# American Institute of Mathematical Sciences

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

## Cylindrical estimates for mean curvature flow in hyperbolic spaces

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

Received  June 2020 Revised  December 2020 Published  March 2021 Early access  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.

Citation: Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and 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. [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. [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. [4] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922. [5] G. Hamilton, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998. [6] G. Hamilton, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, J. Differ. Geom., 84 (1986), 463-480.  doi: 10.1007/BF01388742. [7] G. Hamilton, Deforming hypersurfaces of the the sphere by their mean curvature, Math. Z., 84 (1986), 205-219.  doi: 10.1007/BF01166458. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [4] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922. [5] G. Hamilton, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998. [6] G. Hamilton, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, J. Differ. Geom., 84 (1986), 463-480.  doi: 10.1007/BF01388742. [7] G. Hamilton, Deforming hypersurfaces of the the sphere by their mean curvature, Math. Z., 84 (1986), 205-219.  doi: 10.1007/BF01166458. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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