In a recent preprint [GWW1], Ge, Wang and Wu proposed a family of new masses () for an asymptotically flat manifold and proved a positive mass theorem and some Penrose type inequalities for graphs, at least when . They remarked that their results can be generalized for all . While highly plausible, it does not seem obvious that their calculation can be extended to the case where (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 1A complete -dimensional Riemannian manifold is said to be asymptotically flat of order (with one end) if there is a compact subset such that is diffeomorphic to for some and in the standard coordinates in , the metric satisfies:with

for some constant . Here denotes the ordinary partial derivative and is the Euclidean distance from the origin.

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

If we further assume that and the scalar curvature is integrable in , then in general relativity, the ADM mass ([ADM]) is defined by (under an admissible coordinate system):

Here is the coordinate sphere with Euclidean radius , is the outward unit normal of w.r.t. and is the volume of the standard unit sphere in . (In this note, unless otherwise stated, we will sum over any index (except ) 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 , which they called the Gauss-Bonnet-Chern mass, defined by

Here

is a -tensor whose definition will be given in Subsection 2. We remark that when , then

and is just the ADM mass of . It was proved in [GWW1] that if , then 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 . They remarked that their results can be generalized for all . While this is very reasonable, it is not obvious to us that their calculation can be extended to include the case where (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 be the graph of an asymptotically flat function . Suppose is integrable on , then

In particular, if , then .

The Gauss-Bonnet curvature 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 be a bounded open set in and (not necessarily connected). If is a smooth asymptotically flat function such that each connected component of is in a level set of and as . Furthermore, assume that each connected component of is -convex and star-shaped, and , then

where . Note that is a quantity which is intrinsic to .

**2. Preliminaries **

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

Let us fix some notations first. Let be a hypersurface in an -dimensional Riemannian manifold . Let be a unit normal vector field (chosen to be outward pointing whenever this makes sense) of in . The shape operator of in (w.r.t. ) is then defined by . Here is the connection on . Let , where is a local frame on . In this and the following subsection, any index that appears more than once will be summed over .

Following [Grosjean] and [Reilly2], we define the -th mean curvature of in (w.r.t. ) by

Here is zero if or for some , or if as sets, otherwise it is defined as the sign of the permutation .

Equivalently, is given by where are the principal curvatures. We also define . The normalized -th mean curvature is defined as

For convenience, we define where is the position vector of . The reason for this choice is that the corresponding Minkowski formula holds (cf. [Grosjean], [Reilly2]).

We define the -th Newton tensor of (w.r.t. ) as follows:

** 2.2. Gauss-Bonnet curvatures **

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

Definition 4Let be a Riemannian manifold, then the Gauss-Bonnet curvature is defined asHere we use the convention that is the sectional curvature. We also define

As mentioned before, we define (under an admissible coordinate system):

From now on, we assume is a hypersurface in given by the graph of a smooth function , with the induced metric from , where is a (possibly empty) bounded open subset in . We say that such is graphical. Let be the Euclidean metric on and be the standard connection on . Let be the standard coordinates of , , and . Then the metric on can be expressed as

Following [Lam], we say that is asymptotically flat (of order ) if

for some . It is then easy to see that is an asymptotically flat manifold (of order ).

Assume is non-constant, and let be a regular level set of (i.e. the preimage of a regular value of ). We can naturally regard as a smooth -dimensional hypersurface of . Then is a unit normal of and the second fundamental form of (w.r.t. ) is given by , where . It is easy to see that . Similarly, let be the unit normal of in , then the second fundamental form of (w.r.t. ) is given by .

We will use the convention that the indices be raised and lowered by the metric . For example, and .

Lemma 5If is graphical, then

*Proof:* We compute

The forth equation follows because is anti-symmetric in . By the Gauss equation,

We can without loss of generality assume that at a fixed point, and . Then at this point, we have

Therefore

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

So we have

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

The result follows.

**3. Main results **

The positive mass theorem follows immediately from Lemma 5:

Theorem 6Let be the graph of an asymptotically flat function . Suppose is integrable on , then

In particular, if , then .

*Proof:* By Lemma 5 and the divergence theorem,

Here we have used the fact that .

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

Proposition 7On a regular level set of ,

Here is the -th mean curvature of (w.r.t. ), considered as a hypersurface in .

*Proof:* We can w.l.o.g. assume that . Choose normal coordinates around a fixed point in such that at , is tangential to for and . Then the second fundamental form of w.r.t. is and (w.r.t. ).

We will use the convention that the indices run from to and run from to . We will sum over any index (except ) that appears more than once. For example, .

By our assumption, , , and . Also observe that by (4), if is not or . Write instead of for the moment, then

We have used the fact that . By the Gauss equation,

Therefore, using , in view of (4), we have

By diagonalizing (see e.g. [Grosjean]), we can deduce that

From this the result follows.

Proposition 8Let be a bounded open set in and . If is a smooth asymptotically flat function such that each connected component of is in a level set of and as . Then

*Proof:* By the divergence theorem on and Lemma 5, we have

Here, in the second line, is the unit outward normal of in and is the region bounded by . By Proposition 7, since as , we have

Therefore

We say a hypersurface in is -convex if for all . is -convex if and only if is convex in the usual sense. Also, if , then is actually (strictly) -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 is a star-shaped -convex hypersurface in , then we have

Recall that is the volume of the unit sphere in . The equality holds if and only if is a sphere.

We are now ready to prove the Penrose type inequalities.

Theorem 10Let be a bounded open set in and (not necessarily connected). If is a smooth asymptotically flat function such that each connected component of is in a level set of and as . Furthermore, assume that each connected component of is -convex and star-shaped, thenNote that and . Also, if is even, then is an intrinsic quantity on (i.e. it depends on the induced metric on only). Indeed, by a direct computation using the Gauss equation (similar to (3)),

where is the -th Gauss-Bonnet curvature on and . In particular, if , then

*Proof:* By Theorem 9,

The result then immediately follows from Proposition 8.

*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 is a hyperplane if the equality case of the positive mass theorem holds? Can we deduce that is the Schwarzschild manifold if the equality case of the Penrose type inequalities holds?

Can you recommend me some books that can help me to understand this topic (or also good books on Geometry)? I have recently discovered Besse’s Einstein Manifolds but maybe you can suggest me something more recent or simply with a better tractation. Best Regards.