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

  • 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.

    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

    Related Papers:

  • 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.



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    [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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