Taiwanese Journal of Mathematics

SUBMANIFOLDS OF CONSTANT SCALAR CURVATURE IN A HYPERBOLIC SPACE FORM

Zhong Hua Hou

Full-text: Open access

Abstract

Let $M^n$ be a closed submanifold immersed into a real hyperbolic space form $\Bbb H^{n+p}$ of constant curvature $-1$. Denote by $R$ the normalized scalar curvature of $M^n$ and by $H$ the mean curvature of $M^n$. Suppose that $R$ is constant and bigger than or equal to $-1$. We first extend Cheng-Yau's technique to higher codimensional cases. Then, for $M^n$ with parallel normalized mean curvature vector field, we show that, if $H$ satisfies a certain inequality, then $M^n$ is totally umbilical or the equality part holds. We describe all $M^n$ whose $H$ satisfies this equality.

Article information

Source
Taiwanese J. Math., Volume 3, Number 1 (1999), 55-70.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407054

Digital Object Identifier
doi:10.11650/twjm/1500407054

Mathematical Reviews number (MathSciNet)
MR1676022

Zentralblatt MATH identifier
0972.53035

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Keywords
normalized mean curvature vector field normalized scalar curvature

Citation

Hou, Zhong Hua. SUBMANIFOLDS OF CONSTANT SCALAR CURVATURE IN A HYPERBOLIC SPACE FORM. Taiwanese J. Math. 3 (1999), no. 1, 55--70. doi:10.11650/twjm/1500407054. https://projecteuclid.org/euclid.twjm/1500407054


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