x|^{-\frac \tau 2})$

for some ${\tau> \frac {n-2}2}$. It is then easy to see that ${(M,g)}$ is an asymptotically flat manifold (of order ${\tau}$).

Assume ${f}$ is non-constant, and let ${\Sigma}$ be a regular level set of ${M}$ (i.e. the preimage of a regular value of ${f}$). We can naturally regard ${\Sigma}$ as a smooth ${(n-1)}$-dimensional hypersurface of ${\mathbb{R}^n}$. Then ${\eta= \frac{1}{w}(-\nabla f, 1)}$ is a unit normal of ${M}$ and the second fundamental form of ${M}$ (w.r.t. ${\eta}$) is given by ${A_{ij}= \langle \nabla _i \eta, \partial _j\rangle}$, where ${w= \sqrt{1+|\nabla f|^2}}$. It is easy to see that ${A_{ij}= \frac{-\nabla ^2 _{ij}f}{w}}$. Similarly, let ${\nu= -\frac{\nabla f}{|\nabla f|}}$ be the unit normal of ${\Sigma}$ in ${\mathbb{R}^n}$, then the second fundamental form of ${\Sigma}$ (w.r.t. ${\nu}$) is given by ${A^{\Sigma}_{ij}= \langle \nabla _i \nu, \partial _j\rangle= \frac{-\nabla ^2_{ij} f}{|\nabla f|}}$.

We will use the convention that the indices be raised and lowered by the metric ${g}$. For example, ${f^i = g^{ij}\partial _j f}$ and ${f_i^j= g^{jk}f_{ki}=g^{jk}\partial _k\partial _i f}$.

 Lemma 5 If ${M}$ is graphical, then $\displaystyle \partial _i\left((P_{(k+1)})^{ijlm}\partial _m g_{jl}\right)= \frac{1}{2}L_{k+1}.$

Proof: We compute

$\displaystyle \begin{array}{rl} (P_{(k+1)})^{ijlm}\partial _m g_{jl} =&\displaystyle (P_{(k+1)})^{ijlm}(f_{jm}f_l+f_{lm}f_j)\\ =&\displaystyle g^{lp}g^{pq}(P_{(k+1)})^{ij}_{pq}(f_{jm}f_l+f_{lm}f_j)\\ =&\displaystyle (P_{(k+1)})^{ij}_{pq}(f_j^qf^p+f^{pq}f_j)\\ =&\displaystyle (P_{(k+1)})^{ij}_{pq}f_j^qf^p\\ =&\displaystyle w(P_{(k+1)})^{ji}_{qp}A_j^qf^p. \end{array}$

The forth equation follows because ${(P_{(k+1)})^{ij}_{pq}}$ is anti-symmetric in ${p, q}$. By the Gauss equation,

$\displaystyle \begin{array}{rcl} (P_{(k+1)})^{ji}_{qp} &=& \displaystyle \frac{1}{2^{k+1}}\varepsilon_{j_1\cdots j_{2k}qp}^{i_1\cdots i_{2k} ji}{R_{i_1i_2}}^{j_1j_2}\cdots {R_{i_{2k-1}i_{2k}}}^{j_{2k-1}j_{2k}}\\ &=& \displaystyle \frac{1}{2^{k+1}}\varepsilon_{j_1\cdots j_{2k}qp}^{i_1\cdots i_{2k} ji}(A_{i_1}^{j_1}A_{i_2}^{j_2}- A_{i_1}^{j_2}A_{i_2}^{j_1})\cdots(A_{i_{2k-1}}^{j_{2k-1}}A_{i_{2k}}^{j_{2k}}- A_{i_{2k-1}}^{j_{2k}}A_{i_{2k}}^{j_{2k-1}})\\ &= &\displaystyle\frac{1}{2}\varepsilon_{j_1\cdots j_{2k}qp}^{i_1\cdots i_{2k}ji}A_{i_1}^{j_1}A_{i_2}^{j_2}\cdots A_{i_{2k-1}}^{j_{2k-1}}A_{i_{2k}}^{j_{2k}}. \end{array} \ \ \ \ \ (3)$

Therefore

$\displaystyle \begin{array}{rl} (P_{(k+1)})^{ijlm}\partial _m g_{jl} =&\displaystyle \frac{w}{2}\varepsilon^{i_1\cdots i_{2k}ji}_{j_1\cdots j_{2k}qp}A_{i_1}^{j_1}\cdots A_{i_{2k}}^{j_{2k}}A_j^qf^p\\ =&\displaystyle \frac{(2k+1)!}{2}w (T_{2k+1})_p^if^p. \end{array}$

