arXiv subjects
Publications and source records attributed to Vladimir Marković.
Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $κ>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-κ}$ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.
In this paper, we study Weil--Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $φ:\mathbb S^1\to\mathbb S^1$ is Weil--Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\mathbb D$ has square-integrable Beltrami differential. Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil--Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extension minimizes this energy.
We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman projections, and meromorphic partitions of unity.
Given a pair $(X,\nabla)$, consisting of a closed Riemann surface $X$ and a holomorphic connection $\nabla$ on the trivial principal bundle $X\times\mathrm{SL}_2(\mathbb{C})\to X$, the Riemann-Hilbert map sends $(X,\nabla)$ to its monodromy representation. We compute the derivative of this map, and provide a simple description of the locus where it is injective, recovering in the process several previously obtained results.
Let $Σ^p_g$ denote a surface of genus $g$ and with $p$ punctures. Our main result is that the fraction of degree $n$ covers of $Σ^p_g$ which have the Putman-Wieland property tends to $1$ as $n\to \infty$. In addition, we show that the monodromy of a random cover of $Σ^p_g$ is asymptotically almost surely tangle free.