arXiv research

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

The paper proves the existence of capillary geodesics on Riemannian 2-disks.

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded…

Geodesics on polygons in a unit disk are studied with unique metric properties.

Paper finds conformal metrics on a disk with specific curvatures.

The paper proves ideal triangulations and disk unfolding for singular flat surfaces.

Inverse mean curvature flow converges to a disk in hyperbolic space.

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…

The paper solves a problem in metric geometry for disks with negative curvature.

Rare Teichmüller disks converge to small limit sets.

Minimal surfaces in hyperbolic space have a renormalized area criterion.

We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting themselves at 120 degrees, and reaching orthogonally the boundary of the disk.

Unique geodesics selected by energy minimization in Teichmüller space.

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…

The paper proves the existence and properties of geodesics on convex surfaces.

Sharp inequalities for curved surfaces and cones.

On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform…

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These res…

We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove…

The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $μ(=\frac{1}{\hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of th…

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

For a one-parameter family of simple metrics of constant curvature ($4κ$ for $κ\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also t…

Study calculates first $p$-widths of unit disk.

The study proves inequalities for area and boundary length of disks in convex manifolds.

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the …

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Te…

New functionals defined for free boundary minimal submanifolds in higher dimensions.

It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller …

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

The paper solves curvature prescription on a disk with negative Gaussian curvature.

The study finds minimal surfaces in complex space forms are often totally geodesic.

Loewner inequality proven for curved surfaces.

Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic…

The paper studies Steklov eigenvalues in space forms and warped product manifolds, deriving bounds and monotonicity results.

We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. W…

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …

The paper solves curvature prescription problems on balls and disks.

A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed …

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acousti…

Paper estimates Gaussian curvature of minimal graphs in a specific manifold.

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

Let $h:X \to Y$ be a homeomorphism between hyperbolic surfaces with finite topology. If $h$ is homotopic to a holomorphic map, then every closed geodesic in $X$ is at least as long as the corresponding geodesic in $Y$, by the Schwarz Lemma. The converse holds trivially when $X$ and $Y$ are disks or annuli, and it holds…