We can without loss of generality assume that at a fixed point, ${f_n=|\nabla f|}$ and ${f_1=\cdots =f_{n-1}=0}$. Then at this point, we have

$\displaystyle \begin{array}{rl} f^p = g^{pq}f_q= \delta_{pn}g^{nn}f_n = \frac{1}{w^2}f_n\delta_{pn} =\frac{f_p}{w^2}. \end{array}$

Therefore

$\displaystyle (P_{(k+1)})^{ijlm}\partial _m g_{jl}=\frac{(2k+1)!}{2}\frac{1}{w} (T_{2k+1})_p^if_p.$

By a classical result of Reilly ([Reilly1] Proposition 4.1),

$\displaystyle \partial _i \left(\frac{1}{w} (T_{2k+1})_p^if_p\right)= (2k+2)H_{2k+2}.$

So we have

$\displaystyle \partial _i((P_{(k+1)})^{ijlm}\partial _m g_{jl})=\frac{(2k+2)!}{2}H_{2k+2}.$

On the other hand, by a computation similar to (3), we have

$\displaystyle \begin{array}{rl} L_{k+1}=& \displaystyle \frac{1}{2^{k+1}}\varepsilon_{j_1\cdots j_{2k+2}}^{i_1\cdots i_{2k+2}}{R_{i_1i_2}}^{j_1j_2}\cdots {R_{i_{2k+1}i_{2k+2}}}^{j_{2k+1}j_{2k+2}}\\ =&\displaystyle \varepsilon_{j_1\cdots j_{2k+2}}^{i_1\cdots i_{2k+2}}A_{i_1}^{j_1}\cdots A_{i_{2k+2}}^{j_{2k+2}}\\ =&\displaystyle (2k+2)!H_{2k+2}. \end{array}$

The result follows. $\Box$

3. Main results

The positive mass theorem follows immediately from Lemma 5:

 Theorem 6 Let ${(M^n,g)}$ be the graph of an asymptotically flat function ${f: \mathbb{R}^n\rightarrow \mathbb{R}}$. Suppose ${L_k}$ is integrable on ${(M,g)}$, then $\displaystyle m_k(g)= \frac{c_k(n)}{2}\int_M \frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g.$ In particular, if ${L_k\ge 0}$, then ${m_k(g)\ge 0}$.

Proof: By Lemma 5 and the divergence theorem,

$\displaystyle \begin{array}{rl} m_k=& \displaystyle \lim_{r\rightarrow \infty}c_k(n)\int_{S_r} (P_{(k)})^{ijlm}\partial_m g_{jl}\nu_i dS\\=& \displaystyle c_k(n)\int_{\mathbb{R}^n} \partial _i((P_{(k)})^{ijlm}\partial_m g_{jl})dV_\delta\\ =&\displaystyle \frac{c_k(n) }{2}\int_{\mathbb{R}^n} L_k dV_\delta\\ =&\displaystyle \frac{c_k(n)}{2} \int_{M} \frac{L_k}{\sqrt {1+|\nabla f|^2}}dV_g. \end{array}$

Here we have used the fact that ${dV_\delta= \frac{1}{\sqrt{1+|\nabla f|^2}}dV_g}$. $\Box$

The following proposition is crucial to prove the Penrose type inequalities:

 Proposition 7 On a regular level set ${\Sigma}$ of ${f}$, $\displaystyle (P_{(k+1)})^{ijlm}\partial _m g_{jl} \nu_i= \frac{(2k+1)!|\nabla f|^{2k+2}}{2(1+|\nabla f|^2)^{k+1}} H_{2k+1}^\Sigma.$ Here ${H_k^\Sigma}$ is the ${k}$-th mean curvature of ${\Sigma}$ (w.r.t. ${\nu}$), considered as a hypersurface in ${\mathbb{R}^n}$.

Proof: We can w.l.o.g. assume that ${\nu= -\frac{\nabla f}{|\nabla f|}}$. Choose normal coordinates around a fixed point ${p}$ in ${\mathbb{R}^n}$ such that at ${p}$, ${\partial _a}$ is tangential to ${\Sigma}$ for ${a=1,\cdots, n}$ and ${f_n=-|\nabla f|}$. Then the second fundamental form of ${M}$ w.r.t. ${\frac{1}{w}(-\nabla f, 1)}$ is ${A_{ij}= \frac{-\nabla ^2 _{ij}f}{w}}$ and ${A^{\Sigma}_{ab}= \langle \nabla _a\nu, \partial _b\rangle= \frac{-\nabla ^2_{ab} f}{|\nabla f|}}$ (w.r.t. ${\nu}$).

