M. Magdalena Rodriguez
We describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfaces and the singly periodic Scherk minimal surface of angle $π/2$.
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We give a proof of the classical Schwarz reflection principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(κ, τ)$ for $κ\leqslant 0$ and $τ\geqslant 0$. In our previous paper we proved a reflection principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$.
The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to $+\infty$ or to $-\infty.$ We give a survey on the development of Jenkins-Serrin type problems over domains on Riemannian manifolds. The existence of this type of graphs imposes restrictions on the geometry of the boundary of these domains. We also improve some earlier results by proving Theorem 1.8, and prove the existence of translating Jenkins-Serrin gr…
In this paper we find functions over bounded domains in the 2-dimensional Euclidean space, whose graphs (in the Heisenberg space) has constant mean curvature different from zero and taking on (possibly) infinite boundary values over the boundary of the domain.
The Jenkins-Serrin problem in 3-manifolds with a Killing vector field
本文研究具有 Killing 向量场的 3-流形上的 Jenkins-Serrin 问题，给出解存在的充要条件，并发展散度线理论及最大值原理，最终在欧氏空间和齐次 3-流形中构造新最小曲面。
Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an ingredient in the proof of the spacetime positive mass theorem given by L.-H. Huang, D. Lee, R. Schoen and the first author. We show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to …
We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds
We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.
A version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol$_3$ is proved. Moreover, we construct examples of admissible domains where the results may be applied.
In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that use is to solve a Jenkins-Serrin problem over a strip domain and to consider the conjugate minimal surface to the graph.
We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions defined over bounded domains of the hyperbolic plane.
We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over domains of $M$ bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in $M\times R$ with the conformal structure of $C$.
We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.
In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.
It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a $\mathscr{C}^2$ bounded domain $Ω$ in $M$ and a function $ H = H (x, z) $ continuous in $\overlineΩ\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the strong Serrin condition $(n-1)\mathcal…
Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain Jenkins-Serrin graphs over ideal polygonal domains (with total intrinsic curvature 2 pi(1-k)); we also get properly embedded minimal surfaces which are symmetric with respect to a horizontal slice and have total intrinsic curvature 4 pi(1-k), genus zero and k vertical planar ends.
An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow
本文提出了一种适用于非均凸域的极小曲面方程可解性准则，并将其应用于均曲率流，解决了传统方法无法处理的某些情况。
In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins-Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in $\rm Sol_3$.
In this note we discuss graphs over a domain $Ω\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $Ω$ has peice-wise smooth boundary. Let $γ\subset\partialΩ$ be a smooth connected arc and $Σ$ be a complete graph in $N^2\times \mathbb{R}$ over $Ω$. We show that if $Σ$ is a minimal or translating graph, then $γ$ is a geodesic in $N^2$. Moreover if $Σ$ is a CMC graph, then $γ$ has constant principle curvature in $N^2$. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and CMC graphs …
We study the problem of finding a minimal graph with prescribed boundary data in arbitrary dimension and codimension. Existence, uniqueness, stability and regularity are treated. We first present the well-known results for codimension one: Jenkins-Serrin's existence theorem, convexity properties of the area which give uniqueness and stability and De Giorgi's theorem for regularity. In higher codimension we first discuss the counterexamples of Lawson and Osserman. Then we present the recent results of Mu-Tao Wang: we use the mean curvature flow to show existence for small boundary data and we p…
Existence and non-existence of minimal graphs
本文通过平均曲率流研究任意维数余维数下极小曲面系统的Dirichlet问题，证明了平均凸有界域上极小图的存在性，推广了Jenkins-Serrin判据，并构造了余维数2不可解的反例，还探讨了唯一性。
In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between solutions of infinite boundary value problems for minimal surfaces and those of lightlike line boundary problems for maximal surfaces in the Lorentz-Minkowski spacetime. We also investigate some symmetry relations associated with the above correspondence together with their conju…
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