arXiv research

The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.

Flat stable minimal hypersurfaces in 5 or 6D are always flat.

Paper studies stability of curved surfaces in a half-space.

Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling…

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

Compact hypersurfaces minimize area in convex cones with free boundary.

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the…

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X:M\to \mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersu…

WS diffusion models handle anisotropic Gaussian noise better than conventional methods.

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we …

New method stabilizes private LASSO for high-dimensional data with diverse covariate scales.

SpecGD mitigates misalignment in phase retrieval models with anisotropic inputs.

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this …

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…

Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for which these hypersurfaces are critical points and we prove that, up to translations…

Paper proves anisotropic Minkowski inequality and related inequalities.

New Kelvin transform for anisotropic elliptic problems.

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations

Paper proves rigidity results for anisotropic capillary hypersurfaces.

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentrici…

Unique ancient solutions found for anisotropic curve shortening flow.

Formula proves monotonicity for anisotropic minimal hypersurfaces.

In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…

Motivated by the study of wave fronts in anisotropic media, we propose an incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An anisotropic version of the Laguerre functional is considered. In some circumstances, this functional can be used to determine that two wavefronts observed at distinct time…

We introduce the anisotropic tensor calculus, which is a way of handling with tensors that depend on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application, we show how to define derivations using anisotropic linear conne…

A general approach to formulation of supergravity in higher order anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) is given. We analyze three models of locally anisotropic supergravity.

Characterizes paths minimizing anisotropic lengths in Euclidean space.

Anisotropic minimal graphs over half-spaces are flat.

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…

Graphs with bounded anisotropic mean curvature are regular almost everywhere.

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature s…

Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.

The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures.

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define for hypersurfaces in $\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r; F}$, a generalization of the usual $r$-th mean curvature function. We call a hypersurface is anisotropic mini…

Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

The paper proves stability of Wulff shapes using anisotropic curvature functionals.

Anisotropic curvature flow studied for planar networks.