We will use the convention that the indices ${a, b, c}$ run from ${1}$ to ${n-1}$ and ${i, j, l, m, p, q}$ run from ${1}$ to ${n}$. We will sum over any index (except ${n}$) that appears more than once. For example, ${\displaystyle (P_{(k+1)})^{na}_{na}A_{aa}=\sum_{a=1}^{n-1}(P_{(k+1)})^{na}_{na}A_{aa}}$.

By our assumption, ${f_a=0}$, ${f_n=-|\nabla f|}$, and ${g^{nn}= w^{-2}}$. Also observe that by (4), ${(P_{(k+1)})^{ij}_{lm}=0}$ if ${(l,m)}$ is not ${(i,j)}$ or ${(j,i)}$. Write ${P}$ instead of ${P_{(k+1)}}$ for the moment, then

$\displaystyle \begin{array}{rcl} P^{ijlm}\partial _m g_{jl} \nu_i = P^{njlm}\partial _mg_{jl} \nu_n &=& \displaystyle P^{njlm}\partial _mg_{jl}= g^{lp}g^{mq}P^{nj}_{pq}\partial _mg_{jl}\\ &=& \displaystyle g^{nn}P^{nj}_{nj}\partial _jg_{jn}+ g^{nn}P^{nj}_{jn}\partial _ng_{jj}\\ &=& \displaystyle w^{-2}P^{na}_{na}\partial _ag_{an}+ w^{-2}P^{na}_{an}\partial _ng_{aa}\\ &=& \displaystyle w^{-2}P^{na}_{na}(\partial _ag_{an}-\partial _ng_{aa})\\ &=& \displaystyle w^{-2}P^{na}_{na}(f_{aa}f_n-f_{na}f_a)\\ &=& \displaystyle w^{-2}P^{na}_{na}f_{aa}f_n\\ &=& \displaystyle \frac{|\nabla f|}{w}P^{na}_{na}A_{aa}. \end{array} \ \ \ \ \ (4)$

We have used the fact that ${\partial _ag_{an}-\partial _ng_{aa}= f_{aa}f_n- f_{na}f_a=f_{aa}f_n}$. By the Gauss equation,

$\displaystyle \begin{array}{rl} (P_{(k+1)})^{ij}_{lm} =&\displaystyle \displaystyle \frac{1}{2^{k+1}}\varepsilon_{j_1\cdots j_{2k}lm}^{i_1\cdots i_{2k} ij}{R_{i_1i_2}}^{j_1j_2}\cdots {R_{i_{2k-1}i_{2k}}}^{j_{2k-1}j_{2k}}\\ =&\displaystyle \frac{1}{2^{k+1}}\varepsilon_{j_1\cdots j_{2k}lm}^{i_1\cdots i_{2k} ij}(A_{i_1}^{j_1}A_{i_2}^{j_2}- A_{i_1}^{j_2}A_{i_2}^{j_1})\cdots(A_{i_{2k-1}}^{j_{2k-1}}A_{i_{2k}}^{j_{2k}}- A_{i_{2k-1}}^{j_{2k}}A_{i_{2k}}^{j_{2k-1}})\\ = &\frac{1}{2}\varepsilon_{j_1\cdots j_{2k}lm}^{i_1\cdots i_{2k}ij}A_{i_1}^{j_1}A_{i_2}^{j_2}\cdots A_{i_{2k-1}}^{j_{2k-1}}A_{i_{2k}}^{j_{2k}}. \end{array}$

Therefore, using ${A_{ij}=\frac{A^{\Sigma}_{ij}|\nabla f|}{w}}$, in view of (4), we have

$\displaystyle \begin{array}{rl} (P_{(k+1)})^{ijlm}\partial _m g_{jl} \nu_i =&\displaystyle \frac{|\nabla f|}{2w}\varepsilon_{j_1j_2\cdots j_{2k}na}^{i_1i_2\cdots i_{2k}na} A_{i_1}^{j_1}\cdots A_{i_{2k}}^{j_{2k}}A_{aa}\\ =&\displaystyle \frac{|\nabla f|}{2w}\varepsilon_{b_1b_2\cdots b_{2k}na}^{a_1a_2\cdots a_{2k}na} A_{a_1}^{b_1}\cdots A_{a_{2k}}^{b_{2k}}A_{aa}\\ =&\displaystyle \frac{|\nabla f|}{2w}\varepsilon_{b_1b_2\cdots b_{2k}a}^{a_1a_2\cdots a_{2k}a} A_{a_1}^{b_1}\cdots A_{a_{2k}}^{b_{2k}}A_{aa}\\ =&\displaystyle \frac{|\nabla f| ^{2k+2}}{2w^{2k+2}}\varepsilon_{b_1b_2\cdots b_{2k}a}^{a_1a_2\cdots a_{2k}a} (A^\Sigma)_{a_1}^{b_1}\cdots (A^\Sigma)_{a_{2k}}^{b_{2k}}(A^\Sigma)_{aa}. \end{array}$

