## Locally constrained inverse curvature flows

HTML articles powered by AMS MathViewer

- by Julian Scheuer and Chao Xia PDF
- Trans. Amer. Math. Soc.
**372**(2019), 6771-6803 Request permission

## Abstract:

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order—namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti–de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.## References

- Nicholas D. Alikakos and Alexandre Freire,
*The normalized mean curvature flow for a small bubble in a Riemannian manifold*, J. Differential Geom.**64**(2003), no. 2, 247–303. MR**2029906** - Ben Andrews,
*Contraction of convex hypersurfaces in Euclidean space*, Calc. Var. Partial Differential Equations**2**(1994), no. 2, 151–171. MR**1385524**, DOI 10.1007/BF01191340 - 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 - Kenneth A. Brakke,
*The motion of a surface by its mean curvature*, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR**0485012** - 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 $\mathbb {H}^{n+1}$*(private note). - Simon Brendle, Pei-Ken Hung, and Mu-Tao Wang,
*A Minkowski inequality for hypersurfaces in the anti–de Sitter–Schwarzschild manifold*, Comm. Pure Appl. Math.**69**(2016), no. 1, 124–144. MR**3433631**, DOI 10.1002/cpa.21556 - 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 - Bennett Chow,
*Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature*, J. Differential Geom.**22**(1985), no. 1, 117–138. MR**826427** - Bennett Chow,
*Deforming convex hypersurfaces by the square root of the scalar curvature*, Invent. Math.**87**(1987), no. 1, 63–82. MR**862712**, DOI 10.1007/BF01389153 - 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 - Klaus Ecker and Gerhard Huisken,
*Immersed hypersurfaces with constant Weingarten curvature*, Math. Ann.**283**(1989), no. 2, 329–332. MR**980601**, DOI 10.1007/BF01446438 - Christian Enz,
*The scalar curvature flow in Lorentzian manifolds*, Adv. Calc. Var.**1**(2008), no. 3, 323–343. MR**2458241**, DOI 10.1515/ACV.2008.014 - 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,
*Flow of nonconvex hypersurfaces into spheres*, J. Differential Geom.**32**(1990), no. 1, 299–314. MR**1064876** - Claus Gerhardt,
*Closed Weingarten hypersurfaces in space forms*, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 71–97. MR**1449403** - Claus Gerhardt,
*Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds*, J. Reine Angew. Math.**554**(2003), 157–199. MR**1952172**, DOI 10.1515/crll.2003.003 - Claus Gerhardt,
*Curvature problems*, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR**2284727** - Claus Gerhardt,
*Inverse curvature flows in hyperbolic space*, J. Differential Geom.**89**(2011), no. 3, 487–527. MR**2879249** - Claus Gerhardt,
*Non-scale-invariant inverse curvature flows in Euclidean space*, Calc. Var. Partial Differential Equations**49**(2014), no. 1-2, 471–489. MR**3148124**, DOI 10.1007/s00526-012-0589-x - Claus Gerhardt,
*Curvature flows in the sphere*, J. Differential Geom.**100**(2015), no. 2, 301–347. MR**3343834** - Frederico Girão and Neilha M. Pinheiro,
*An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere*, Ann. Global Anal. Geom.**52**(2017), no. 4, 413–424. MR**3735905**, DOI 10.1007/s10455-017-9562-4 - 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,
*A fully-nonlinear flow and quermassintegral inequalities (in Chinese)*, Sci. Sin. Math.**48**(2018), no. 1, 147–156, DOI 10.1360/N012017-00009. - Pengfei Guan, Junfang Li, and Mu Tao Wang,
*A volume preserving flow and the isoperimetric problem in warped product spaces*, Trans. Am. Math. Soc.**372**(2019), 2777–2798, DOI 10.1090/tran/7661. - Gerhard Huisken,
*Flow by mean curvature of convex surfaces into spheres*, J. Differential Geom.**20**(1984), no. 1, 237–266. MR**772132** - Gerhard Huisken,
*Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature*, Invent. Math.**84**(1986), no. 3, 463–480. MR**837523**, DOI 10.1007/BF01388742 - Gerhard Huisken,
*The volume preserving mean curvature flow*, J. Reine Angew. Math.**382**(1987), 35–48. MR**921165**, DOI 10.1515/crll.1987.382.35 - Gerhard Huisken and Tom Ilmanen,
*The inverse mean curvature flow and the Riemannian Penrose inequality*, J. Differential Geom.**59**(2001), no. 3, 353–437. MR**1916951** - Gerhard Huisken and Carlo Sinestrari,
*Convexity estimates for mean curvature flow and singularities of mean convex surfaces*, Acta Math.**183**(1999), no. 1, 45–70. MR**1719551**, DOI 10.1007/BF02392946 - Gerhard Huisken and Carlo Sinestrari,
*Mean curvature flow with surgeries of two-convex hypersurfaces*, Invent. Math.**175**(2009), no. 1, 137–221. MR**2461428**, DOI 10.1007/s00222-008-0148-4 - N. V. Krylov,
*Nonlinear elliptic and parabolic equations of the second order*, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR**901759**, DOI 10.1007/978-94-010-9557-0 - Matthias Makowski and Julian Scheuer,
*Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere*, Asian J. Math.**20**(2016), no. 5, 869–892. MR**3622318**, DOI 10.4310/AJM.2016.v20.n5.a2 - James McCoy,
*The surface area preserving mean curvature flow*, Asian J. Math.**7**(2003), no. 1, 7–30. MR**2015239**, DOI 10.4310/AJM.2003.v7.n1.a2 - James A. McCoy,
*Mixed volume preserving curvature flows*, Calc. Var. Partial Differential Equations**24**(2005), no. 2, 131–154. MR**2164924**, DOI 10.1007/s00526-004-0316-3 - Julian Scheuer,
*Non-scale-invariant inverse curvature flows in hyperbolic space*, Calc. Var. Partial Differential Equations**53**(2015), no. 1-2, 91–123. MR**3336314**, DOI 10.1007/s00526-014-0742-9 - Julian Scheuer,
*The inverse mean curvature flow in warped cylinders of non-positive radial curvature*, Adv. Math.**306**(2017), 1130–1163. MR**3581327**, DOI 10.1016/j.aim.2016.11.003 - Julian Scheuer,
*Isotropic functions revisited*, Arch. Math. (Basel)**110**(2018), no. 6, 591–604. MR**3803748**, DOI 10.1007/s00013-018-1162-4 - John I. E. Urbas,
*On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures*, Math. Z.**205**(1990), no. 3, 355–372. MR**1082861**, DOI 10.1007/BF02571249 - John I. E. Urbas,
*An expansion of convex hypersurfaces*, J. Differential Geom.**33**(1991), no. 1, 91–125. MR**1085136** - 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 - Yong Wei and Changwei 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. MR**3393689**, DOI 10.2140/pjm.2015.277.219 - 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

