arXiv research

Geodesics found in spacetime satisfy curvature conditions.

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from t…

Study heat content on RCD(K,N) spaces with specific boundary conditions.

Paper constructs new non-Anosov Partially Hyperbolic Geodesic flows using conformal deformations.

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 supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate …

Given a geodesic line $γ$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $γ$. Then we extend the result to $γ$ being a hypercycle i.e. a geodesic on a hypers…

This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

Let $M$ be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure $m_F$ associated to a potential $F$. We compute the Hausdorff dimension of the conditional measures of $m_F$. We study the $m_F$-almost sure asymptotic penetration behaviour of locally geodesic lines of…

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain n…

Given any K and N we show that there exists a compact geodesic metric measure space satisfying locally the CD(0,4) condition but failing CD(K,N) globally. The space with this property is a suitable non convex subset of R^2 equipped with the l^\infty-norm and the Lebesgue measure. Combining many such spaces gives a (non…

The study shows that ergodic measures are not generic on non-positively curved manifolds.

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. In particular, positive topological entropy implies chaotic…

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of wh…

Study geodesic trees and exceptional directions in FPP on hyperbolic groups.

Study introduces new curvature conditions for Lorentzian spaces using Rényi entropy.

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, …

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …

Random walks on mapping class groups identified with geodesic laminations.

New method calculates geodesic distances in Gaussian random field manifolds.

In this article, a proof of the interpolation inequality along geodesics in $p$-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature…

Geodesics in curved spaces spread evenly over time.

Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolut…

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they…

Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.

Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones whic…

Study counts ergodic measures in surface lamination strata.

Study ergodic properties of geodesic flows on specific manifolds without conjugate points.

Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces.

The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian…

The study proves sub-Riemannian manifolds cannot satisfy $\mathrm{CD}$ conditions unless they are Riemannian.

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.

We associate certain probability measures on $\R$ to geodesics in the space $\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in the finite dimensional symmetric space of hermitian norms on $H^0(X, kL)$. We prove that the measures associated to the finite dimensional spaces converge weakly to t…

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that the quali…

New findings show different cost functions yield equivalent curvature bounds.

Characterizes geodesic completeness for landmark spaces.

This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.

Study geodesic distances and convexity in contact sets.

Study equilibrium measures on manifolds without conjugate points with visibility covering.

Geodesics of the same type on curved surfaces are randomly distributed.

In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based …

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which as…

New characterization of geodesic currents via curve functionals.