Curvature estimates for minimal submanifolds of higher codimension

Abstract

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved.

Keywords

Curvature estimates / Minimal submanifold / Bernstein type theorem

References

[1] Bombieri E., De Giorgi E., Guusti E.. Minimal cones and the Bernstein problem. Invent. Math., 1969, 7(3): 243-268

[2] Bernstein S.. Sur un théorème de géométrie et ses application aux équations aux dérivés partielles du type elliptique. Comm. de la Soc. Math. de Kharkov (2éme Sér.), 1915, 15: 38-45

[3] Cheng S. Y., Li P., Yau S. T.. Heat equations on minimal submanifolds and their applications. Amer. J. Math., 1984, 106(5): 1033-1065

[4] do Carmo M., Peng C. K.. Stable complete minimal surfaces in ℝ³ are planes. Bull. Amer. Math. Soc., 1979, 1(6): 903-906

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

[6] Fischer-Colbrie D., Schoen R.. The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Comm. Pure Appl. Math., 1980, 33(2): 199-211

[7] Fujimoto H.. Modified defect relations for Gauss map of minimal surfaces. J. Diff. Geom., 1989, 29(2): 245-262

[8] Fischer-Colbrie D.. Some rigidity theorems for minimal submanifolds of the sphere. Acta Math., 1980, 145(1): 29-46

[9] Ecker K., Huisken G.. A Bernstein result for minimal graphs of controlled growth. J. Diff. Geom., 1990, 31(2): 397-400

[10] Heinz E.. Über die lösungen der minimalflächengleichung. Nachr. Akad. Wiss. Göttingen Math. Phys., 1952, K1II: 51-56

[11] Hildebrandt S., Jost J., Widman K. O.. Harmonic mappings and minimal submanifolds. Invent. Math., 1980, 62(2): 269-298

[12] Jost J., Xin Y. L.. Bernstein type theorems for higher codimension. Calc. Var. PDE, 1999, 9(4): 277-296

[13] Lawson H. B., Osserman R.. Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math., 1977, 139(1): 1-17

[14] Li A. M., Li J. M.. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math., 1992, 58(6): 582-594

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

[16] Ni L.. Gap theorems for minimal submanifolds in ℝ ⁿ⁺¹. Comm. Analy. Geom., 2001, 9(3): 641-656

[17] Osserman R.. A Survey of Minimal Surfaces, 1969, New York: Van Nostrand Reinhold

[18] Ruh E. A., Vilms J.. The tension field of Gauss map. Trans. Amer. Math. Soc., 1970, 149(2): 569-573

[19] Schoen R., Simon L., Yau S. T.. Curvature estimates for minimal hypersurfaces. Acta Math., 1975, 134(1): 275-288

[20] Simons J.. Minimal varieties in Riemannian manifolds. Ann. of Math., 1968, 88(1): 62-105

[21] Smoczyk K., Wang G. F., Xin Y. L.. Bernstein type theorems with flat normal bundle. Calc. Var. PDE, 2006, 26(1): 57-67

[22] Solomon B.. On the Gauss map of an area-minimizing hypersurface. J. Diff. Geom., 1984, 19(1): 221-232

[23] Wong Y. C.. Differential geometry of Grassmann manifolds. Proc. NAS, 1967, 57(3): 589-594

[24] Xavier F.. The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere. Ann. of Math., 1981, 113(1): 211-214

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

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

[27] Xin Y. L.. Bernstein type theorems without graphic condition. Asia J. Math., 2005, 9(1): 31-44

[28] Xin Y. L.. Mean curvature flow with convex Gauss image. Chin. Ann. Math., 2008, 29B(2): 121-134