The area minimizing problem in conformal cones, II
- PDF / 388,525 Bytes
- 30 Pages / 612 x 792 pts (letter) Page_size
- 91 Downloads / 192 Views
. ARTICLES .
https://doi.org/10.1007/s11425-020-1792-3
The area minimizing problem in conformal cones, II Qiang Gao1 & Hengyu Zhou2,3,∗ 1
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China; of Mathematics and Statistics, Chongqing University, Chongqing 401331, China; 3Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Chongqing 401331, China
2College
Email: [email protected], [email protected] Received July 16, 2020; accepted September 30, 2020
Abstract
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. Keywords MSC(2010)
area minimizing problem, conformal cones, mean curvature equation 49Q20, 53C21, 53A10, 35A01, 35J25
Citation: Gao Q, Zhou H Y. The area minimizing problem in conformal cones, II. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-020-1792-3
1
Introduction
In this paper we continue to study the area minimizing problem with prescribed boundary in a class of conformal cones similar to [15]. A conformal cone in this paper is defined as follows. Definition 1.1. Let N be an n-dimensional open Riemannian manifold with a metric σ, R be the real line with the metric dr2 and ϕ(x) be a C 2 positive function on N . In this paper we call Mϕ := {N × R, ϕ2 (x)(σ + dr2 )}
(1.1)
as a conformal product manifold. Let Ω be a C 2 bounded domain with compact closure in N . We refer Ω × R in Mϕ as a conformal cone, denoted by Qϕ . Let ψ(x) be a C 1 function on ∂Ω and Γ be its graph in ∂Ω × R. The area minimizing problem in a ¯ ϕ , the closure of Qϕ , to realize conformal cone Qϕ is to find an n-integer multiplicity current in Q min{M(T ) | T ∈ G, ∂T = Γ},
(1.2)
* Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
2
Gao Q et al.
Sci China Math
where M is the mass of integer multiplicity currents in Mϕ , and G denotes the set of n-integer multiplicity ¯ ϕ , i.e., for any T ∈ G, its support spt(T ) is contained in Ω ¯ × [a, b] for currents with compact support in Q some finite numbers a < b (see Subsection 4.1 for more details). The main reason to study the conformal product manifold Mϕ in Definition 1.1 is that the hyperbolic space is a special case of Mϕ (see Remark 6.5). Wi
Data Loading...
