Global pinching theorems for minimal submanifolds in spheres

• Autorzy: Kairen Cai
• Języki publikacji: EN

Abstrakty

EN

Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n+p}(1)$. By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and $||σ||_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if $||σ||_{n/2} < C$, then M is a minimal submanifold in the sphere $S^{n+p-1}(1+H²)$ with sectional curvature 1+H².

Kategorie tematyczne

• 53C20: Global Riemannian geometry, including pinching
• 53C40: Global submanifolds

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2003
• Tom: 96
• Numer: 2
• Strony: 225-234

Daty

wydano

2003

Twórcy

autor

Kairen Cai

• Department of Mathematics, Hangzhou Teachers' College, 96 Wen Yi Road, Hangzhou 310036, P.R. China

Identyfikatory

DOI

10.4064/cm96-2-7