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Penrose-type inequalities with a Euclidean background

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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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Notes

  1. Other names for this type of estimate are the Aleksandrov–Fenchel inequality and the isoperimetric inequality for quermassintegrals [27]. Minkowski gave the first proof, for convex regions in \(\mathbb {R}^3\) [25].

  2. These results are essentially due to Federer and Fleming [12,13,14]. See also the appendix of [30].

References

  1. Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 2(122), 997–1006 (1961)

    Article  MathSciNet  Google Scholar 

  2. Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)

    Article  MathSciNet  Google Scholar 

  3. Bray, H.: Negative point mass singularities in general relativity. Global problems in mathematical relativity. (Isaac Newton Institute, University of Cambridge, August 30, 2005). http://www.newton.ac.uk/webseminars/pg+ws/2005/gmr/0830/bray/. Accessed 1 Dec 2017

  4. Bray, H.: On the positive mass, Penrose, and ZAS inequalities in general dimension. In: Bray, H.L., Minicozzi W.P., II (eds.) Surveys in Geometric Analysis and Relativity. Adv. Lect. Math. (ALM), vol. 20, pp. 1–27. Int. Press, Somerville (2011)

  5. Bray, H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)

    Article  MathSciNet  Google Scholar 

  6. Bray, H., Iga, K.: Superharmonic functions in \(\mathbf{R}^n\) and the Penrose inequality in general relativity. Commun. Anal. Geom. 10(5), 999–1016 (2002)

    Article  Google Scholar 

  7. Bray, H., Jauregui, J.: A geometric theory of zero area singularities in general relativity. Asian J. Math. 17(3), 525–559 (2013)

    Article  MathSciNet  Google Scholar 

  8. Bray, H., Khuri, M.: P.D.E’.s which imply the Penrose conjecture. Asian J. Math. 15(4), 557–610 (2011)

    Article  MathSciNet  Google Scholar 

  9. Bray, H., Lee, D.A.: On the Riemannian Penrose inequality in dimensions less than eight. Duke Math. J. 148(1), 81–106 (2009)

    Article  MathSciNet  Google Scholar 

  10. Chruściel, P.: Boundary conditions at spatial infinity from a Hamiltonian point of view. In: Bergmann, P.G., De Sabbata, V. (eds.) Topological Properties and Global Structure of Space–Time (Erice, 1985). NATO Adv. Sci. Inst. Ser. B Phys., vol. 138, pp. 49–59. Plenum, New York (1986)

  11. de Lima, L., Girão, F.: A rigidity result for the graph case of the Penrose inequality. arXiv:1205.1132 (2012)

  12. Federer, H.: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing at chains modulo two with arbitrary codimension. Bull. Am. Math. Soc. 76, 767–771 (1970)

    Article  Google Scholar 

  13. Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 2(72), 458–520 (1960)

    Article  MathSciNet  Google Scholar 

  14. Fleming, W.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo 2(11), 69–90 (1962)

    Article  MathSciNet  Google Scholar 

  15. Freire, A., Schwartz, F.: Mass-capacity inequalities for conformally at manifolds with boundary. Commun. Part. Differ. Equ. 39(1), 98–119 (2014)

    Article  Google Scholar 

  16. Gibbons, G.W.: Collapsing shells and the isoperimetric inequality for black holes. Class. Quantum Gravity 14(10), 2905–2915 (1997)

    Article  MathSciNet  Google Scholar 

  17. Guan, P., Li, J.: The quermassintegral inequalities for \(k\)-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)

    Article  MathSciNet  Google Scholar 

  18. Huang, L.-H., Wu, D.: The equality case of the Penrose inequality for asymptotically at graphs. Trans. Am. Math. Soc. 367(1), 31–47 (2015)

    Article  Google Scholar 

  19. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)

    Article  MathSciNet  Google Scholar 

  20. Jauregui, J.: Invariants of the harmonic conformal class of an asymptotically at manifold. Commun. Anal. Geom. 20(1), 163–201 (2012)

    Article  Google Scholar 

  21. Jauregui, J.: Mass estimates, conformal techniques, and singularities in general relativity. Ph.D. Thesis, Duke University (2010)

  22. Lam, M.-K.G.: The graphs cases of the Riemannian positive mass and Penrose inequalities in all dimensions. arXiv:1010.4256 (2010)

  23. Lam, M.-K.G.: The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. Ph.D. Thesis, Duke University (2011)

  24. Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)

    Article  MathSciNet  Google Scholar 

  25. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  26. Robbins, N.: Zero area singularities in general relativity and inverse mean curvature flow. Class. Quantum Gravity 27(2), 025011 (2010)

    Article  MathSciNet  Google Scholar 

  27. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  28. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)

    Article  MathSciNet  Google Scholar 

  29. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Giaquinta, M. (ed.) Topics in Calculus of Variations (Montecatini Terme, 1987): Lecture Notes in Math., vol. 1365, pp. 120–154. Springer, Berlin (1989)

