arXiv research

This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.

Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.

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

A prime geodesic theorem for singular geodesics in a locally symmetric space is proved. As an application, an asymptotic formula for units in number fields is given.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

Study shortest geodesics on flat cone spheres with conical singularities.

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.

New families of non-singular geodesic orbit nilmanifolds discovered.

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

We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…

The paper examines geometric invariants near a specific type of singular point.

In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensiona…

New singularity concept in GR: volume singularities.

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

The paper proves a conjecture about spacetimes and singularities.

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) an…

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

Note on minimal maps' uniqueness via singular values.

Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.

In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points.…

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to natura…

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.

Study of Ricci flow convergence on surfaces with boundary.

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular loc…

Study on extremal subsets in geodesically complete spaces with curvature constraints.

This paper calculates interaction strength for translation surfaces with multiple singularities.

Simplified proof of cosmic singularity theorem using new mathematical techniques.

Study magnetic geodesic flows on spheres, describing their bifurcations.

We investigate the rudiments of Riemannian geometry on orbit spaces $M/G$ for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space $M/G$ and they can hit strata which are more singular only at the end points. This is phrased as convexity …

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…

Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.

We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincaré-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with geodesics away from the conformal singularity set and are shown to satisfy natural "b…

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly …

Study counts geodesics on modular surface, linking to necklace counting.

The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of $\R^n$ with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the calculus of variation in the large

Paper proves uniqueness of minimal maps in curved spaces.

We prove that the asymptotic completion of a developable Möbius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip contains a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.

Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{1,1}$ geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some a…

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number theory (class number asymptotics) it is, however, necessary to consider this case, t…

Study geodesics in sub-Riemannian manifolds, resolving open questions.

Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.

Proves Gannon-Lee theorem for $C^1$ spacetimes.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

Study wall singularities in spaces with upper curvature bounds.

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…