arXiv research

New examples show non-integer Hausdorff dimensions in collapsing spaces.

Hausdorff reflection keeps space shape intact.

Study describes limits of surfaces in a mathematical space.

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove…

Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.

Paper studies Hausdorff dimensions of specific limit sets for groups on curved spaces.

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respe…

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.

The study characterizes lamination limits and homeomorphisms in 3D handlebodies.

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

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…

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

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover.

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

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

Newly discovered Eguchi-Hanson metric arises from edge metrics.

Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.

Study shows local topologies of certain geometric spaces.

Paper shows limits of Heisenberg manifolds are flat tori.

Study flat manifolds' collapsed limits as flat orbifolds.

Let R be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T-infinity limits over R, that do not in general fall under the scope of Marker and Steinhorn's definability-of-types theorem. We prove that if R admits analytic cell decomposition, then every T-infinity limit over R is def…

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …

We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curv…

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

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

Triangle comparison for Kaehler manifolds with curvature bounds.

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

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…

Extends arguments to limit structure in Calabi-Yau degenerations.

The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and…

This paper studies limits of aspherical manifolds with specific curvature conditions.

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

Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…

I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…

In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.

Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.

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

New examples show strong Kato limits can be branching and not satisfy known conditions.

Study limits of curved spaces with boundaries.