Contents Online
Mathematical Research Letters
Volume 19 (2012)
Number 3
On Yau rigidity theorem for minimal submanifolds in spheres
Pages: 511 – 523
DOI: https://dx.doi.org/10.4310/MRL.2012.v19.n3.a1
Authors
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
Published 8 November 2012