Bulletin of the Korean Mathematical Society (대한수학회보)

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ALMOST RIGIDITY OF CONVEX HYPERSURFACES VIA THE EXTINCTION TIME OF MEAN CURVATURE FLOW

  • Received : 2020.07.10
  • Accepted : 2021.02.09
  • Published : 2021.07.31
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Abstract

We prove that if a compact convex hypersurface of ℝn+1 has almost maximal extinction time when it is evolved by the mean curvature flow, then it must be nearly round in the C0-norm.

Keywords

Acknowledgement

The author is partially supported by NSFC (Nos. 11701580 and 11521101) and Guangdong Natural Science Foundation 2019A1515011804.

References

  1. R. H. Bamler and D. Maximo, Almost-rigidity and the extinction time of positively curved Ricci flows, Math. Ann. 369 (2017), no. 1-2, 899-911. https://doi.org/10.1007/s00208-016-1494-y
  2. J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189-237. https://doi.org/10.2307/2118589
  3. T. H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 175-191. https://doi.org/10.1007/s002220050049
  4. K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547-569. https://doi.org/10.1007/BF01232278
  5. R. S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215-226. http://projecteuclid.org/euclid.jdg/1214456010 https://doi.org/10.4310/jdg/1214456010
  6. G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. http://projecteuclid.org/euclid.jdg/1214438998 https://doi.org/10.4310/jdg/1214438998
  7. J. Langer, A compactness theorem for surfaces with Lp-bounded second fundamental form, Math. Ann. 270 (1985), no. 2, 223-234. https://doi.org/10.1007/BF01456183