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Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d \sigma < C_2, $ then there exists a…

Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or…

We construct an explicit intertwining operator $\lcal$ between the Schr\"odinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface…

Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that…

In this paper we give a new, and shorter, proof of Huber's theorem which affirms that for a connected open Riemann surface endowed with a complete conformal Riemannian metric, if the negative part of its Gaussian curvature has finite mass,…

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the…

Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure…

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the…

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak…

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincar\'e-Bendixson theorem describing recurrence properties and $\omega$-limit sets of geodesics for a meromorphic connection…

We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We…

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence $\{S_n \rightarrow S\}$ of finite Galois covers of a hyperbolic Riemann Surface…

In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in…

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…

Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical…

This paper addresses an open problem recently posed by V. Kozlov: a rigorous proof of the non-integrability of the geodesic flow on the cubic surface $x y z=1$. We prove this is the case using the Morales-Ramis theorem and Kovacic…

Consider a simply connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic and…

We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many…