Minkowski inequalities and constrained inverse curvature flows in warped spaces

References

[1] V. Agostiniani, M. Fogagnolo and L. Mazzieri, Minkowski inequalities via nonlinear potential theory, preprint (2019), https://arxiv.org/abs/1906.00322. 10.1007/s00205-022-01756-6Search in Google Scholar

[2] C. B. Allendoerfer, Steiner’s formulae on a general S n + 1 , Bull. Amer. Math. Soc. 54 (1948), 128–135. 10.1090/S0002-9904-1948-08966-2Search in Google Scholar

[3] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. 10.1515/CRELLE.2007.051Search in Google Scholar

[4] S. Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247–269. 10.1007/s10240-012-0047-5Search in Google Scholar

[5] S. Brendle, P. Guan and J. Li, An inverse type hypersurface flow in space forms, private note. Search in Google Scholar

[6] S. Brendle, P.-K. Hung and M.-T. Wang, A Minkowski inequality for hypersurfaces in the anti–de Sitter–Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144. 10.1002/cpa.21556Search in Google Scholar

[7] D. Chen, H. Li and T. Zhou, A Penrose type inequality for graphs over Reissner–Nordström–anti-deSitter manifold, J. Math. Phys. 60 (2019), no. 4, Article ID 043503. 10.1063/1.5050865Search in Google Scholar

[8] L. L. de Lima and F. 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. 10.1007/s00023-015-0414-0Search in Google Scholar

[9] C. Enz, The scalar curvature flow in Lorentzian manifolds, Adv. Calc. Var. 1 (2008), no. 3, 323–343. 10.1515/ACV.2008.014Search in Google Scholar

[10] E. Gallego and G. Solanes, Integral geometry and geometric inequalities in hyperbolic space, Differential Geom. Appl. 22 (2005), no. 3, 315–325. 10.1016/j.difgeo.2005.01.006Search in Google Scholar

[11] F. Gao, D. Hug and R. Schneider, Intrinsic volumes and polar sets in spherical space, Math. Notae 41 (2001/02), 159–176. Search in Google Scholar

[12] Y. Ge, G. Wang and J. Wu, The GBC mass for asymptotically hyperbolic manifolds, Math. Z. 281 (2015), no. 1–2, 257–297. 10.1007/s00209-015-1483-ySearch in Google Scholar

[13] Y. Ge, G. Wang, J. Wu and C. Xia, A Penrose inequality for graphs over Kottler space, Calc. Var. Partial Differential Equations 52 (2015), no. 3–4, 755–782. 10.1007/s00526-014-0732-ySearch in Google Scholar

[14] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), no. 1, 299–314. 10.4310/jdg/1214445048Search in Google Scholar

[15] C. Gerhardt, Curvature Problems, Ser. Geom. Topol. 39, International Press, Somerville, 2006. Search in Google Scholar

[16] C. Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. 10.4310/jdg/1335207376Search in Google Scholar

[17] P. Guan and J. Li, The quermassintegral inequalities for 𝑘-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. 10.1016/j.aim.2009.03.005Search in Google Scholar

[18] P. Guan and J. Li, A mean curvature type flow in space forms, Int. Math. Res. Not. IMRN 2015 (2015), no. 13, 4716–4740. 10.1093/imrn/rnu081Search in Google Scholar

[19] P. Guan and J. Li, Isoperimetric inequalities and hypersurface flows, preprint (2019), http://www.math.mcgill.ca/guan/Guan-Li-2019S1.pdf. Search in Google Scholar

[20] P. Guan, J. Li and M.-T. Wang, A volume preserving flow and the isoperimetric problem in warped product spaces, Trans. Amer. Math. Soc. 372 (2019), no. 4, 2777–2798. 10.1090/tran/7661Search in Google Scholar

[21] Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint (2020), https://arxiv.org/abs/2002.10643. 10.1007/s00208-020-02076-4Search in Google Scholar

[22] G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. 10.4310/jdg/1090349447Search in Google Scholar

[23] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Math. Appl. (Soviet Series) 7, D. Reidel, Dordrecht, 1987. 10.1007/978-94-010-9557-0Search in Google Scholar

[24] K.-K. Kwong and P. Miao, A new monotone quantity along the inverse mean curvature flow in R n , Pacific J. Math. 267 (2014), no. 2, 417–422. 10.2140/pjm.2014.267.417Search in Google Scholar

[25] B. Lambert and J. Scheuer, Isoperimetric problems for spacelike domains in generalized Robertson–Walker spaces, J. Evol. Equ. (2020), 10.1007/s00028-020-00584-z. 10.1007/s00028-020-00584-zSearch in Google Scholar

[26] M. Makowski and J. Scheuer, Rigidity results, inverse curvature flows and Alexandrov–Fenchel type inequalities in the sphere, Asian J. Math. 20 (2016), no. 5, 869–892. 10.4310/AJM.2016.v20.n5.a2Search in Google Scholar

[27] S. McCormick, On a Minkowski-like inequality for asymptotically flat static manifolds, Proc. Amer. Math. Soc. 146 (2018), no. 9, 4039–4046. 10.1090/proc/14047Search in Google Scholar

[28] H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), no. 4, 447–495. 10.1007/BF01445180Search in Google Scholar

[29] J. Natário, A Minkowski-type inequality for convex surfaces in the hyperbolic 3-space, Differential Geom. Appl. 41 (2015), 102–109. 10.1016/j.difgeo.2015.05.002Search in Google Scholar

[30] L. A. Santaló, On parallel hypersurfaces in the elliptic and hyperbolic 𝑛-dimensional space, Proc. Amer. Math. Soc. 1 (1950), 325–330. 10.1090/S0002-9939-1950-0036532-6Search in Google Scholar

[31] J. Scheuer, Inverse curvature flows in Riemannian warped products, J. Funct. Anal. 276 (2019), no. 4, 1097–1144. 10.1016/j.jfa.2018.08.021Search in Google Scholar

[32] J. Scheuer, The Minkowski inequality in de Sitter space, preprint (2019), https://arxiv.org/abs/1909.06837. 10.2140/pjm.2021.314.425Search in Google Scholar

[33] J. Scheuer and C. Xia, Locally constrained inverse curvature flows, Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803. 10.1090/tran/7949Search in Google Scholar

[34] J. Steiner, Jacob Steiner’s gesammelte Werke. Vo. 2, Cambridge University, Cambridge, 2013. Search in Google Scholar

[35] J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), no. 3, 355–372. 10.1007/BF02571249Search in Google Scholar

[36] G. Wang and C. Xia, Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space, Adv. Math. 259 (2014), 532–556. 10.1016/j.aim.2014.01.024Search in Google Scholar

[37] Z. Wang, A Minkowski-Type Inequality for Hypersurfaces in the Reissner–Nordstrom-anti-deSitter Manifold, ProQuest LLC, Ann Arbor, 2015; Thesis (Ph.D.)–Columbia University. Search in Google Scholar

[38] Y. Wei, On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 46. 10.1007/s00526-018-1342-xSearch in Google Scholar

[39] Y. Wei and C. Xiong, Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere, Pacific J. Math. 277 (2015), no. 1, 219–239. 10.2140/pjm.2015.277.219Search in Google Scholar

[40] H. Zhou, Inverse mean curvature flows in warped product manifolds, J. Geom. Anal. 28 (2018), no. 2, 1749–1772. 10.1007/s12220-017-9887-zSearch in Google Scholar