arXiv research

A 6-regular triangulation for hyperbolic plane created.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

Proves conjecture about geodesic foliations in Riemannian planes.

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has 16 balanced vertices and does not contain proper geodesic subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with 4 boundary vertices that is not a tree.

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.

Classifies geodesic flows on projective plane with potential field.

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

Geodesics on extended Siegel-Jacobi upper half-plane determined.

The study quantifies geodesic divergence on Riemannian planes with bounded geometry.

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained earlier by McMullen, Moh…

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…

A 2D Riemannian space has only 2 injective geodesics.

The paper characterizes Ricci solitons on the Poincaré upper half plane.

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round $2$-sphere. In the first half of this paper we survey some r…

The Poincaré series for surfaces with boundary extends to the complex plane.

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

The paper finds two types of metric lines in curve spaces.

Classifies geodesic-preserving bijections in Thurston geometries.

Geometric structures over algebras describe geodesics and spaces.

The paper defines catenary curves in spheres and hyperbolic planes.

The study reveals conditions for infinite closed geodesics on specific surfaces.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theore…

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

New metrics derived from geodesics simplify semi-Riemannian geometry.

Planes are the only calibrated submanifolds with flat normal bundles.

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh.…

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic…

We study totally geodesic planes in hyperbolic 3-manifolds $M$ having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal $PSL(2,R)-$invariant subset of $M$ is either an immersed totally geodesic surface or all of $M$. We also show that for an arbitrary infinite volume hyperboli…

We prove that, generically, magnetic geodesics on surfaces will turn away from points with lightlike tangent planes, and we motivate our result with numerical solutions for closed magnetic geodesics.

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function al…

The study provides a criterion for fractional-linear integrals of geodesics on surfaces.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connected…

Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.

Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not …

We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.

We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.

Penrose limit results for specific 3-surfaces in space-time.

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces i…

We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the beh…

Improved rigidity of Delaunay triangulated plane.

We use the $η$-invariant of Atiyah-Patodi-Singer to compute the Eells-Kuiper invariant for the Eells-Kuiper quaternionic projective plane. By combining with a known result of Bérard-Bergery, it shows that every Eells-Kuiper quaternionic projective plane carries a Riemannian metric such that all geodesics passing throug…