arXiv research

Proves singular set of certain integral hypercurrents has measure zero.

Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.

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

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that…

The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is on…

Study shows how geodesics behave in hyperbolic manifolds and proves measure singularity.

This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.

The study bounds Hausdorff measure of flat singular points in area-minimizing currents.

In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure an…

The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.

Injectivity proven for measure homology of certain wild spaces.

Study bounds singular set of minimal hypersurfaces with index control.

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

Refined theorem on linear perturbations with applications in singularity theory and optimization.

We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $μ$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan me…

New measures on orbit spaces for orthogonal groups identified.

Study weak super Ricci flow through neckpinch in metric measure spaces.

The Cannon-Thurston map's measures become singular with respect to sphere measures.

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

The paper extends entropy concepts to Monge-Ampère measures with prescribed singularities.

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak no…

Introduces LLC, a new complexity measure for DNNs based on SLT.

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on …

Paper shows equivalence of two curvature notions on singular surfaces.

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components …

The study examines conditions for scalar curvature and Dirac operators on singular spaces.

3D Ricci flows have bounded diameter before Type I singularities.

Bound on singular points for area-minimizing surfaces.

Study shows a subset of foliations on $\mathbb{P}^n$ has all singular points linearizable.

We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval, when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strate…

Study maximizes eigenvalues in dimensions 3 and above.

The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix bur…

Lipschitz-volume rigidity holds for smooth manifolds but fails for singular spaces.

We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, …

New scalars measure failure of CC metrics to solve singular Yamabe problem.

We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown t…

The paper measures and limits the extent of non-smooth points in Alexandrov spaces.

We introduce thermodynamic response functions for singular Bayesian models.

Study absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.

We study singularity structure of Yang-Mills flow in dimensions $n \geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratifi…

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…

Study simplicial volume via foliated simplices and duality.

Monotonicity and rigidity of W-entropy proved in singular spaces.

We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the cont…

Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.

The paper studies harmonic map flows and proves rectifiability of singular sets.