arXiv research

Suppose (X,ω) is a compact Kähler manifold. In the present work we propose a simple construction for weak geodesic rays in the space of Kähler metrics that seems to be tied together with properties of the class E(X,ω). As an application of our construction, we prove a characterization of E(X,ω) in terms of envelopes.

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…

Analyzes projections of test configurations to vector fields, proving moment convergence.

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ra…

Study compares different stability notions in Kähler geometry.

Study extends geodesic ray transform results to orientable surfaces.

Paper proves existence of constant scalar curvature Kähler metrics under certain conditions.

We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Ampère equation on the deformation to the normal cone of the submanifold. We use this to est…

Regularity results for geodesic X-ray transform on nonsmooth manifolds

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

Study proves uniqueness for ray transform on surfaces with obstacles.

Functions with constant geodesic X-ray transform are restricted to manifolds with specific geometrical properties.

Unique geodesics selected by energy minimization in Teichmüller space.

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…

New asymmetric metric on Teichmüller space for surfaces.

Injectivity theorems for geodesic ray transform on curved 2D manifolds.

Sharp stability estimate for geodesic ray transform on simple manifolds.

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

New Teichmüller geodesic rays found with unique foliations.

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

Injectivity of geodesic ray transform for piecewise constants on compact manifolds.

Study characterizes geodesic ray transform on surfaces, isolating and separating sub-ranges.

Study geodesic X-ray transforms on curved manifolds using Carleman estimates.

Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.

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

Study X-ray transform on Anosov manifolds with improved stability estimates.

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.

Study characterizes X-ray transform kernel for periodic slabs and related manifolds.

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

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

Bound skinning map diameter on hyperbolic 3-manifold rays.

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…

Study shows X-ray transform injectivity for hyperbolic manifolds.

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…

The geodesic X-ray transform on disks of constant curvature is characterized and decomposed.

Study inverse problems for twisted geodesic flows on manifolds.

Article examines stability of geodesic X-ray transforms and artifacts.

Proves $C^{1,1}$ regularity for complex Monge-Ampère equations and geodesic rays.

Reconstructs geodesic ray transforms on simple Riemannian surfaces with connections.

Study X-ray transform on conic spaces, proving injectivity under certain conditions.

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.

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

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

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