Asymptotic Plateau Problem

On the asymptotic Plateau's problem for CMC hypersurfaces in hyperbolic space

It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.

H^2xR中的渐近Plateau问题：高曲线

Asymptotic Plateau Problem in H^2xR: Tall Curves

本文研究$\BHH$中面积最小化曲面的渐近Plateau问题，并为有限曲线提供了相当完整的解答。

On the asymptotic Plateau problem in Cartan--Hadamard manifolds

We prove dynamical stability of a natural class of hypersurface laminations defined over Cartan--Hadamard manifolds of pinched curvature. We achieve this by providing a complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds, proposed by Labourie in [25], together with its natural higher-dimensional generalisation.

On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,τ)$

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,τ)$. As one of our main results, we present sufficient conditions for a curve $Γ$ in $\partial_{\infty} \mathbb{E}(-1,τ)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,τ)$ having $Γ$ as its asymptotic boundary.

On the asymptotic Plateau problem in hyperbolic space

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant $σ_{n-1}$ curvature, i.e. the existence of a complete hypersurface in $\mathbb{H}^{n+1}$ satisfying $σ_{n-1}(κ)=σ\in (0,n)$ with a prescribed asymptotic boundary $Γ$. The key ingredient is the curvature estimates. Previously, this is only known for $σ_0<σ<n$, where $σ_0$ is a positive constant.

$\u210D^{n+1}$中渐近Plateau问题半凸解的曲率估计

Curvature estimates for semi-convex solutions of asymptotic Plateau problem in $\mathbb{H}^{n+1}$

本文研究双曲空间中的渐近$σ_k$ Plateau问题，通过推导Hessian方程的新关键凹性不等式，建立了满足条件的半凸完备超曲面的$C^2$估计，拓展了相关曲率估计理论。

Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $k = n$ when the asymptotic boundary $Γ$ bounds a uniformly convex domain, and for $k < n$ when $Γ$ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work \cite{Sui2019, Sui-Sun}, and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.

$n$维渐近锥流形中的一维渐近平坦问题

One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds

研究$n$维渐近锥流形上一维渐近平坦问题，在指数映射为真的条件下，证明任意对跖点间存在具有给定渐近边界且Morse指标不超过$n-1$的测地线。

On the asymptotic Plateau problem in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ contained between two bounded entire minimal graphs separated by vertical distance less than $\sqrt{1+4τ^2}π$ have multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a …

双曲空间 $\mathbb{H}^3$ 中渐近 Plateau 问题的不变解

Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$

研究双曲空间 $\mathbb{H}^3$ 中渐近 Plateau 问题，核心方法是证明由特定极小曲面构成的叶状结构存在，主要贡献是给出该问题在等距变换单参数子群作用下不变曲线的解。

通过等距超平面求解渐近 Plateau 问题

Asymptotic Plateau problem via equidistant hyperplanes

本研究在 $\n\mathbb{H}^{n+1}$ 中构造出满足广泛曲率方程、具有指定渐近边界的严格局部凸完备超曲面，通过等距超平面上的测地图逼近，提出渐近 Plateau 问题解的新构造方法，其解在非唯一情形下可异于经典解。

On the asymptotic Plateau's problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symm…

Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(κ)=σ\in(0, 1)$ with a prescribed asymptotic boundary $Γ$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $Γ$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $Γ$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature…

具有预定平均曲率超曲面的渐近 Plateau 问题

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

本文证明了在满足严格凸性条件且截面曲率具有负常数上界的 Cartan-Hadamard 流形中，具有预定平均曲率超曲面的渐近 Plateau 问题解的存在性，并在低维情形下得到了多重性结果。

双曲空间中渐近Plateau问题的唯一性与非唯一性

Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space

该研究针对三维双曲空间中的渐近Plateau问题，证明了解的唯一性等价于稳定极小圆盘类中的唯一性，给出了满足唯一性的曲线条件，并构造出存在不可数多稳定极小圆盘解的拟圆周实例。

Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any…

On the asymptotic Plateau problem for CMC hypersurfaces in hyperbolic space

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $Γ\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\} $ then, given $\left\vert H\right\vert <1,$ there is a complete, properly embedded, CMC $H$ hypersurface $Σ$ of $\mathbb{H}^{n+1}$ such that $\partial_{\infty }S=Γ\cup\left\{ x_{n+1}=+\infty\right\} .$ This result can be seen as a limit case of the existence theorem proved by B. Guan and J. Spruck in \cite{GS} on CMC $…