    Chapter  Google Scholar 

  30. Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)

    Article  MathSciNet  Google Scholar 

  31. Schoen, R., Yau, S.-T.: Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity. Proc. Natl. Acad. Sci. USA 76(3), 1024–1025 (1979)

    Article  MathSciNet  Google Scholar 

  32. Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities. arXiv:1704.05490 (2017)

  33. Schwartz, F.: A volumetric Penrose inequality for conformally at manifolds. Ann. Henri Poincaré 12, 67–76 (2011)

    Article  MathSciNet  Google Scholar 

  34. Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62(1), 79–125 (2002)

    Article  MathSciNet  Google Scholar 

  35. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981)

    Article  MathSciNet  Google Scholar 

Download references

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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Correspondence to Jeffrey L. Jauregui.

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

$$\begin{aligned} {{\mathrm{cap}}}(\varOmega ) \ge \left( \frac{V}{\beta _n}\right) ^{\frac{n-2}{n}}, \end{aligned}$$
(22)

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

$$\begin{aligned} (n-2)\omega _{n-1} {{\mathrm{cap}}}(\varOmega )&= \int _{\mathbb {R}^n {\setminus }\varOmega } |\nabla \varphi |^2 \mathrm{d}V. \end{aligned}$$

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,

$$\begin{aligned} \int _{\mathbb {R}^n {\setminus }\varOmega } |\nabla \varphi |^2 \mathrm{d}V = \int _0^1 \int _{\varSigma _t} |\nabla \varphi |^2 \frac{1}{|\nabla \varphi |} \mathrm{d}A_t \mathrm{d}t, \end{aligned}$$
(23)

where \(\mathrm{d}A_t\) is the area form on \(\varSigma _t\). By the Schwarz inequality, for almost all \(t \in [0,1)\),

$$\begin{aligned} |\varSigma _t|^2 \le \left( \int _{\varSigma _t} |\nabla \varphi | \mathrm{d}A_ t\right) \left( \int _{\varSigma _t} \frac{1}{ |\nabla \varphi |} \mathrm{d}A_ t\right) , \end{aligned}$$
(24)

where \(|\varSigma _t|\) is the area of \(\varSigma _t\). Combining (23) and (24) produces:

$$\begin{aligned} \int _{\mathbb {R}^n {\setminus }\varOmega } |\nabla \varphi |^2 \mathrm{d}V \ge \int _0^1 \frac{|\varSigma _t|^2}{\int _{\varSigma _t} \frac{1}{|\nabla \varphi |} \mathrm{d}A_t} \mathrm{d}t. \end{aligned}$$

Let V(t) be the volume in \(\mathbb {R}^n\) of the region bounded by \(\varSigma _t\), so again by the co-area formula:

$$\begin{aligned} V(t) = {{\mathrm{vol}}}(\varOmega ) + \int _0^t \int _{\varSigma _s} \frac{1}{|\nabla \varphi |} \mathrm{d}A_s \mathrm{d}s, \end{aligned}$$

and therefore

$$\begin{aligned} V'(t) =\int _{\varSigma _t} \frac{1}{|\nabla \varphi |} \mathrm{d}A_t \end{aligned}$$

for almost all \(t \in [0,1)\). Combining the above gives

$$\begin{aligned} (n-2)\omega _{n-1} {{\mathrm{cap}}}(\varOmega )&\ge \int _0^1 \frac{|\varSigma _t|^2}{V'(t)} \mathrm{d}t\\&\ge \int _0^1 \frac{(\omega _{n-1})^2 \left( \frac{V(t)}{\beta _n}\right) ^{\frac{2(n-1)}{n}}}{V'(t)} \mathrm{d}t, \end{aligned}$$

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

$$\begin{aligned} (n-2)\omega _{n-1} {{\mathrm{cap}}}(\varOmega ) \ge \int _0^1 \frac{\omega _{n-1}R(t)^{n-1}}{R'(t)} \mathrm{d}t, \end{aligned}$$
(25)

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 \(t \in [0,1)\):

$$\begin{aligned} (n-2)\omega _{n-1} {{\mathrm{cap}}}(\varOmega )&\ge \int _0^1 \int _{{\tilde{\varSigma }}_t}|\nabla {\tilde{\varphi }}| d{\tilde{A}}_t \mathrm{d}t&(\text {by}~(25) )\\&= \int _{\mathbb {R}^n {\setminus }{\tilde{\varOmega }}} |\nabla {\tilde{\varphi }}|^2 \mathrm{d}V&(\text {co-area formula}). \end{aligned}$$

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

$$\begin{aligned} {{\mathrm{cap}}}(\varOmega ) \ge {{\mathrm{cap}}}({\tilde{\varOmega }}) - \frac{\epsilon }{(n-2)\omega _{n-1}}. \end{aligned}$$

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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