The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×R
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摘要:
In this paper,we discuss graphs over a domain Ω (C) N2 in the product manifold N2×R.Here N2 is a complete Riemannian surface and Ω has piecewise smooth boundary.Let γ (C) (e)Ω be a smooth connected arc and ∑ be a complete graph in N2×R over Ω.We show that if ∑ is a minimal or translating graph,then γ is a geodesic in N2.Moreover if ∑ 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.