A geometric capacitary inequality for sub-static manifolds with harmonic potentials

Virginia Agostiniani; Lorenzo Mazzieri; Francesca Oronzio; Virginia Agostiniani; Lorenzo Mazzieri; Francesca Oronzio

doi:10.3934/mine.2022013

Mathematics in Engineering

2022, Volume 4, Issue 2: 1-40. doi: 10.3934/mine.2022013

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A geometric capacitary inequality for sub-static manifolds with harmonic potentials

1.
Dipartimento di Fisica, Università di Trento, Via Sommarive 14, 38123 Povo (TN), Italy
2.
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo (TN), Italy
3.
Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Cintia, Monte S. Angelo 80126 Napoli, Italy

Received: 09 January 2021 Accepted: 19 May 2021 Published: 02 July 2021

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $ \{F_{\beta}\} $ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $ \beta = \frac{n-2}{n-1} $ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.
- sub–static metrics,
- splitting theorem,
- Schwarzschild solution,
- overdetermined boundary value problems
Citation: Virginia Agostiniani, Lorenzo Mazzieri, Francesca Oronzio. A geometric capacitary inequality for sub-static manifolds with harmonic potentials[J]. Mathematics in Engineering, 2022, 4(2): 1-40. doi: 10.3934/mine.2022013

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Abstract

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $ \{F_{\beta}\} $ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $ \beta = \frac{n-2}{n-1} $ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.

References

[1]	V. Agostiniani, M. Fogagnolo, L. Mazzieri, Minkowski inequalities via nonlinear potential theory, 2020, arXiv: 1906.00322.
[2]	V. Agostiniani, L. Mazzieri, On the geometry of the level sets of bounded static potentials, Commun. Math. Phys., 355 (2017), 261–301. doi: 10.1007/s00220-017-2922-x
[3]	V. Agostiniani, L. Mazzieri, Monotonicity formulas in potential theory, Calc. Var., 59 (2020), 1–32. doi: 10.1007/s00526-019-1640-y
[4]	R. Bartnik, The mass of an asymptotically flat manifold, Commun. Pure Appl. Math., 39 (1986), 661–693. doi: 10.1002/cpa.3160390505
[5]	L. Benatti, M. Fogagnolo, L. Mazzieri, Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature, 2021, arXiv: 2101.06063v4.
[6]	A. L. Besse, Einstein manifolds, Berlin: Springer-Verlag, 2008.
[7]	S. Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. math. IHÉS, 117 (2013), 247–269. doi: 10.1007/s10240-012-0047-5
[8]	P. Chruściel, Boundary conditions at spatial infinity from a Hamiltonian point of view, In: Topological properties and global structure of space-time, New York: Plenum Press, 1986, 49–59.
[9]	A. Dirmeier, Growth conditions for conformal transformations preserving Riemannian completeness, 2012, arXiv: 1202.5437.
[10]	S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, 3 Eds., Berlin: Springer-Verlag, 2004.
[11]	N. Garofalo, F. H. Lin, Monotonicity properties of variational integrals, ${A}_{p}$-weights, and unique continuation, Indiana U. Math. J., 35 (1986), 245–268. doi: 10.1512/iumj.1986.35.35015
[12]	D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer-Verlag, 2001.
[13]	F. Girão, D. Rodrigues, Weighted geometric inequalities for hypersurfaces in sub-static manifolds, B. Lond. Math. Soc., 52 (2020), 121–136. doi: 10.1112/blms.12312
[14]	A. Grigor'yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, Volume 47, American Mathematical Society, 2009.
[15]	R. Hardt, H. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili, Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359–373.
[16]	R. Hardt, L. Simon, Nodal sets for solutions of elliptic equations, J. Differ. Geom., 30 (1989), 505–522.
[17]	S. Hirsch, P. Miao, A positive mass theorem for manifolds with boundary, Pac. J. Math., 306 (2020), 185–201. doi: 10.2140/pjm.2020.306.185
[18]	A. Kasue, Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Jpn., 35 (1983), 117–131.
[19]	J. Li, C. Xia, An integral formula and its applications on sub-static manifolds, J. Differ. Geom., 113 (2019), 493–518.
[20]	C. Mantoulidis, P. Miao, L. F. Tam, Capacity, quasi-local mass, and singular fill-ins, J. reine angew. Math., 768 (2020), 55–92.
[21]	S. McCormick, On a Minkowski-like inequality for asymptotically flat static manifolds, P. Am. Math. Soc., 146 (2018), 4039–4046. doi: 10.1090/proc/14047
[22]	P. Miao, L. F. Tam, Evaluation of the ADM mass and center of mass via the Ricci tensor, P. Am. Math. Soc., 144 (2016), 753–761.
[23]	P. Petersen, Riemannian geometry, 3 Eds., New York: Springer, 2016.
[24]	S. Pigola, G. Veronelli, The smooth riemannian extension problem, 2016, arXiv: 1606.08320.
[25]	W. Rudin, Real and complex analysis, 3 Eds., New York: McGraw-Hill, 1987.
[26]	M. E. Taylor, Measure theory and integration, Providence: American Mathematical Society, 2006.
[27]	Z. Wang, A Minkowski-type inequality for hypersurfaces in the Reissner-Nordström-anti-deSitter manifold, 2015. Available from: https://academiccommons.columbia.edu/doi/10.7916/D86H4GGN.

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