In this paper,we discuss graphs over a domain ??N^2 in the product manifold N^2×R.Here N^2is a complete Riemannian surface and?has piecewise smooth boundary.Letγ???be a smooth connected arc andΣbe a complete gr...In this paper,we discuss graphs over a domain ??N^2 in the product manifold N^2×R.Here N^2is a complete Riemannian surface and?has piecewise smooth boundary.Letγ???be a smooth connected arc andΣbe a complete graph in N^2×R over?.We show that ifΣis a minimal or translating graph,thenγis a geodesic in N^2.Moreover ifΣis a CMC graph,thenγhas constant principal curvature in N^2.This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and constant mean curvature(CMC)graphs in N^2×R.展开更多
In this paper we continue to study the connection among the area minimizing problem,certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivatio...In this paper we continue to study the connection among the area minimizing problem,certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from[15].These cones are certain generalizations of hyperbolic spaces.We describe the structure of area minimizing n-integer multiplicity currents in bounded C^2 conformal cones with prescribed C^1 graphical boundary via a minimizing problem of these area functionals.As an application we solve the corresponding Dirichlet problem of minimal surface equations under a mean convex type assumption.We also extend the existence and uniqueness of a local area minimizing integer multiplicity current with star-shaped infinity boundary in hyperbolic spaces into a large class of complete conformal manifolds.展开更多
基金supported by the Open Fund of the State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation(Southwest Petroleum University,PLN2021-17)the Science and Technology Project of Southwest Petroleum University(2021JBGS08)Sichuan Science and Technology Program(2022YFSY0040)。
基金supported by National Natural Science Foundation of China (Grant Nos. 11261378 and 11521101)
文摘In this paper,we discuss graphs over a domain ??N^2 in the product manifold N^2×R.Here N^2is a complete Riemannian surface and?has piecewise smooth boundary.Letγ???be a smooth connected arc andΣbe a complete graph in N^2×R over?.We show that ifΣis a minimal or translating graph,thenγis a geodesic in N^2.Moreover ifΣis a CMC graph,thenγhas constant principal curvature in N^2.This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and constant mean curvature(CMC)graphs in N^2×R.
基金supported by National Natural Science Foundation of China(Grant No.11771456).supported by National Natural Science Foundation of China(Grant No.11801046)the Fundamental Research Funds for the Central Universities of China(Grant No.2019CDXYST0015)。
文摘In this paper we continue to study the connection among the area minimizing problem,certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from[15].These cones are certain generalizations of hyperbolic spaces.We describe the structure of area minimizing n-integer multiplicity currents in bounded C^2 conformal cones with prescribed C^1 graphical boundary via a minimizing problem of these area functionals.As an application we solve the corresponding Dirichlet problem of minimal surface equations under a mean convex type assumption.We also extend the existence and uniqueness of a local area minimizing integer multiplicity current with star-shaped infinity boundary in hyperbolic spaces into a large class of complete conformal manifolds.
