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The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ

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Abstract

In this paper, we discuss graphs over a domain Ω ⊂ N2 in the product manifold N2×ℝ. Here N2 is a complete Riemannian surface and Ω has piecewise smooth boundary. Let γ ⊂ ∂Ω be a smooth connected arc and Σ be a complete graph in N2×ℝ over Ω. We show that if Σ is a minimal or translating graph, then γ is a geodesic in N2. Moreover if Σ is a CMC graph, then γ has constant principal curvature in N2. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and constant mean curvature (CMC) graphs in N2×ℝ.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11261378 and 11521101). The author is very grateful to the encouragement from Professor Lixin Liu. The author also thanks the referees for careful readings and helpful suggestions.

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  1. Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

    Hengyu Zhou

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  1. Hengyu Zhou
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Correspondence to Hengyu Zhou.

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Zhou, H. The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ. Sci. China Math. 62, 585–596 (2019). https://doi.org/10.1007/s11425-017-9159-8

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