Baris Coskunuzer
Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.
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We give existence and nonuniqueness results for simple planar curves with prescribed geodesic curvature.
We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyperbolic n-space, is dense in the space of all codimension-1 closed submanifolds at infinity. In dimension 3, we also prove that the set of uniqueness curves in asymptotic sphere for area minimizing planes is generic in the set of Jordan curves at infinity. We also give some nonuniq…
We construct a family of inequivalent Calabi-Yau metrics on $\mathbf{C}^3$ asymptotic to $\mathbf{C} \times A_2$ at infinity, in the sense that any two of these metrics cannot be related by a scaling and a biholomorphism. This provides the first example of families of Calabi-Yau metrics asymptotic to a fixed tangent cone at infinity, while keeping the underlying complex structure fixed. We propose a refinement of a conjecture of Székelyhidi addressing the classification of such metrics.
This is a survey of old and recent results about the asymptotic Plateau problem. Our aim is to give a fairly complete picture of the field, and present the current situation.
We show that if a Jordan curve C in the asymptotic sphere contains a smooth point, there is an embedded H-plane in H^3 asymptotic to C for any H in [0,1).
We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2xR. In particular, we identify the collection of Jordan curves in the asymptotic boundary of H2xR, which bounds an area minimizing surface in H2xR. Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different.
On Almost-Fuchsian submanifodls of Hadamard spaces and the asymptotic Plateau problem
本文研究了负曲率空间中具有小曲率的极小子流形，证明了在曲率负被夹的Hadamard空间中，第二基本形受限的完备极小子流形能使渐近Plateau问题对一类无穷远球面唯一可解。
For two disjoint rectifiable star-shaped Jordan curves (including round circles) in the asymptotic boundary of hyperbolic 3-space, if the distance (see Definition 1.8) between these two Jordan curves are bounded from above by some constant, then there exists an annulus-type area minimizing (or equivalently least area) minimal surface asymptotic to these two Jordan curves. The main results of this paper are Theorem 1.7 and Theorem 1.11.
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.
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。
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.
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.
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。
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.
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。
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 …
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.
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。
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。
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…
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…
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。
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。
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…
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 $…
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。
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 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。
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。
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.
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.
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。
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…
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.
Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.
We study the number of solutions of the asymptotic Plateau problem in H^3. By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of C^3 Jordan curves in S^2_{infty}(H^3) such that any curve in this set bounds a unique least area plane in H^3.
A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem
本文介绍了基于PINNs的机器学习框架在渐近普拉托问题中的应用方法，提出两项优化技术降低训练成本，为微分几何研究者提供PINNs部署的实用指南。
Hypersurfaces of constant sum Hessian curvature in Hyperbolic space
本文研究双曲空间中常和Hessian曲率的渐近Plateau问题，旨在寻找满足特定曲率方程和渐近边界条件的完备超曲面。
Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity
本文推广双曲空间经典渐近普拉托问题，通过构造常数量曲率图极限并引入新技术建立二阶先验估计，证明所有可能曲率值下存在具指定无穷远渐近边界的光滑完备常数量曲率超曲面。
In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution to the Plateau problem. But what happens if a soap film spans a loop of inextensible but flexible wire? We consider this simple query that couples Plateau's problem to Euler's Elastica: a special class of twist-free curves of given length that minimize their total squared curvatu…
Tensile minimal surfaces and thread boundary problems
针对张拉结构中由缆索围成的极小曲面问题，利用Plateau共轭方法构造了两族嵌入极小曲面（张拉悬链面和张拉k-noid），并证明其存在性与对称性，为建筑网格壳设计提供数学基础。
Nonabelian Yang-Mills-Higgs and Plateau's problem in codimension three
本文研究SU(2)-杨-米尔斯-希格斯能量在大质量极限下的渐近行为，证明其Γ-收敛到余维三面积泛函，并在广义磁单极方程场景中推导得能量测度极限与校准流的关系，为相关几何预测提供证据。
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