arXiv research

Solves complex Plateau problem in higher dimensions using numerical invariants.

The paper flattens a non-degenerate CR singular point in complex space.

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.

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…

Spherical Plateau problem studies minimal surfaces in quotients of spheres.

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.

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…

Solves area minimizing surface problem in metric spaces with bounded genus.

Solutions found for specific curvature problems in geometric settings.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

Theoretical guarantees for permutation-equivariant QNNs avoid barren plateaus.

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 regularit…

We prove that any finite dimensional Alexandrov space with a lower curvature bound is locally Lipschitz contractible. As applications, we obtain a sufficient condition for solving the Plateau problem in an Alexandrov space considered by Mese and Zulkowski.

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…

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

Solves Plateau's problem for curves with self-intersections.

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

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

Solves Plateau problem for surfaces in pinched curvature manifolds.

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

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

Solves Plateau's Problem in Heisenberg group for graphs.

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

Paper finds invariant solutions for Plateau problem in hyperbolic space.

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…

The paper proves existence of minimal homotopies for immersed planar curves.

This research solves Plateau's problem for CRPC surfaces.

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

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 proves uniqueness of weak solutions for Plateau flow.

In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type mi…

New energy model avoids self-intersections in curve optimization.

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

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

Generalizes embeddedness result for extreme curves.

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.

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.

Unique minimal surfaces near quadratic cones are identified.

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 …

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…

Paper estimates curvature of semi-convex hypersurfaces in hyperbolic 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 exis…

The paper solves a partial Plateau problem using $H$-mass.

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 paper proves existence and partial regularity for Legendrian area-minimizing currents.

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…