arXiv research

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in \cite{Amano14}. We also consider the infimum of the asymptotic distance in shifting base points of the rays along the geodesics. We show that th…

Unique geodesics selected by energy minimization in Teichmüller space.

In this paper, we obtain the explicit limit value of the Teichmüller distance between two Teichmüller geodesic rays which are determined by Jenkins-Strebel differentials having a common end point on the augmented Teichmüller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This …

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness …

Study on Teichmüller rays' asymptotic behavior and distances.

This paper studies convergence of horospheres in CAT(0) spaces.

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also sho…

New Teichmüller geodesic rays found with unique foliations.

Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\M(S)$ of Riemann surfaces: Teichmüller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactif…

The study examines the asymptotic behavior of extremal length in Teichmüller space.

The paper shows how sublinear biLipschitz equivalences affect Morse boundaries of metric spaces.

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.

Study proper sampling for X-ray transforms on simple surfaces.

Study shows orbits on a specific surface without intersecting geodesics.

The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M, proved in Math. Ann. 165 (1966), 309--317.

Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.

We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincaré disc: a point on the unit circle). We prove here that, if two minimal rays have a common asymptotic direction…

Study extends geodesic ray transform results to orientable surfaces.

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times\partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suita…

Regularity results for geodesic X-ray transform on nonsmooth manifolds

In this paper, we first prove a folklore conjecture on a greatest lower bound of the Calabi energy in all Kähler manifold. Similar result in algebriac setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound estimate of the K energy in terms of the geodesic distance and the Calabi energy. This is…

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

Study proves uniqueness for ray transform on surfaces with obstacles.

New metric spaces for geodesic rays in cohomology classes.

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit…

In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the $n$-sphere. We also bound the length of certain minimizing geodesic nets.

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

Bounds on geodesic distances on Stiefel manifold derived from new metrics.

Local X-ray transform works well near boundaries in hyperbolic spaces.

We study first passage percolation (FPP) on a Gromov-hyperbolic group $G$ with boundary $\partial G$ equipped with the Patterson-Sullivan measure $ν$. We associate an i.i.d.\ collection of random passage times to each edge of a Cayley graph of $G$, and investigate classical questions about the asymptotics of first pass…

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,α}, 0<α<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions with constant geodesic X-ray transform always …

We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmue…

We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermor…

Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmuller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmuller quasi-geodesic and stays in a bounded neighborhood of a Teichmulle…

Study inverse problems for twisted geodesic flows on manifolds.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration.

For smooth test configurations, there always exist C^{1,1} geodesic rays in Kahler metric space parallel to the algebraic ray. The $\yen$ invariant agrees with Futaki invariant, at least under nice assumptions. Explicit examples in Toric cases are calculated. On simple test configurations, Donaldson's correspondence be…

Geodesic rays prove key aspects of cscK metrics existence and stability.

Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.

Study geodesic ray transform on 2D manifolds with conjugate points.

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the …

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces…