Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1525-1536.

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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   

  1. 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;
  • Supported by:
    the NSFC(11801496);the NSFC(11926352);the Fok Ying-Tung Education Foundation (China);the Hubei Key Laboratory of Applied Mathematics (Hubei University)


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