Taiwanese Journal of Mathematics

A Classification Theorem for Complete PMC Surfaces with Non-negative Gaussian Curvature in $M^n(c) \times \mathbb{R}$

Zhong Hua Hou and Wang Hua Qiu

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Abstract

Let $M^{n}(c)$ be an $n$-dimensional space form with constant sectional curvature $c$. Alencar-do Carmo-Tribuzy [5] classified all parallel mean curvature (abbrev. PMC) surfaces with non-negative Gaussian curvature $K$ in $M^n(c) \times \mathbb{R}$ with $c \lt 0$. Later on, Fetcu-Rosenberg [28] generalized their results for $c \neq 0$. However, the classification to PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ is still open. In this paper, we give a complete classification to the PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ whose tangent plane spans the constant angle with factor $\mathbb{R}$.

Article information

Source
Taiwanese J. Math., Volume 20, Number 1 (2016), 205-226.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874429

Digital Object Identifier
doi:10.11650/tjm.20.2016.5766

Mathematical Reviews number (MathSciNet)
MR3462875

Zentralblatt MATH identifier
1357.53067

Subjects
Primary: 53B25: Local submanifolds [See also 53C40] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
non-negative Gaussian curvature parallel mean curvature product spaces

Citation

Hou, Zhong Hua; Qiu, Wang Hua. A Classification Theorem for Complete PMC Surfaces with Non-negative Gaussian Curvature in $M^n(c) \times \mathbb{R}$. Taiwanese J. Math. 20 (2016), no. 1, 205--226. doi:10.11650/tjm.20.2016.5766. https://projecteuclid.org/euclid.twjm/1498874429


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