Related papers: Minimal entropy and geometric decompositions in di…

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also…

On a real analytic Riemannian manifold a Grauert tube is an uniquely adapted complex structure defined on the tangent bundle. It is called entire if it may be defined on the whole tangent bundle. Here, we show that the geodesic flow of an…

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully…

In 2004, Taubes introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this…

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group.…

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this…

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "\`a la Margulis" for the systole and a volume estimate for…

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing…

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the…

Topological entropy is not lower semi-continous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive…

In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions.…

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and…

We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+\epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent…

We show the equivalences of several notions of entropy, like a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform…

We introduce a description of a minimal surface in a space with boundary, as the world-hypersurface that the entangling surface traces. It does so by evolving from the boundary to the interior of the bulk under an appropriate geometric…

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system…

Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem…

Two flows defined on a smooth manifold are equivalent if there exists a homeomorphism of the manifold that sends each orbit of one flow onto an orbit of the other flow while preserving the time orientation. The topological entropy of a flow…

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of…

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a…