arXiv research

Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

Study on a metric for disk automorphisms with maximal modulus.

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in $R^n$ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.

Paper solves open problem in complex Finsler geometry.

Established a Hardy inequality on Finsler manifolds.

We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to calculate these functions for spherically symmetric Finsler surfa…

Uniform diameter bound for reflection group disk patterns.

Uniform convexity in divisible domains leads to hyperbolic geometry.

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…

The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.

Study cash-subadditive risk measures without quasi-convexity.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis-Vázquez improvement, if Finsler manifolds are of strictly ne…

Unified framework for robust risk measures beyond convexity.

In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately …

Random walks generate quasi-convex subgroups in acylindrically hyperbolic groups.

The paper defines quasi-convex subsets in spaces with lower curvature bound.

We show that a map with Hölder exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov fo…

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …

We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…

Paper infers intrinsic dimension from quasi-convex measurements.

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

Geodesic flows on specific manifolds are structurally stable.

Study on travel time formulas in a lake with wind flow.

In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Del…

Optimal Liouville theorem for minimal disks in any codimension.

This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.

Suppose $τ$ is a train track on a surface $S$. Let $C(τ)$ be the set of isotopy classes of simple closed curves carried by $τ$. Masur and Minsky [2004] prove $C(τ)$ is quasi-convex inside the curve complex $C(S)$. We prove the complement, $C(S) - C(τ)$, is quasi-convex.

Paper introduces combinatorial Ricci flows on infinite disk triangulations.

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Ma…

Paper introduces quasi-logconvex risk measures and their properties.

The study characterizes subgroups of mapping tori of free groups.

Study potential theory to detect completeness of Finsler manifolds.

The paper studies Harnack inequalities on Finsler metric measure spaces.

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincaré condition and the measure contraction property. Using a compari…

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials p…

The main point of this paper is to prove the following useful result: If the almost everywhere 2-jet of a locally quasi-convex function u satisfies a degenerate elliptic constraint F, then u is F-subharmonic, i.e., u is a viscosity F-subsolution. This AE Theorem makes otherwise difficult results transparent. Some insta…

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-…

Study growth rates of subgroups in groups with a constricting element.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation $Δu - e^{2u} = K_0(z)$ on general open surfaces. A few oth…

This work analyzes minimum-time navigation on Riemannian manifolds using Finsler geometry.

This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that…