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. LiuH. W. XuF. 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. LiuH. W. XuF. 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.

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.

[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. LiuH. W. XuF. 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. LiuH. W. XuF. 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.

[1]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[2]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012

[3]

Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010

[4]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[5]

Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076

[6]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441

[7]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153

[8]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure and Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549

[9]

Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367

[10]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[11]

Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256

[12]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036

[13]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

[14]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure and Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[15]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[16]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

[17]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116

[18]

Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1

[19]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963

[20]

Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (175)
  • HTML views (110)
  • Cited by (0)

Other articles
by authors

[Back to Top]