双曲空间中常$H_{k}$曲率超曲面的渐近Plateau问题

The asymptotic Plateau problem for Hypersurfaces of constant $H_{k}$ curvature in hyperbolic space

本研究针对双曲空间中常$H_k$曲率超曲面的渐近Plateau问题，证明了满足指定曲率条件与无穷远渐近边界的光滑完备$k$-凸超曲面的存在性，将既有存在性定理中常数$σ$的取值范围拓展至完整的$(0,1)$区间。

Embeddedness of the solutions to the H-Plateau Problem

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve in the unit sphere bounds an embedded H-disk in B.

双曲空间中 $p$-凸超曲面的渐近 Plateau 问题

The Asymptotic Plateau Problem for $p$-Convex Hypersurfaces in Hyperbolic Space

本文解决了双曲空间中由主曲率 $p$ 重和的正规化几何平均定义的曲率的渐近 Plateau 问题，将已知情形推广到所有 $n\ge3$、$2\le p\le n-1$，并证明了唯一完备可容许垂直图的存在性，关键是通过一致曲率估计和两个三阶导数下界实现。

叶状Plateau问题与曲面子群的渐近计数

Foliated Plateau problems and asymptotic counting of surface subgroups

本文研究三维流形中曲面子群依$k$-曲面面积的渐近计数，确定下界并证明刚性，通过解决Cartan--Hadamard流形中的一般叶状Plateau问题克服了现有证明的局限。

Embedded Plateau Problem

We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of D. The same result is also valid for homogenously regular manifolds with sufficiently convex boundary.

Hyperbolic Plateau problems

We consider surfaces of constant Gaussian curvature immersed in 3-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is 3-dimensional hyperbolic space. This allows us to prove results of existence of solutions to the asymptotic Plateau problem, as defined by Labourie, and the continuous dependence of these solutions on the data.

Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the pseudo-hyperbolic space which are limits of positive curves. We also discuss a compact Plateau problem. The required compactness arguments rely on an analysis of the pseudo-holomorphic curves defined by the Gauss lifts of the maximal surfaces.

$3$-凸超曲面在$\mathbb{H}^5$中的渐近平台问题

Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

本文研究了5维双曲空间中$3$-凸超曲面的渐近平台问题，通过拉格朗日乘数法证明了满足特定曲率方程和渐近边界的超曲面存在性。

Minimal Surfaces in H^2xR

We give a fairly complete solution to the asymptotic Plateau Problem for minimal surfaces in H^2xR. In particular, we identify the collection of finite Jordan curves in the asymptotic cylinder which bounds a minimal surface in H^2xR.

The Perron Method and the Non-Linear Plateau Problem

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in $\Bbb{R}^{n+1}$.

The non-linear Plateau problem in non-positively curved manifolds

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.

Soap Film Solutions to Plateau's Problem

Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator to solve Plateau's problem for orientable, embedded surfaces. But integra…

The Dirichlet problem for minimal surfaces equation and Plateau problem at infinity

In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in term of lines of divergence in the domain. Using this second result, we build some solutions of the Dirichlet problem on unbounded domain. We then give a new proof of the result of C. Cos\'ın and A. Ros concerning the Plateau problem at infinity for horizontal ends.

高维仿射Bernstein问题和Plateau问题

The higher dimensional affine Bernstein and Plateau problems

本文证明了所有维数下仿射Plateau问题的内部正则性，给出了维数十及以上仿射Bernstein问题的光滑反例，并在增长条件下证明了所有维数的仿射Bernstein定理。

Complex Plateau problem in non Kähler manifolds

We consider a complex Plateau problem for strongly pseudoconvex contours in non Kähler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a conjecture.

The Plateau problem for polygonal boundary curves in Minkowski 3-space

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one …

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

二次锥附近 Plateau 问题的唯一性

Uniqueness in the Plateau problem near Quadratic cones

本文研究单位球内边界为极小化二次锥链环小扰动的极小超曲面，证明其由边界条件唯一确定，从而得到该类边界条件下 Plateau 问题解的唯一性。

Multiple Solutions to the Planar Plateau Problem

We give existence and nonuniqueness results for simple planar curves with prescribed geodesic curvature.

Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(σ_n/σ_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(κ) = σ\in (1,\infty)$ with a prescribed compact future asymptotic boundary $Γ$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly con…

The Plateau problem for convex curvature functions

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated.

Plateau's problem for singular curves

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.

Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas.

Kohn-Rossi Cohomology and its application to the Complex Plateau Problem III

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension 2n-1 in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn-Rossi cohomology groups on $X$ in 1981. For n=2 and $N\ge n+1$, the problem has been open for over 30 years. In this paper we introduce a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this …

The Plateau problem for marginally outer trapped surfaces

We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in this paper are flexible and can be adapted to other non-variational existence problems.

Regarding the Euler-Plateau Problem with Elastic Modulus

We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily pl…

Equivariant Plateau Problems

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(Σ,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $θ:π_1(Σ,P)\toπ_1(M,Q)$ be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and sufficient conditions for the existence of complex projective structures with specified holonomy to manifolds of non-constant negative curvature, we obtain necessary conditions on $θ$ for the existence of a so called $θ$-equivariant Plateau problem over $Σ$, which is equivalent t…

Smooth solution to higher dimensional complex Plateau problem

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. For $n\ge 3$, Yau solved the complex Plateau problem of hypersurface type by checking a bunch of Kohn-Rossi cohomology groups in 1981. In this paper, we generalize Yau's conjecture on some numerical invariant of every isolated surface singularity defined by Yau and the author to any dimension and prove that the conjecture is true for local complete intersection singularities of dimension $n\ge 3$. As a direct application, we solved complex Plateau problem of hypersurface type for an…

在Heisenberg群中内在图的Plateau问题

Plateau's Problem for intrinsic graphs in the Heisenberg Group

本文利用几何构造，在Heisenberg群H^1中解决内在图的Plateau问题，通过小性条件证明了边界或Lipschitz边界数据的最小性，并应用这些技术获得H-周长最小化者的新的正则性结果。

Plateau's problem in Finsler 3-space

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.

The Plateau problem at infinity for horizontal ends and genus 1

In this paper, we study Alexandrov-embedded r-noids with genus 1 and horizontal ends. Such minimal surfaces are of two types and we build several examples of the first one. We prove that if a polygon bounds an immersed polygonal disk, it is the flux polygon of an r-noid with genus 1 of the first type. We also study the case of polygons which are invariant under a rotation. The construction of these surfaces is based on the resolution of the Dirichlet problem for the minimal surface equation on an unbounded domain.

The periodic Plateau problem and its application

Given a noncompact disconnected complete periodic curve $Γ$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $Γ$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\partial T$ along their boundary. It is also proved that every Platonic solid of $\mathbb R^3$ contains five types of free boundary embedded minimal surfaces of genus zero.

Plateau 问题的层级以及通过 Allen–Cahn 方程逼近 Plateau 定律

A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation

本文提出带体积约束与张成条件的 Allen–Cahn 扩散界面 Plateau 问题，证明两类奇异极限分别收敛到毛细作用与经典 Plateau 问题，并由此说明 Plateau 型奇点可被 Allen–Cahn 极小解逼近。

Minimal Surfaces in $\mathbb{H}_2\times \mathbb{R}$: Nonfillable Curves

We study the asymptotic Plateau problem in $\mathbb{H}_2\times \mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite curves in the asymptotic boundary.

具有常主曲率比率的曲面族与Plateau问题

Families of surfaces with constant ratio of principal curvatures and Plateau's problem

本文提出构造包含给定极小曲面的常主曲率比率曲面族的方法，部分解决CRPC曲面的Plateau问题，并展示迷向几何方法及逐次逼近法的应用。

校准流的Plateau问题的唯一性

Uniqueness in the Plateau problem for calibrated currents

该研究针对校准流的有向Plateau问题，结合面积最小化流的边界正则性理论与Frank Morgan风格的唯一延拓论证，证明了具连通$C^{3,α}$边界的紧支撑光滑校准积分流的解唯一性，还推广至连续校准积分平链，并在余维1情形给出以校准形式正则性表述的唯一性充分条件。

A Multiple-Valued Plateau Problem

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$ with sum $Q$ and any collection of $K$ disjoint Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary data with these multiplicities which extends to a Dirichlet minimizing $Q$-valued function with minimal Dirichlet energy among a…