arXiv research

Minimal surfaces' boundary points are always smooth.

Proves smoothness of minimal surfaces near polyhedral boundaries.

The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.

Study improves boundary smoothness for area-minimizing currents with complex boundaries.

Proves optimal regularity for sphere minimizers in 3-sphere.

Characterizes minimizing curves in Riemannian manifolds.

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solution…

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…

Explains the history and challenges of minimal surfaces.

The paper proves properties of curves in Riemannian manifolds.

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…

In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized $vector$ minimization problem are equiva…

Minimal partitions with minimal perimeter found in metric spaces.

The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.

A theorem simplifies mass-minimizing flat chains' regularity.

Examples of area-minimizing graphs with low regularity in a specific group.

Study on high-codimensional minimal surfaces in hyperbolic space.

New geometric insights reveal properties of adversarial training problems.

ERM with $f$-divergence regularization yields unique solution.

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…

New regularization method reduces support of empirical risk minimization solutions.

Study geodesics in sub-Riemannian manifolds, resolving open questions.

Diagonal linear networks converge to lasso regularization path during training.

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $Ω$. We prove existence, regularity and some structural properties of minimizers. In particular, when $Ω$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a…

Optimal regularity theory for stable minimal hypersurfaces with small singular set.

Study uses property elicitation to understand how fairness regularizers affect optimal decisions.

Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledg…

In this work, we study data preconditioning, a well-known and long-existing technique, for boosting the convergence of first-order methods for regularized loss minimization. It is well understood that the condition number of the problem, i.e., the ratio of the Lipschitz constant to the strong convexity modulus, has a h…

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.

An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2π/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been for…

Proves a function's locally least gradient property if its level sets are minimal laminations.

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

Geometric proof shows regularity of anisotropic minimal surfaces in 2D.

The study shows that certain graphs are regular at boundary points.

Improved optimal regularity for harmonic almost complex structures.

Optimizes biharmonic map regularity using stratification methods.

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.

AMP regularization improves deep learning models by favoring flat minima.

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

Analyzes surfaces minimizing mean curvature variation using PDEs.

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…

We show $C^{1,α}$-regularity for energy minimizing maps from a 2-dimensional Riemannian manifold into a Finsler space $(\R^n, F)$ with a Finsler structure $F(u,X)$.

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear e…

Develops theory for stable capillary minimal hypersurfaces in half-space.