arXiv research

Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.

Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.

Solves area-minimizing surface problem for finite curves in H^2xR.

Solves Plateau problem for surfaces in pinched curvature manifolds.

Paper finds invariant solutions for Plateau problem in hyperbolic space.

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.

Paper estimates curvature of semi-convex 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.

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.

Smooth solutions found for a curvature problem in hyperbolic space.

Geodesic lines with specific boundaries found on a special type of manifold.

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 s…

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

Study non-fillable curves in a hyperbolic surface with a real line.

Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.

Study of constant curvature hypersurfaces in hyperbolic space.

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 th…

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perime…

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…

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 …

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 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 hyper…

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 ${\wideti…

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 …

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 study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

Study counts surface subgroups in curved 3D manifolds.

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.

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 …

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) …

Study approximates Plateau's laws using the Allen-Cahn equation.

Unique solutions found for Plateau problems in smooth and continuous calibrations.

Spherical Plateau problem studies minimal surfaces in quotients of spheres.

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never …

Solves Plateau's Problem in Heisenberg group for graphs.

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…

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …

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 …

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^…

This research solves Plateau's problem for CRPC surfaces.

Paper proves uniqueness of weak solutions for Plateau flow.

Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.

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 L…

New energy model avoids self-intersections in curve optimization.

Generalizes embeddedness result for extreme curves.

Gradient-free optimizers are ineffective on barren plateaus in quantum computing.

Following on from ``Hyperbolic Plateau problems'' (by the same author), we provide a complete geometric description of solutions to the Plateau problem $(S,φ)$ when $S$ is a compact Riemann surface with a finite number of points removed.

Paper bounds surface diameter and solves Plateau-Douglas problem.