arXiv research

Metric measure boundary vanishes on certain spaces without boundary.

Study solves Yamabe problems on metric measure spaces with or without boundary.

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.

Study on free boundary problems in RCD spaces, proving existence and regularity.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

Two different spacetimes can mimic each other's boundary measurements.

New method counts boundary pieces in ReLU classifiers for better complexity measure.

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

New method measures generalizability of deep neural networks based on decision boundary complexity.

The paper proves a Liouville theorem for specific harmonic maps with free boundary.

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …

Gradient estimate for harmonic functions with boundary condition proved.

This paper improves boundary regularity of harmonic maps in metric measure spaces.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

Global EQG sums boundary states over manifold diffeomorphism classes.

New varifolds with capillary boundary properties studied.

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …

The paper examines the structure and stability of boundaries in noncollapsed RCD spaces.

One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said …

Study Poisson boundaries of building lattices and generalize rigidity results.

Sharp heat equation gradient estimates on compact manifolds.

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…

Paper develops a fractal dimension-based generalization measure.

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary r…

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

Volterra square-root process boundary behavior and martingale measures

Study optimal transport on simplex boundary, proving transport map and potential regularity.

Study on the structure of classifier boundaries in DNA sequencing.

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

We give sufficient conditions for a parametrised family of probability measures on a Riemannian manifold with boundary to be represented by random maps of class $C^k$. The conditions allow for the probability densities to approach zero towards the boundary of the manifold. We also formulate two obstructions to regular …

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

Measures neural network decision boundary volume to predict model performance.

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

Paper shows ergodicity and irreducibility of mapping class group boundary representation.

Study gradient flow of phase transitions with fixed contact angle.

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …

Formula identifies boundary flux for Kähler manifolds under parallel deformation.

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.

Unique hyperbolic manifolds identified by boundary pleating.

We show under weak hypotheses that $\partial X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $Γ$. In particular, we show that if $Γ$ admits a nonelementary proper action on $X$, and $μ$ is a generating prob…

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.