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Vladimir Markovic

Publications and source records attributed to Vladimir Markovic.

At least 19 recordsLinked to original sources
arXiv2026

Holomorphic curves in compact quotients of SL(2,C)

We prove that every discrete faithful representation of the surfcae group into SL(2,C) is the monodromy of a holomorphic connection on the trivial rank-2 vector bundle over a Riemann surface. As an application, we answer the question posed by Ghys and Huckleberry-Winkelmann (known as the Margulis' problem) by proving that every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two. The main tools we use are the Non-Abelian Hodge correspondence, the WKB analysis, and the Morgan-Shalen compactification.

math.GT↗
arXiv2026

Caratheodory metrics on Teichmuller spaces

Let $S$ be an arbitrary Riemann surface whose Teichm\"uller space $T(S)$ has dimension at least two. A long standing problem is to determine whether the Carath\'eodory metric $d_C$ agrees with the Teichm\"uller metric $d_T$ on $T(S)$. It was shown that $d_C\ne d_T$ when $S$ is a closed surface of genus at least two. In this paper we study the general case, and prove that $d_C\ne d_T$ on $T(S)$ except possibly on the following seven Teichm\"uller spaces: $T^1_{0,0}$, $T^1_{0,1}$, $T^2_{0,0}$, $T^1_{0,2}$, $T^2_{0,1}$, $T^3_{0,0}$, and $T^3_{0,1}$.

math.GT↗
arXiv2023

Geometrically and topologically random surfaces in a closed hyperbolic three manifold

We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M. In particular, we describe PSL(2,R) invariant measures on the Grassmann bundle G(M) which arise as limits of random minimal surfaces. It is showed that if M contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring.

math.GT↗
arXiv2023

Minimal surfaces and the new main inequality

We establish the new main inequality as a minimizing criterion for minimal maps to products of $\mathbb{R}$-trees, and the infinitesimal new main inequality as a stability criterion for minimal maps to $\mathbb{R}^n$. Along the way, we develop a new perspective on destabilizing minimal surfaces in $\mathbb{R}^n$, and as a consequence we reprove the instability of some classical minimal surfaces; for example, the Enneper surface.

math.DG↗
arXiv2022

Unstable minimal surfaces in $\mathbb{R}^n$ and in products of hyperbolic surfaces

We prove that every unstable equivariant minimal surface in $\mathbb{R}^n$ produces a maximal representation of a surface group into $\prod_{i=1}^n\textrm{PSL}(2,\mathbb{R})$ together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in $\mathbb{R}^n$ to a surface in a product of $\mathbb{R}$-trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

math.DG↗
arXiv2019

Non-realizability of the Torelli group as area-preserving homeomorphisms

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms.

math.GT↗
arXiv2019

Homogenization of random quasiconformal mappings and random Delauney triangulations

In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.

math.CV↗
arXiv2018

Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces

It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller disks on which the two metrics agree, and we conjecture that the Carathéodory and Teichmüller metrics agree on a Teichmüller disk if and only if the Teichmüller disk is generated by a differential with no odd-order zeros. Using dynamical results of Minsky, Smillie, and Weiss, we show that it suffices to consider disks generated by Jenkins-Strebel differentials. We then prove a complex-analytic criterion characterizing Jenkins-Strebel differentials which generate disks on which the metrics coincide. Finally, we use this criterion to prove the conjecture for the Teichmüller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Carathéodory and Teichmüller metrics are different to the case of compact surfaces with punctures.

math.GT↗
arXiv2015

Heat flows on hyperbolic spaces

In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converges to a quasi-isometric harmonic map. This implies the Schoen-Li-Wang conjecture that every quasiconformal map of $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to a harmonic quasi-isometry of the $n$-dimensional hyperbolic space.

math.DG↗
arXiv2014

Homology of curves and surfaces in closed hyperbolic 3-manifolds

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $π_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $π_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques developed earlier by Kahn and Markovic about good pants constructions, but we only distill geometric and topological ingredients from their papers so no hard analysis is involved in this paper.

math.GT↗
arXiv2014

The good pants homology and the Ehrenpreis conjecture

We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces (the good pants are pairs of pants whose cuffs have the length nearly equal to some large number R). Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpreis conjecture.

math.GT↗
arXiv2013

The Moduli space of Riemann Surfaces of Large Genus

Let $\mathcal{M}_{g,ε}$ be the $ε$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,ε}$ is bounded below by $(c_1g)^{2g}$ and bounded above by $(c_2g)^{2g}$, where the constants $c_1,c_2$ depend only on $ε$ and $r$, and in particular not on $g$. Using the counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.

math.GT↗
arXiv2012

Criterion for Cannon's Conjecture

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of $G$ are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contains "enough" quasi-convex surface subgroups.

math.GT↗
arXiv2011

Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

math.GT↗
arXiv2010

Decomposing diffeomorphisms of the sphere

We prove that any diffeomorphism of the sphere S^n to itself can be decomposed into bi-Lipschitz mappings of small isometric distortion and which move points a small amount in the spherical metric.

math.CV↗
arXiv2009

Classification of continuously transitive circle groups

Let G be a closed transitive subgroup of Homeo(S^1) which contains a non-constant continuous path f: [0,1] --> G. We show that up to conjugation G is one of the following groups: SO(2,R), PSL(2,R), PSL_k(2,R), Homeo_k(S^1), Homeo(S^1). This verifies the classification suggested by Ghys [Enseign. Math. 47 (2001) 329-407]. As a corollary we show that the group PSL(2,R) is a maximal closed subgroup of Homeo(S^1) (we understand this is a conjecture of de la Harpe). We also show that if such a group G < Homeo(S^1) acts continuously transitively on k-tuples of points, k>3, then the closure of G is Homeo(S^1) (cf Bestvina's collection of `Questions in geometric group theory').

math.GR↗