arXiv research

Study complex lines in symplectic geometry, generalizing previous results.

Study of hyperbolic behavior in complex manifolds with specific vector bundles.

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

Defines new metric space sections with Ahlfors-David regularity.

The Ahlfors Laplacian is applied to solve geometric and relativistic problems.

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known…

The paper decomposes spacelike hypersurface properties for general relativistic vacuum equations.

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe [23] that generalizes the CR Schwarzian derivative studied earlier by the second-named author [21]. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spheric…

We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a …

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.

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

This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-…

The affine-additive group is hyperbolic with a non-vanishing 4-capacity.

Analyzes metric spaces homeomorphic to manifolds, proving rigidity and inequalities.

New Grunsky operator for disk maps to complex plane.

The $p$--modulus ${\rm mod}_p(\mathcal{F})$ of a foliation $\mathcal{F}$ on a Riemannian manifold $M$ is a generalization of extremal length of plane curves introduced by L. Ahlfors. We study the variation $t\mapsto{\rm mod}_p(\mathcal{F}_t)$ of the modulus. In particular, we consider product of moduli of orthogonal fo…

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applicati…

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $φ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a ge…

This dissertation contains a comprehensive study of the topology of 2-manifolds and a complementary analysis of the work done by Edwin E. Moise, L. V. Ahlfors and Ian Richards. Our aim is to study the well known classification of surfaces. Here we present the technical tools needed for proving rigorously the classifica…

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature…

In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:$ if $f(Δ)\cap \{0,1,\infty \}=\emptyset ,$ then% A(f,Δ)\leq hL(f,\partial Δ),% where $Δ$ is the disk $|z|<1$ in $\mathbb{C},$ $S$ is the unit Riemann sphere,…

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spac…

The article approximates solutions to the Beltrami equation using similarity surfaces.

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…

We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we us…

Geometric data uniquely determines convex subsets in hyperbolic manifolds.

We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of t…

We study isometric maps between Teichmüller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but also exhibit many differences. The focus here is on recent results proved by the …

In this article we extend a euclidean result of David and Semmes to the Heisenberg group by giving a sufficient condition for a $k$-Ahlfors-regular subset to have big pieces of bilipschitz images of subsets of $\R^k$. This Carleson type condition measures how well the set can be approximated by the Heisenberg $k$-plane…

Attempts to build a discrete theory for rational maps on the sphere via circle packing have foundered on discretization effects in locating branch points. The authors remove this impediment by introducing generalized branch points. A generalized branch point need no longer be attached to an individual circle, but with …

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, …

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci…

Constructs flows on manifolds with small curvature, proving Euclidean topology.

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conf…

New theorem proves rigidity of circle packings in hyperbolic geometry.

In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part …

Paper removes singularities from compact area minimizers in positive scalar curvature manifolds.

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…

Study on hyperconvex representations of surface groups and their geometric properties.

A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in $\mathbb{R}^2$, or equivalently, in $S^2$. A carpet is called thin if its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its $Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a $(1,Q)$-Poin…

Holomorphic solutions vary in Sobolev spaces for Beltrami equations.

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entrop…