## Additional Information

**Julian Scheuer**- Affiliation: Albert-Ludwigs-Universität, Mathematisches Institut, Abteilung Reine Mathematik, Ernst-Zermelo-Stra\normalfont{ß}e 1, 79104 Freiburg, Germany
- MR Author ID: 1104274
- Email: julian.scheuer@math.uni-freiburg.de
**Chao Xia**- Affiliation: School of Mathematical Sciences, Xiamen University, 361005 Xiamen, People’s Republic of China
- MR Author ID: 922365
- Email: chaoxia@xmu.edu.cn
- Received by editor(s): August 20, 2017
- Published electronically: August 28, 2019
- Additional Notes: The first author was being supported by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation), research grant “Quermassintegral preserving local curvature flows”, number SCHE 1879/3-1. The research of the second author was supported in part by NSFC (Grant No. 11501480, 11871406), the Natural Science Foundation of Fujian Province of China (Grant No. 2017J06003) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180009). Part of this work was done while he was visiting the Institute of Mathematics at Albert-Ludwigs-Universität Freiburg. He would like to thank the Institute for its hospitality.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 6771-6803 - MSC (2010): Primary 53C21, 53C24, 53C44
- DOI: https://doi.org/10.1090/tran/7949
- MathSciNet review: 4024538