Mean curvature flow with convex Gauss image

Abstract

In this paper, the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied. The confinable property of the Gauss image under the mean curvature flow is proved, which in turn helps one to obtain the curvature estimates. Then the author proves a long time existence result. The asymptotic behavior of these solutions when t → ∞ is also studied.

Keywords

Mean curvature flow / Convex Gauss image / Curvature estimates / Long time existence

References

[1] Chen J., Li J.. Mean curvature flow of surfaces in 4-manifolds. Adv. Math., 2002, 163: 287-309

[2] Chen J., Li J.. Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math., 2004, 156(1): 25-51

[3] Chen J., Tian G.. Two-dimensional graphs moving by mean curvature flow. Acta Math. Sinica, Engl. Ser., 2000, 16: 541-548

[4] Chen Q., Xu S. L.. Rigidity of compact minimal submanifolds in a unit sphere. Geom. Dedicata, 1993, 45(1): 83-88

[5] Fischer-Colbrie D.. Some rigidity theorems for minimal submanifolds of the sphere. Acta Math., 1980, 145: 29-46

[6] Ecker K., Huisken G.. Mean curvature evolution of entire graphs. Ann. of Math., 1989, 130(3): 453-471

[7] Ecker K., Huisken G.. Interior estimates for hypersurfaces moving by mean curvature. Invent. Math., 1991, 105(3): 547-569

[8] Hildebrandt S., Jost J., Widman K. O.. Harmonic mappings and minimal submanifolds. Invent. Math., 1980, 62: 269-298

[9] Huisken G.. Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom., 1984, 20(1): 237-266

[10] Huisken G.. Asymptotic behavior for singularities of the mean curvature flow. J. Diff. Geom., 1990, 31(1): 285-299

[11] Jost J., Xin Y. L.. Bernstein type theorems for higher codimension. Calculus Var. PDE, 1999, 9: 277-296

[12] Li A. M., Li J. M.. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math., 1992, 58: 582-594

[13] Moser J.. On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math., 1961, 14: 577-591

[14] Simons J.. Minimal varieties in Riemannian manifolds. Ann. of Math., 1968, 88: 62-105

[15] Smoczyk K.. Harnack inequality for the Lagrangian mean curvature flow. Calculus Var. PDE, 1999, 8: 247-258

[16] Smoczyk K.. Angle theorems for Lagrangian mean curvature flow. Math. Z., 2002, 240: 849-863

[17] Smoczyk K., Wang M.-T.. Mean curvature flows for Lagrangian submanifolds with convex potentials. J. Diff. Geom., 2002, 62: 243-257

[18] Wang M.-T.. Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math., 2002, 148(3): 525-543

[19] Wang M.-T.. Gauss maps of the mean curvature flow. Math. Research Lett., 2003, 10: 287-299

[20] Wong Y.-C.. Differential geometry of Grassmann manifolds. Proc. N. A. S., 1967, 57: 589-594

[21] Xin Y. L.. Geometry of Harmonic Maps. Progress in Nonlinear Differential Equations and Their Applications, 1996, Boston: Birkhäuser

[22] Xin Y. L.. Minimal Submanifolds and Related Topics, 2003, Singapore: World Scientific Publication