Geometric inequalities for static convex domains in hyperbolic space

Hu, Yingxiang; Li, Haizhong

doi:10.1090/tran/8628

Geometric inequalities for static convex domains in hyperbolic space
HTML articles powered by AMS MathViewer

by Yingxiang Hu and Haizhong Li PDF

Trans. Amer. Math. Soc. 375 (2022), 5587-5615 Request permission

Abstract:

We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.

References

Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. MR 1267897
Ben Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. MR 2339467, DOI 10.1515/CRELLE.2007.051
Ben Andrews, Xuzhong Chen, and Yong Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 7, 2467–2509. MR 4269419, DOI 10.4171/jems/1059
Ben Andrews, Yingxiang Hu, and Haizhong Li, Harmonic mean curvature flow and geometric inequalities, Adv. Math. 375 (2020), 107393, 28. MR 4170217, DOI 10.1016/j.aim.2020.107393
Ben Andrews, James McCoy, and Yu Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 611–665. MR 3070558, DOI 10.1007/s00526-012-0530-3
Ben Andrews and Yong Wei, Quermassintegral preserving curvature flow in hyperbolic space, Geom. Funct. Anal. 28 (2018), no. 5, 1183–1208. MR 3856791, DOI 10.1007/s00039-018-0456-9
Simon Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247–269. MR 3090261, DOI 10.1007/s10240-012-0047-5
Simon Brendle, Pengfei Guan, and Junfang Li, An inverse curvature type hypersurface flow in $\mathbb H^{n+1}$, preprint.
Simon Brendle and Mu-Tao Wang, A Gibbons-Penrose inequality for surfaces in Schwarzschild spacetime, Comm. Math. Phys. 330 (2014), no. 1, 33–43. MR 3215576, DOI 10.1007/s00220-014-1972-6
Esther Cabezas-Rivas and Vicente Miquel, Volume preserving mean curvature flow in the hyperbolic space, Indiana Univ. Math. J. 56 (2007), no. 5, 2061–2086. MR 2359723, DOI 10.1512/iumj.2007.56.3060
Levi Lopes de Lima and Frederico Girão, An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality, Ann. Henri Poincaré 17 (2016), no. 4, 979–1002. MR 3472630, DOI 10.1007/s00023-015-0414-0
Yuxin Ge, Guofang Wang, and Jie Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237–260. MR 3263518
Yuxin Ge, Guofang Wang, and Jie Wu, The GBC mass for asymptotically hyperbolic manifolds, Math. Z. 281 (2015), no. 1-2, 257–297. MR 3384870, DOI 10.1007/s00209-015-1483-y
Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. MR 2879249
Pengfei Guan, Curvature measures, isoperimetric type inequalities and fully nonlinear PDEs, Fully nonlinear PDEs in real and complex geometry and optics, Lecture Notes in Math., vol. 2087, Springer, Cham, 2014, pp. 47–94. MR 3203559, DOI 10.1007/978-3-319-00942-1_{2}
Pengfei Guan and Junfang Li, The quermassintegral inequalities for $k$-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. MR 2522433, DOI 10.1016/j.aim.2009.03.005
Pengfei Guan and Junfang Li, A mean curvature type flow in space forms, Int. Math. Res. Not. IMRN 13 (2015), 4716–4740. MR 3439091, DOI 10.1093/imrn/rnu081
Pengfei Guan and Junfang Li, Isoperimetric type inequalities and hypersurface flows, J. Math. Study 54 (2021), no. 1, 56–80. MR 4206465, DOI 10.4208/jms.v54n1.21.03
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Yingxiang Hu and Haizhong Li, Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Paper No. 55, 20. MR 3916118, DOI 10.1007/s00526-019-1488-1
Yingxiang Hu, Haizhong Li, and Yong Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, Math. Ann. (2020), available at https://doi.org/10.1007/s00208-020-02076-4.
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
Haizhong Li, Yong Wei, and Changwei Xiong, A geometric inequality on hypersurface in hyperbolic space, Adv. Math. 253 (2014), 152–162. MR 3148549, DOI 10.1016/j.aim.2013.12.003
Luis A. Santaló, Integral geometry and geometric probability, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. With a foreword by Mark Kac. MR 2162874, DOI 10.1017/CBO9780511617331
Julian Scheuer and Chao Xia, Locally constrained inverse curvature flows, Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803. MR 4024538, DOI 10.1090/tran/7949
Guofang Wang and Chao Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532–556. MR 3197666, DOI 10.1016/j.aim.2014.01.024
Chao Xia, A Minkowski type inequality in space forms, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 96, 8. MR 3523663, DOI 10.1007/s00526-016-1037-0