Abstract
The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.
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Acknowledgements
The author would like to thank Fernando Schwartz and Hubert Bray for helpful discussions and suggestions. He also would like to note that Schwartz’s paper [33] was the original inspiration for this work. Much of the work for this paper was carried out while the author was affiliated with the University of Pennsylvania. The author would also like to thank the referee for a number of insightful comments and suggestions that improved the paper.
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Appendix: The Poincaré–Faber–Szegö capacity–volume inequality
Appendix: The Poincaré–Faber–Szegö capacity–volume inequality
For reference, we include a proof of the capacity–volume inequality used in the Proof of Theorem 10. The following is based entirely on the dimension three case of [25]. In this Appendix, all geometric quantities are with respect to the Euclidean metric.
Theorem 11
(Poincaré–Faber–Szegö) Let \(\varOmega \) be a bounded open set in \(\mathbb {R}^n\) with smooth boundary such that \(\mathbb {R}^n {\setminus }\varOmega \) is connected. Then
where \({{\mathrm{cap}}}(\varOmega )\) and V are the capacity (see (18)) and volume of \(\varOmega \), respectively.
Proof
Let \(0 \le \varphi <1\) be the unique function on \(\mathbb {R}^n {\setminus }\varOmega \) that vanishes on \(\partial \varOmega \), is harmonic on \(\mathbb {R}^n {\setminus }\varOmega \), and approaches 1 at infinity. Then
For \(t \in [0,1)\), let \(\varSigma _t\) be the level set \(\varphi ^{-1}(t)\). Note that \(\varSigma _t\) is smooth for almost every t, and \(|\nabla \varphi |\ne 0\) on \(\varSigma _t\) for such t. By the co-area formula,
where \(\mathrm{d}A_t\) is the area form on \(\varSigma _t\). By the Schwarz inequality, for almost all \(t \in [0,1)\),
where \(|\varSigma _t|\) is the area of \(\varSigma _t\). Combining (23) and (24) produces:
Let V(t) be the volume in \(\mathbb {R}^n\) of the region bounded by \(\varSigma _t\), so again by the co-area formula:
and therefore
for almost all \(t \in [0,1)\). Combining the above gives
where we have used the isoperimetric inequality on the second line. Let R(t) be the radius of the sphere that has volume equal to V(t), i.e., \(V(t)=\beta _n R(t)^n\). Then \(V'(t) = n\beta _n R(t)^{n-1} R'(t)\) for almost all \(t \in [0,1)\), so
having used the fact \(n\beta _n = \omega _{n-1}\).
Now, let \({\tilde{\varOmega }}\) be the open ball about the origin with the same volume as \(\varOmega \). Let \({\tilde{\varSigma }}_t\) be the sphere about the origin of radius R(t), with area form \(d {\tilde{A}}_t\). Note \({\tilde{\varSigma }}_0 = \partial {\tilde{\varOmega }}\). Let \({\tilde{\varphi }} : \mathbb {R}^n \setminus {\tilde{\varOmega }} \rightarrow \mathbb {R}\) be the function that equals t on \({\tilde{\varSigma }}_t\). Note that \({\tilde{\varphi }}\) is continuous, since \(R^{-1}\) is continuous (which holds because V, and hence R, is strictly increasing), \({\tilde{\varphi }} = 0\) on \(\partial {\tilde{\varOmega }}\), and \({\tilde{\varphi }} \rightarrow 1\) at infinity. We continue inequality (25), using the fact that \(\omega _{n-1} R(t)^{n-1} = \int _{{\tilde{\varSigma }}_t} d{\tilde{A}}_t\) and the observation that \(|\nabla {\tilde{\varphi }}| = \frac{1}{R'(t)}\) on \({\tilde{\varSigma }}_t\) for almost all \(t \in [0,1)\):
Thus, \({\tilde{\varphi }}\) is in the Sobolev space \(W^{1,2}_{\text {loc}}(\mathbb {R}^n {\setminus } {\tilde{\varOmega }})\). Note also that \({\tilde{\varphi }}\) is smooth near \(\partial {\tilde{\varOmega }}\) and is smooth outside a compact set. (These facts follow from \(|\nabla \varphi | \ne 0\) near \(\partial \varOmega \), as \(\partial _\nu (\varphi )>0\) on \(\partial \varOmega \) by the maximum principle, as well as \(|\nabla \varphi | \ne 0\) near infinity, as \(\varphi \) admits an expansion \(\varphi (x) = 1 + \frac{c}{|x|^{n-2}} + O(|x|^{1-n})\) into spherical harmonics near infinity.) If \({\tilde{\varphi }}\) is not smooth on \(\mathbb {R}^n {\setminus } {\tilde{\varOmega }}\), proceed as follows. Let \(U \subset \mathbb {R}^n {\setminus } {\tilde{\varOmega }}\) be a smooth open set whose closure is compact and disjoint from \(\partial {\tilde{\varOmega }}\), where U contains all points where \({\tilde{\varphi }}\) is not smooth. Using the density of smooth functions in \(W^{1,2}(U)\), given any \(\epsilon >0\), there exists a smooth function that agrees with \({\tilde{\varphi }}\) near \(\partial {\tilde{\varOmega }}\) and outside a compact set, such that
But is a valid competitor in the definition of the capacity of \({\tilde{\varOmega }}\), so that
Combining, we have
It is straightforward to check that equality holds in (22) for round balls. Thus, since \(\varOmega \) has the same volume as \({\tilde{\varOmega }}\), and \(\epsilon \) can be made arbitrarily small, the proof is complete. \(\square \)
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Jauregui, J.L. Penrose-type inequalities with a Euclidean background. Ann Glob Anal Geom 54, 509–527 (2018). https://doi.org/10.1007/s10455-018-9611-7
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DOI: https://doi.org/10.1007/s10455-018-9611-7