arXiv research

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

We formulate the theory of nearly autoparallel maps (generalizing conformal transforms) of locally anisotropic spaces and define the nearly autoparallel integration as the inverse operation to both covariant derivation and deformation of connections by nearly autoparallel maps. By using this geometric formalism we cons…

The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.

We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorph…

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold whic…

We present a geometric approach to the field theory with higher order anisotropic interactions. The concepts of higher order space, or locally anisotropic, space (in brief, h-space, or la-space) are introduced as general ones for various types of higher order extensions of Lagrange and Finsler geometry and higher dimen…

The horizon and geodesic structure of static configurations generated by anisotropic conformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic extension of such off--diagonal vacuum metrics and conclude that for small deformations there are different classes of vacuum solutions of …

We review the geometric setting of the field theory with locally anisotropic interactions. The concept of locally anisotropic space is introduced as a general one for various type of extensions of Lagrange and Finsler geometry and higher dimension (Kaluza--Klein type) spaces. The problem of definition of spinors on gen…

The paper studies a curvature flow on hypersurfaces in R^(n+1).

We synthesize and extend the previous ideas about appearance of both noncommutative and Finsler geometry in string theory with nonvanishing B--field and/or anholonomic (super) frame structures \cite{vstring,vstr2,vnonc,vncf}. There are investigated the limits to the Einstein gravity and string generalizations containin…

Continuum mechanics theory describes skin's complex anisotropic behavior.

In this paper we use the anholonomic frames method to construct exact solutions for vacuum 5D gravity with metrics having off-diagonal components. The solutions are in general anisotropic and possess interesting features such as an anisotropic warp factor with respect to the extra dimension, or a gravitational scaling/…

Solves time-minimizing navigation on a mountain slope using Riemann-Finsler geometry.

We study Finsler black holes induced from Einstein gravity as possible effects of quantum spacetime noncommutativity. Such Finsler models are defined by nonholonomic frames not on tangent bundles but on (pseudo) Riemannian manifolds being compatible with standard theories of physics. We focus on noncommutative deformat…

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…

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…

The paper extends LDDMM framework to include Lie group actions in large deformation shape registration.

Paper proves anisotropic Minkowski inequality and related inequalities.

New Kelvin transform for anisotropic elliptic problems.

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

Paper proves rigidity results for anisotropic capillary hypersurfaces.

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

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

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.

Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

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

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.

Study adiabatic limits of calibrated submanifolds in Riemannian geometry.

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.

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.