Mathematical Research Letters

Volume 19 (2012)

Number 3

On Yau rigidity theorem for minimal submanifolds in spheres

Pages: 511 – 523

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n3.a1

Authors

Juan-Ru Gu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Hong-Wei Xu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

In this paper, we investigate the well-known Yau rigidity theorem forminimal submanifolds in spheres. Using the parameter method of Yauand the DDVV inequality verified by Lu and Ge–Tang, we prove that if$M$ is an $n$-dimensional oriented compact minimal submanifold inthe unit sphere $S^{n+p}$, and if $K_{M}\geq\frac{p\cdot sgn(p-1)}{2(p+1)},$ then $M$ is either a totally geodesic sphere,one of the Clifford minimal hypersurfaces in $S^{n+1}$, or theVeronese surface in $S^4$. Here ${\rm sgn}(\cdot)$ is the standard signfunction. We also extend the rigidity theorem above to the casewhere $M$ is a compact submanifold with parallel mean curvature in aspace form.

Keywords

minimal submanifold, Yau rigidity theorem, sectional curvature, mean curvature

2010 Mathematics Subject Classification

53C24, 53C40, 53C42

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