乘积流形$M^{n}\times{\Bbb R}$中一类常平均曲率方程解的存在性和唯一性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1525-1536.

乘积流形 $M^{n}\times{\Bbb R}$ 中一类常平均曲率方程解的存在性和唯一性

高雅(),毛井*(),宋春兰

湖北大学数学与统计学学院 & 应用数学湖北省重点实验室武汉 430062

收稿日期:2020-01-31 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 毛井 E-mail:Echo-gaoya@outlook.com;jiner120@163.com
作者简介:高雅, E-mail:Echo-gaoya@outlook.com
基金资助:
国家自然科学基金(11801496);国家自然科学基金(11926352);霍英东教育基金会青年教师基金;应用数学湖北省重点实验室基金

Existence and Uniqueness of Solutions to the Constant Mean Curvature Equation with Nonzero Neumann Boundary Data in Product Manifold $M^{n}\times{\Bbb R}$

Ya Gao(),Jing Mao*(),Chunlan Song

Faculty of Mathematics and Statistics & Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan 430062

Received:2020-01-31 Online:2020-12-26 Published:2020-12-29
Contact: Jing Mao E-mail:Echo-gaoya@outlook.com;jiner120@163.com
Supported by:
the NSFC(11801496);the NSFC(11926352);the Fok Ying-Tung Education Foundation (China);the Hubei Key Laboratory of Applied Mathematics (Hubei University)

摘要/Abstract

摘要：

该文证明了乘积流形 $M^{n}\times{\Bbb R}$ 中具有非零Neumann边值条件的常平均曲率方程解的存在性和唯一性, 这里 $M^{n}$ 是Ricci曲率非负的 $n$ 维完备黎曼流形, $n\geq2$ , ${\Bbb R}$ 是1维的欧氏空间.等价地, 这个结论给出了定义在紧致严格凸域 $\Omega\subset M^{n}$ 上的具有非退化Neumann边值条件的常平均曲率图超曲面的存在性.

关键词: 常平均曲率, Neumann边值条件, 凸性, Ricci曲率, 乘积流形

Abstract:

In this paper, we can prove the existence and uniqueness of solutions to the constant mean curvature (CMC for short) equation with nonzero Neumann boundary data in product manifold $M^{n}\times{\Bbb R}$ , where $M^{n}$ is an $n$ -dimensional $(n\geq2)$ complete Riemannian manifold with nonnegative Ricci curvature, and ${\Bbb R}$ is the Euclidean 1-space. Equivalently, this conclusion gives the existence of CMC graphic hypersurfaces defined over a compact strictly convex domain $\Omega\subset M^{n}$ and with nonzero Neumann boundary data.

Key words: Constant mean curvature, Neumann boundary condition, Convexity, Ricci curvature, Product manifold

中图分类号:

O186.1

高雅,毛井,宋春兰. 乘积流形 $M^{n}\times{\Bbb R}$ 中一类常平均曲率方程解的存在性和唯一性[J]. 数学物理学报, 2020, 40(6): 1525-1536.

Ya Gao,Jing Mao,Chunlan Song. Existence and Uniqueness of Solutions to the Constant Mean Curvature Equation with Nonzero Neumann Boundary Data in Product Manifold $M^{n}\times{\Bbb R}$ [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1525-1536.

参考文献 9

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2	Alschuler S J , Wu L F . Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc Var Partial Differential Equations, 1994, 2: 101- 111
3	Gao Y, Gong Y J, Mao J. Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold ${\Bbb M}.{n}\times{\Bbb R}$ . 2020, arXiv: 2001.09860
4	Ma X N , Wang P H , Wei W . Constant mean curvature surfaces and mean curvature flow with nonzero Neumann boundary conditions on strictly convex domains. J Funct Anal, 2018, 274: 252- 277
5	Ma X N , Xu J J . Gradient estimates of mean curvature equations with Neumann boundary condition. Adv Math, 2016, 290: 1010- 1039
6	Mazet L , Rodríguez M M , Rosenberg H . Periodic constant mean curvature surfaces in ${\Bbb H}.{2}\times{\Bbb R}$ . Asian J Math, 2014, 18: 829- 858
7	Meeks W H , Rosenberg H . Stable minimal surfaces in $M\times{\Bbb R}$ . J Differential Geom, 2004, 68: 515- 534
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9	Wang J , Wei W , Xu J J . Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. Commun Pure Appl Anal, 2019, 18 (6): 3243- 3265

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