A sharp isoperimetric property of the renormalized area of a minimal surface in hyperbolic space

Bernstein, Jacob

doi:10.1090/proc/15960

A sharp isoperimetric property of the renormalized area of a minimal surface in hyperbolic space
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Proc. Amer. Math. Soc. 150 (2022), 4487-4502 Request permission

Abstract:

We prove an inequality bounding the renormalized area of a complete minimal surface in hyperbolic space in terms of the conformal length of its ideal boundary.

References

Spyridon Alexakis and Rafe Mazzeo, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds, Comm. Math. Phys. 297 (2010), no. 3, 621–651. MR 2653898, DOI 10.1007/s00220-010-1054-3
Spyros Alexakis and Rafe Mazzeo, Complete Willmore surfaces in $\Bbb {H}^3$ with bounded energy: boundary regularity and bubbling, J. Differential Geom. 101 (2015), no. 3, 369–422. MR 3415767
Jacob Bernstein, Colding Minicozzi entropy in hyperbolic space, Nonlinear Anal. 210 (2021), Paper No. 112401, 16. MR 4255506, DOI 10.1016/j.na.2021.112401
Jacob Bernstein and Lu Wang, Relative expander entropy in the presence of a two-sided obstacle and applications, Adv. Math. 399 (2022), Paper No. 108284, 48. MR 4385133, DOI 10.1016/j.aim.2022.108284
J. Bernstein and L. Wang, Topological uniqueness for self-expanders of small entropy, Preprint, 2019.
Robert L. Bryant, Surfaces in conformal geometry, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 227–240. MR 974338, DOI 10.1090/pspum/048/974338
Qing Chen, On the area growth of minimal surfaces in $H^n$, Geom. Dedicata 75 (1999), no. 3, 263–273. MR 1689272, DOI 10.1023/A:1005167519637
Qing Chen and Yi Cheng, Asymptotic behavior and area growth of minimal surfaces in $\textbf {H}^n$, Osaka J. Math. 36 (1999), no. 2, 397–407. MR 1736484
Qing Chen and Yi Cheng, Chern-Osserman inequality for minimal surfaces in $H^n$, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2445–2450. MR 1664325, DOI 10.1090/S0002-9939-99-05334-4
Jaigyoung Choe and Robert Gulliver, The sharp isoperimetric inequality for minimal surfaces with radially connected boundary in hyperbolic space, Invent. Math. 109 (1992), no. 3, 495–503. MR 1176200, DOI 10.1007/BF01232035
Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI 10.1512/iumj.1983.32.32046
C. Robin Graham and Edward Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nuclear Phys. B 546 (1999), no. 1-2, 52–64. MR 1682674, DOI 10.1016/S0550-3213(99)00055-3
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. MR 3152944, DOI 10.4007/annals.2014.179.2.6
M. T. Nguyen, Weighted monotonicity theorems and applications to minimal surfaces in hyperbolic space, arXiv:2105.12625 [math], 2021.
Shinsei Ryu and Tadashi Takayanagi, Aspects of holographic entanglement entropy, J. High Energy Phys. 8 (2006), 045, 48. MR 2249925, DOI 10.1088/1126-6708/2006/08/045
Ryu Shinsei and Tadashi Takayanagi, Holographic derivation of entanglement entropy from the AdS/CFT correspondence, Phys. Rev. Lett. 96 (2006), no. 18, 181602.
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Yongzhe Zhang, Superconvexity of the heat kernel on hyperbolic space with applications to mean curvature flow, Proc. Amer. Math. Soc. 149 (2021), no. 5, 2161–2166. MR 4232206, DOI 10.1090/proc/15379