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March  2021, 20(3): 1199-1211. doi: 10.3934/cpaa.2021016

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 Published  February 2021

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.

Keywords: Mean curvature flow, cylindrical estimates, Hyperbolic spaces, singularity, gradient estimate.
Mathematics Subject Classification: Primary: 53C42; Secondary: 35B40.
Citation: Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
References:
[1]

S. Brendle and G. Huisken, Mean curvature flow with surgery of convex surfaces in $\mathbb{R}^3$, Invent. Math., 203 (2016), 615-654.  doi: 10.1007/s00222-015-0599-3.  Google Scholar

[2]

S. Brendle and G. Huisken, A fully nonlinear flow for two-convex hypersurfaces, Invent. Math., 2 (2017), 559-613.  doi: 10.1007/s00222-017-0736-2.  Google Scholar

[3]

E. Codá Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math., 176 (2012), 825-863.  doi: 10.4007/annals.2012.176.2.3.  Google Scholar

[4]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922.  Google Scholar

[5]

G. Hamilton, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[6]

G. Hamilton, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, J. Differ. Geom., 84 (1986), 463-480.  doi: 10.1007/BF01388742.  Google Scholar

[7]

G. Hamilton, Deforming hypersurfaces of the the sphere by their mean curvature, Math. Z., 84 (1986), 205-219.  doi: 10.1007/BF01166458.  Google Scholar

[8]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta. Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.  Google Scholar

[9]

G. Huisken and C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math., 175 (2009), 137-221.  doi: 10.1007/s00222-008-0148-4.  Google Scholar

[10]

L. Lei and H. W. Xu, A new version of Huisken's convergence theorem for mean curvature flow in spheres, preprint, arXiv: math/1505.07217. Google Scholar

[11]

L. Lei and H. W. Xu, An optimal convergence theorem for mean curvature flow arbitrary codimension in hyperbolic spaces, preprint, arXiv: math/1503.06747. Google Scholar

[12]

L. Lei and H. W. Xu, Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem, preprint, arXiv: math/1506.06371v2. Google Scholar

[13]

K. F. Liu, H. W. Xu, F. Ye and E. T. Zhao, The extension and convrgence of mean curvature flow in higher codimension, Trans. Amer. Math. Soc., 175 (2009), 137-221.  doi: 10.1090/tran/7281.  Google Scholar

[14]

K. F. Liu, H. W. Xu, F. Ye and E. T. Zhao, Mean curvature flow of higher codimension in hyperbolic spaces, Commun. Anal. Geom., 21 (2013), 651-669.  doi: 10.4310/CAG.2013.v21.n3.a8.  Google Scholar

[15]

K. F. Liu, H. W. Xu and E. T. Zhao, Mean curvature flow of higher codimension in Riemmanian manifolds, preprint, arXiv: math/1204.0107. doi: 10.4310/CAG.2013.v21.n3.a8.  Google Scholar

[16]

H. T. Nguyen, Convexity and cylindrical estimates for mean curvature flow in the sphere, Trans. Amer. Math. Soc., 367 (2015), 4517-4536.  doi: 10.1090/S0002-9947-2015-05927-3.  Google Scholar

[17]

G. Pipoli and Carlo Sinestrari, Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes, Ann. Glob. Anal. Geom., 51 (2017), 179-188.  doi: 10.1007/s10455-016-9530-4.  Google Scholar

show all references

References:
[1]

S. Brendle and G. Huisken, Mean curvature flow with surgery of convex surfaces in $\mathbb{R}^3$, Invent. Math., 203 (2016), 615-654.  doi: 10.1007/s00222-015-0599-3.  Google Scholar

[2]

S. Brendle and G. Huisken, A fully nonlinear flow for two-convex hypersurfaces, Invent. Math., 2 (2017), 559-613.  doi: 10.1007/s00222-017-0736-2.  Google Scholar

[3]

E. Codá Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math., 176 (2012), 825-863.  doi: 10.4007/annals.2012.176.2.3.  Google Scholar

[4]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922.  Google Scholar

[5]

G. Hamilton, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[6]

G. Hamilton, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, J. Differ. Geom., 84 (1986), 463-480.  doi: 10.1007/BF01388742.  Google Scholar

[7]

G. Hamilton, Deforming hypersurfaces of the the sphere by their mean curvature, Math. Z., 84 (1986), 205-219.  doi: 10.1007/BF01166458.  Google Scholar

[8]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta. Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.  Google Scholar

[9]

G. Huisken and C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math., 175 (2009), 137-221.  doi: 10.1007/s00222-008-0148-4.  Google Scholar

[10]

L. Lei and H. W. Xu, A new version of Huisken's convergence theorem for mean curvature flow in spheres, preprint, arXiv: math/1505.07217. Google Scholar

[11]

L. Lei and H. W. Xu, An optimal convergence theorem for mean curvature flow arbitrary codimension in hyperbolic spaces, preprint, arXiv: math/1503.06747. Google Scholar

[12]

L. Lei and H. W. Xu, Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem, preprint, arXiv: math/1506.06371v2. Google Scholar

[13]

K. F. Liu, H. W. Xu, F. Ye and E. T. Zhao, The extension and convrgence of mean curvature flow in higher codimension, Trans. Amer. Math. Soc., 175 (2009), 137-221.  doi: 10.1090/tran/7281.  Google Scholar

[14]

K. F. Liu, H. W. Xu, F. Ye and E. T. Zhao, Mean curvature flow of higher codimension in hyperbolic spaces, Commun. Anal. Geom., 21 (2013), 651-669.  doi: 10.4310/CAG.2013.v21.n3.a8.  Google Scholar

[15]

K. F. Liu, H. W. Xu and E. T. Zhao, Mean curvature flow of higher codimension in Riemmanian manifolds, preprint, arXiv: math/1204.0107. doi: 10.4310/CAG.2013.v21.n3.a8.  Google Scholar

[16]

H. T. Nguyen, Convexity and cylindrical estimates for mean curvature flow in the sphere, Trans. Amer. Math. Soc., 367 (2015), 4517-4536.  doi: 10.1090/S0002-9947-2015-05927-3.  Google Scholar

[17]

G. Pipoli and Carlo Sinestrari, Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes, Ann. Glob. Anal. Geom., 51 (2017), 179-188.  doi: 10.1007/s10455-016-9530-4.  Google Scholar

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