By diagonalizing ${A^{\Sigma}}$ (see e.g. [Grosjean]), we can deduce that

$\displaystyle \varepsilon_{b_1b_2\cdots b_{2k}a}^{a_1a_2\cdots a_{2k}a} (A^\Sigma)_{a_1}^{b_1}\cdots (A^\Sigma)_{a_{2k}}^{b_{2k}}(A^\Sigma)_{aa}=(2k+1)!H_{2k+1}^{\Sigma}.$

From this the result follows. $\Box$

 Proposition 8 Let ${\Omega}$ be a bounded open set in ${\mathbb{R}^n}$ and ${\Sigma= \partial \Omega}$. If ${f:\mathbb{R}^n\setminus \Omega\rightarrow \mathbb{R}}$ is a smooth asymptotically flat function such that each connected component of ${\Sigma}$ is in a level set of ${f}$ and ${|\nabla f(x)|\rightarrow \infty}$ as ${x\rightarrow \Sigma}$. Then $\displaystyle m_k(g)=\frac{c_k(n)}{2} \int_M\frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g+ \frac{(2k-1)!c_k(n)}{2}\int_\Sigma H_{2k-1}^\Sigma dS.$

Proof: By the divergence theorem on ${\mathbb{R}^n}$ and Lemma 5, we have

$\displaystyle \begin{array}{rl} m_k(g)=&\displaystyle \lim_{r\rightarrow \infty}c_k(n)\int_{S_r} P_{(k)}^{ijlm}\partial _m g_{jl}\nu_idS\\ =&\displaystyle \frac{c_k(n)}{2} \int_{\mathbb{R}^n \setminus \Omega}L_kdV_{\delta}+ c_k(n) \int_{\Sigma} P_{(k)}^{ijlm}\partial _m g_{jk}\nu_idS\\\displaystyle =&\displaystyle \frac{c_k(n)}{2} \int_M\frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g + c_k(n)\int_{\Sigma} P_{(k)}^{ijlm}\partial _m g_{jk}\nu_idS. \end{array}$

Here, in the second line, ${\nu}$ is the unit outward normal of ${\Sigma}$ in ${\mathbb{R}^n}$ and ${\Omega\subset \mathbb{R}^n}$ is the region bounded by ${\Sigma}$. By Proposition 7, since ${\frac{|\nabla f|^{2k}}{(1+|\nabla f|^2)^{k}}\rightarrow 1}$ as ${x\rightarrow \Sigma}$, we have

$\displaystyle \int_{\Sigma} P_{(k)}^{ijlm}\partial _m g_{jk}\nu_idS=\frac{(2k-1)!}{2}\int_\Sigma H_{2k-1}^\Sigma dS.$

Therefore

$\displaystyle m_k(g)=\frac{c_k(n)}{2} \int_M\frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g+ \frac{(2k-1)!c_k(n)}{2}\int_\Sigma H_{2k-1}^\Sigma dS.$

$\Box$

We say a hypersurface ${\Sigma}$ in ${\mathbb{R}^n}$ is ${k}$-convex if ${\sigma_m\ge 0}$ for all ${m \le k}$. ${\Sigma}$ is ${n}$-convex if and only if ${\Sigma}$ is convex in the usual sense. Also, if ${\sigma_k>0}$, then ${\Sigma}$ is actually (strictly) ${k}$-convex (cf. e.g. [BC]).

The following beautiful inequalities are known as the quermassintegral inequalities or the Alexandrov-Fenchel inequalities, which generalize the well-known isoperimetric inequality.

 Theorem 9 ([Guan-Li]) Suppose ${\Sigma}$ is a star-shaped ${k}$-convex hypersurface in ${\mathbb{R}^n}$, then we have $\displaystyle \left(\frac{\int_\Sigma \sigma_{-1}dS}{\omega_{n-1}}\right)^{\frac{1}{n}}\le \left(\frac{\int_\Sigma \sigma_0dS}{\omega_{n-1}}\right)^{\frac{1}{n-1}}\le \cdots \le \left(\frac{\int_\Sigma \sigma_k dS}{\omega_{n-1}}\right)^{\frac{1}{n-k-1}}.$ Recall that ${\omega_{n-1}}$ is the volume of the unit sphere in ${\mathbb{R}^n}$. The equality holds if and only if ${\Sigma}$ is a sphere.

