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.
Let us first recall some definitions. We will adopt the following definition of an asymptotically flat manifold:
Definition 1 A 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:
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.
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. )
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.
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 .
2.1. 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 .
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
We define the -th Newton tensor of (w.r.t. ) as follows:
2.2. Gauss-Bonnet curvatures
Definition 4 Let be a Riemannian manifold, then the Gauss-Bonnet curvature is defined as
Here we use the convention that is the sectional curvature. We also define
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 .
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
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 6 Let be the graph of an asymptotically flat function . Suppose is integrable on , then
In particular, if , then .
Here we have used the fact that .
The following proposition is crucial to prove the Penrose type inequalities:
Proposition 7 On 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 8 Let 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
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 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, then
Note 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?