Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the υ-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the υ-function. Under such restrictions, curvature estimates in terms of υ-function composed with the Gauss map can be carried out.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Chen, J. Y. and Li, J. Y., Mean curvature flow of surfaces in 4-manifolds, Adv. Math., 163, 2001, 287–309.
Chen, J. Y. and Li, J. Y., Singularity of mean curvature flow of Lagrangian submanifolds, Invent. Math., 156(1), 2004, 25–51.
Chen, J. Y. and Tian, G., Moving symplectic curves in Kähler-Einstein surfaces, Acta Math. Sin. (Engl. Ser.), 16, 2000, 541–548.
Ecker, K. and Huisken, G., Mean curvature evolution of entire graphs, Ann. of Math. (2), 130(3), 1989, 453–471.
Ecker, K. and Huisken, G., Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105, 1991, 547–569.
Jost, J. and Xin, Y. L., Bernstein type theorems for higher codimension, Calc. Var. Part. Diff. Eqs., 9, 1999, 277–296.
Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 20(1), 1984, 237–266.
Huisken, G., Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31(1), 1990, 285–299.
Smoczyk, K., Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Part. Diff. Eqs., 8, 1999, 247–258.
Smoczyk, K., Angle theorems for Lagrangian mean curvature flow, Math. Z., 240, 2002, 849–863.
Smoczyk, K. and Wang, M.-T., Mean curvature flows for Lagrangian submanifolds with convex potentials, J. Diff. Geom., 62, 2002, 243–257.
Wang, M.-T., Gauss maps of the mean curvature flow, Math. Res. Lett., 10, 2003, 287–299.
Wong, Y.-C., Differential geometry of Grassmann manifolds, Proc. Natl. Acad. Sci. USA, 57, 1967, 589–594.
Xin, Y. L., Minimal Submanifolds and Related Topics, World Scientific Publishing, Singapore, 2003.
Xin, Y. L., Geometry of Harmonic Maps, Progress in Nonlinear Differential Equations and Their Applications, Vol. 23, Birkhäuser, Boston, 1996.
Xin, Y. L., Mean curvature flow with convex Gauss image, Chin. Ann. Math., 29B(2), 2008, 121–134.
Xin, Y. L., Curvature estimates for submanifolds with prescribed Gauss image and mean curvature, Calc. Var. Part. Diff. Eqs., 37, 2010, 385–405.
Xin, Y. L. and Yang, L., Convex functions on Grassmannian manifolds and Lawson-Osserman problem, Adv. Math., 219, 2008, 1298–1326.
Xin, Y. L. and Yang, L., Curvature estimates for minimal submanifolds of higher codimension, Chin. Ann. Math., 30B(4), 2009, 379–396.
Project supported by the National Natural Science Foundation of China and the Science Foundation of the Ministry of Education of China.
About this article
Cite this article
Xin, Y., Yang, L. Mean curvature flow via convex functions on Grassmannian manifolds. Chin. Ann. Math. Ser. B 31, 315–328 (2010). https://doi.org/10.1007/s11401-009-0173-7
- Mean curvature flow
- Convex function
- Gauss map
2000 MR Subject Classification