Cylindrical estimates for mean curvature flow in hyperbolic spaces

Zhengchao Ji

doi:10.3934/cpaa.2021016

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2021, Volume 20, Issue 3: 1199-1211. Doi: 10.3934/cpaa.2021016

This issue Previous Article The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space Next Article Existence of solution and asymptotic behavior for a class of parabolic equations

Cylindrical estimates for mean curvature flow in hyperbolic spaces

Zhengchao Ji

Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

Received: June 2020

Revised: December 2020

Early access: February 2021

Published: March 2021

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Abstract

We consider the mean curvature flow of a closed hypersurface in hyperbolic space. Under a suitable pinching assumption on the initial data, we prove a priori estimate on the principal curvatures which implies that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes the estimates obtained in the previous works of Huisken, Sinestrari and Nguyen on the mean curvature flow of hypersurfaces in Euclidean spaces and in the spheres.

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Mathematics Subject Classification: Primary: 53C42; Secondary: 35B40.

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References

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