Related papers: Limiting distributions of curves under geodesic fl…

We prove full convergence of gradient-flows of the arc-length restricted tangent point energies in the Hilbert-case towards critical points. This is done through a {\L}ojasiewicz-Simon gradient inequality for these energies. In order to do…

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the…

Let $N$ be a manifold of dimension $m$ with a flat vector bundle given by a representation $\rho:\pi_1(N) \rightarrow \mathrm{GL}(n, \mathbf{R})$ where $\pi_1(N)$ is finitely generated. The holonomy group $\rho$ is a $k$-partially…

We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an…

In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce…

This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and…

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed…

In this paper, we classify helix (spacelike, timelike and null) curves, directed by the geodesic flow vector field, on the (3-dimensional) unit tangent bundle of a pseudo-Riemannian surface of constant Gaussian curvature endowed with a…

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…

Static manifolds with boundary were recently introduced to mathematics. This kind of manifold appears naturally in the prescribed scalar curvature problem on manifolds with boundary when the mean curvature of the boundary is also…

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We…

We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the…

We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in…

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely…

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a…

In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries, constructed and described explicitly by Benoist. These manifolds are…

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the…