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

Abstract

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

CLC Number:

O186.1

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.

Trendmd

References 9

1	Abresch U , Rosenberg H . A Hopf differential for constant mean curvature surfaces in ${\Bbb S}.{2}\times{\Bbb R}$ and ${\Bbb H}.{2}\times{\Bbb R}$. Acta Math, 2004, 193: 141- 174
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
8	Rosenberg H , Schulze F , Spruck J . The half-space property and entire positive minimal graphs in $M\times{\Bbb R}$. J Differential Geom, 2013, 95: 321- 336
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

[1]	Gui Bao,Qian Sun. Existence and Multiplicity of Sign-Changing Solutions for a Bi-Harmonic Equation with Sublinear Nonlinearity [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1186-1191.
[2]	Miaokun Wang,Zaiyin He,Yuming Chu. Concavity of the Complete p-Elliptic Integral of the Second Kind According to Hölder Mean [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1269-1281.
[3]	Miaokun Wang,Yuming Chu,Songliang Qiu. The Simple Proof and Generalization of a Conjecture Concerning Generalized Legendre Identity [J]. Acta mathematica scientia,Series A, 2020, 40(3): 662-666.
[4]	Yue He,Hailong Her. An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds [J]. Acta mathematica scientia,Series A, 2020, 40(2): 257-270.
[5]	Yin Li, Huang Liguo. Two Inequalities Related to Generalized Elliptic Integrals [J]. Acta mathematica scientia,Series A, 2018, 38(1): 46-53.
[6]	Qi Yi, Shi Qingtian. Estimations of Convexity Radius and the Norm of Pre-Schwarz Derivative for Close to Convex Harmonic Mappings [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1057-1066.
[7]	He Yue. New Estimates of Lower Bound for the First Eigenvalue on Compact Manifolds with Positive Ricci Curvature [J]. Acta mathematica scientia,Series A, 2016, 36(2): 215-230.
[8]	Song Hongxue, Yan Qinglun. Multiple Solutions for Quasilinear Elliptic Exterior Problem with Neumann Boundary Conditions [J]. Acta mathematica scientia,Series A, 2015, 35(5): 845-854.
[9]	Xiang Ni, Shi Juhua, Wang Yu'e. The Neumann Boundary Value Problems of Two Dimensional Monge-Ampère Equations [J]. Acta mathematica scientia,Series A, 2015, 35(4): 794-802.
[10]	ZUO Zhan-Fei. Normal Structure and the Generalized Modulus of U-convexity in Banach Spaces [J]. Acta mathematica scientia,Series A, 2013, 33(2): 327-332.
[11]	CHU Yu-Ming, ZHANG Xiao-Ming, SHI Huan-Nan. Gautschi-type Inequalities and Their Applications [J]. Acta mathematica scientia,Series A, 2012, 32(4): 698-708.
[12]	QIAO Jin-Jing, WANG Xian-Tao. On p-harmonic Univalent Mappings [J]. Acta mathematica scientia,Series A, 2012, 32(3): 588-600.
[13]	XU Jin-Ju. A Remark on the Convexity of Hypersurface with Prescribed Curvature Equations [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1176-1180.
[14]	RAO Ro-Feng. New Composite Implicit Iterative Scheme With Errors [J]. Acta mathematica scientia,Series A, 2009, 29(3): 823-831.
[15]	Wang Peihe;Shen Chunli. A Compensatory Phenomenon to the Classical Sphere Theorem [J]. Acta mathematica scientia,Series A, 2008, 28(3): 465-471.

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 9

Related Articles 15

Metrics

Comments

Recommended 10