arXiv research

This paper introduces Hausdorff measure and its applications in fractal geometry.

New geometric measure simplifies complex analysis.

The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.

We prove here that the Poincaré exponent of a geometrically finite group od isometries of the 3-dimensionnal hyperbolic space coincides with the Hausdorff dimension of its limit set. We also compare the natural measures supported by this set: the Patterson measure and the Hausdorff and packing measures corresponding to…

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…

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

We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that C^1 submanifo…

Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In this note we completely characterize Berlanga's topology on measure homology of…

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

The paper studies the dimension of limit sets using variational principles and stationary measures.

Anosov groups limit sets are Ahlfors regular, with applications in Teichmüller spaces.

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

Study uses equivariant topology to measure distances between G metric spaces.

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects…

Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.

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

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of …

We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.

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

Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily…

Paper shows limits of Heisenberg manifolds are flat tori.

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

One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and measure the Hausdorff distance between their images. The best matching corresponds…

Introduces fractional k-dimensional measure bridging fractional length and area.

We establish an area-type formula for the intrinsic spherical Hausdorff measure of every regular curve embedded in an arbitrary graded group.

Schwarzschild 3-manifold stability proven for 3D Penrose inequality.

Study tests if a probability measure is near a real algebraic variety.

Study Poincaré inequality in metric spaces via separating sets.

The paper studies limit sets on $\mathbb{P}(\mathbb{R}^3)$ using stationary measures.

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

Study on metric spaces with properties (ETR), (LBD) and their convergence.

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…

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

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanif…

Study of irreversible metric-measure spaces, proving convergence and stability results.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows

Study area and coarea formulas for graphs and submanifolds in Carnot groups.

This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distributi…

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

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 $S$.

New upper bound found for nodal sets of Laplace eigenfunctions.

This paper studies rectifiability in Carnot groups and proves geometric area formulas.