We are now ready to prove the Penrose type inequalities.

 Theorem 10 Let ${\Omega}$ be a bounded open set in ${\mathbb{R}^n}$ and ${\Sigma= \partial \Omega}$ (not necessarily connected). If ${f:\mathbb{R}^n\setminus \Omega\rightarrow \mathbb{R}}$ is a smooth asymptotically flat function such that each connected component of ${\Sigma}$ is in a level set of ${f}$ and ${|\nabla f(x)|\rightarrow \infty}$ as ${x\rightarrow \Sigma}$. Furthermore, assume that each connected component of ${\Sigma}$ is ${(2k-1)}$-convex and star-shaped, then $\displaystyle \begin{array}{rl} m_k(g)-\frac{c_k(n)}{2} \int_M\frac{L_k}{\sqrt{1+|\nabla f|^2}}dV_g =&\displaystyle \frac{1}{2^k}\frac{\int_\Sigma \sigma^\Sigma_{2k-1}dS}{\omega_{n-1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{2k-2}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{2k-3}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+2}}\\ \ge&\displaystyle \cdots\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_2dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-3}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_1dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_0dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{-1}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n}}. \end{array}$ Note that ${\int_\Sigma \sigma_0^\Sigma dS= \mathrm{Area}(\Sigma)}$ and ${\int_\Sigma\sigma_{-1}^\Sigma dS= n \mathrm{Vol}(\Omega)}$. Also, if ${m}$ is even, then ${\int_\Sigma \sigma_m dS}$ is an intrinsic quantity on ${\Sigma}$ (i.e. it depends on the induced metric on ${\Sigma}$ only). Indeed, by a direct computation using the Gauss equation (similar to (3)), $\displaystyle \sigma^\Sigma_{2m}=\frac{H^\Sigma_{2m}}{{{n-1}\choose {2m}}}= \frac{(n-1-2m)!}{(n-1)!}L_m^\Sigma= \frac{1}{P(n-1, 2m)}L_m^\Sigma$ where ${L_m^\Sigma}$ is the ${m}$-th Gauss-Bonnet curvature on ${\Sigma}$ and ${P(n-1, l)= \frac{(n-1)!}{(n-1-l)!}}$. In particular, if ${L_k(g)\ge 0}$, then $\displaystyle \begin{array}{rl} m_k (g) \ge\frac{1}{2^k}\left(\frac{\int_\Sigma L_{k-1}^\Sigma dS}{P(n-1, 2k-2)\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+1}} \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma L_{k-2}^\Sigma dS}{P(n-1, 2k-4)\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+3}}\\ \ge&\displaystyle \cdots\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma L_1^\Sigma dS}{P(n-1, 2)\omega_{n-1}}\right)^{\frac{n-2k}{n-3}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\mathrm{Area}(\Sigma)}{\omega_{n-1}}\right)^{\frac{n-2k}{n-1}}. \end{array}$

Proof: By Theorem 9,

$\displaystyle \begin{array}{rl} \frac{(2k-1)!c_k (n)}{2}\int_\Sigma H^{\Sigma}_{2k-1}dS =&\displaystyle \frac{(2k-1)!(n-2k)!}{2^k(n-1)!\omega_{n-1}}{{n-1}\choose {2k-1}}\int_\Sigma \sigma^\Sigma_{2k-1}dS\\ =&\displaystyle \frac{1}{2^k}\frac{\int_\Sigma \sigma^\Sigma_{2k-1}dS}{\omega_{n-1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{2k-2}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{2k-3}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2k+2}}\\ \ge&\displaystyle \cdots\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_2dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-3}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_1dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-2}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_0dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n-1}}\\ \ge&\displaystyle \frac{1}{2^k}\left(\frac{\int_\Sigma \sigma^\Sigma_{-1}dS}{\omega_{n-1}}\right)^{\frac{n-2k}{n}}. \end{array}$

The result then immediately follows from Proposition 8. $\Box$

Question: What can we say about the rigidity case of the positive mass theorem and the Penrose type inequalities? i.e. Can we deduce that $(M,g)$ is a hyperplane if the equality case of the positive mass theorem holds? Can we deduce that $(M,g)$ is the Schwarzschild manifold if the equality case of the Penrose type inequalities holds?

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