Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

Zheng, Yizhong

doi:10.1090/proc/15367

Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow
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Proc. Amer. Math. Soc. 149 (2021), 2105-2119 Request permission

Abstract:

We show that the isoperimetric profile $h_{g(t)}(\xi )$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(\xi )$ is uniform Lipschitz continuous and hence $h_{g(t)}(\xi )$ is uniform locally Lipschitz continuous.

References

Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
Ben Andrews and Paul Bryan, Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere, Calc. Var. Partial Differential Equations 39 (2010), no. 3-4, 419–428. MR 2729306, DOI 10.1007/s00526-010-0315-5
Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084, DOI 10.24033/asens.1514
Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
Bennett Chow, The Ricci flow on the $2$-sphere, J. Differential Geom. 33 (1991), no. 2, 325–334. MR 1094458
Paul E. Ehrlich, Continuity properties of the injectivity radius function, Compositio Math. 29 (1974), 151–178. MR 417977
Alfred Gray, The volume of a small geodesic ball of a Riemannian manifold, Michigan Math. J. 20 (1973), 329–344 (1974). MR 339002
Sylvestre Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, Astérisque 163-164 (1988), 5–6, 31–91, 281 (1989) (French, with English summary). On the geometry of differentiable manifolds (Rome, 1986). MR 999971
Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419
M. Miranda Jr., D. Pallara, F. Paronetto, and M. Preunkert, Heat semigroup and functions of bounded variation on Riemannian manifolds, J. Reine Angew. Math. 613 (2007), 99–119. MR 2377131, DOI 10.1515/CRELLE.2007.093
Francesco Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. MR 2976521, DOI 10.1017/CBO9781139108133
Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
S. Nardulli and F.Russo, On the Hamilton’s isoperimetric ratio in complete Riemannian manifolds of finite volume, arXiv:1502.05903, 2015
Stefano Nardulli and Pierre Pansu, A complete Riemannian manifold whose isoperimetric profile is discontinuous, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 537–549. MR 3801288, DOI 10.1007/s00028-017-0410-6
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
Manuel Ritoré, Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions, Rev. Mat. Iberoam. 33 (2017), no. 1, 239–250. MR 3615450, DOI 10.4171/RMI/935
Manuel Ritoré and César Rosales, Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4601–4622. MR 2067135, DOI 10.1090/S0002-9947-04-03537-8
Panos Papasoglu and Eric Swenson, A surface with discontinuous isoperimetric profile and expander manifolds, Geom. Dedicata 206 (2020), 43–54. MR 4091536, DOI 10.1007/s10711-019-00475-9