極小超曲面上的曲率估計

A Note on Curvature Estimates for Minimal Hypersurfaces

李邱勇(Chiu-Yung Li)

指導教授 ： 宋瓊珠

國立清華大學/理學院/數學系/碩士(2011年)

https://doi.org/10.6843/NTHU.2011.00704

摘要

In this note, we use Simons' identity for the Laplacian of the second fundamental form
for minimal hypersurfaces to obtain a number of new estimates for the curvatures of stable
minimal hypersurfaces M which are immersed in a Riemannian manifold N. Under suitable
assumptions on N, Schoen, Simon and Yau found a pointwise bound for the principal cur-
vatures of M, provided dim(M) 6 5. In this case, this pointwise bound implies Bernstein's
theorem for n 6 5. Schoen, Simon and Yau also gave a simplied proof of Simons' result: no
non-trivial 6-dimensional stable minimal cones in R7 implies Bernstein's theorem holds for
n = 6.

關鍵字

超曲面

並列摘要

無資料

並列關鍵字

hypersurface ； second fundamental form

參考文獻

[1] Almgren, F. J., Some interior regularity theorems for minimal surfaces and an extension of

Bernstein's theorem, Ann. of Math. 84 (1966) 277-292.

[2] Bernstein S., Sur un theoreme de geometrie et ses applications aux equations aux derivees

[3] Bombieri, E., De Giorgi E., and Guisti, Minimal cones and the Bernstein problem, Invent.

[4] De Giorge, E., Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19