arXiv research

Study extends geodesic curvature formula to higher dimensions.

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

Extends Polydisk Theorem to Cartan-Hartogs domains.

Study geodesic extendibility on metric spaces and map them to a half-space.

Geodesic algorithms extended to arbitrary ellipsoids.

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…

New metric on geodesic currents connects different surface genera.

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

Researchers extend Chen, Erchenko, and Gogolev's result to more cases.

Research extends geodesic length function study to three holed sphere.

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…

Given a geodesic line $γ$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $γ$. Then we extend the result to $γ$ being a hypercycle i.e. a geodesic on a hypers…

Characterizes parabolic flute surfaces with specific parameters.

Lower bounds on geodesic lengths for spheres with Willmore energy.

We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.

Extends Masur's divergence theorem to complex tori and Kummer surfaces.

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

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

Geodesic graphs for special Finsler metrics on spheres are studied.

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…

Study extends geodesic ray transform results to orientable surfaces.

Extends curve functions to geodesic currents with a simple criterion.

Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.

Quantizes geodesics in Kähler and Sasaki geometry.

Study counts geodesic surfaces in knot complements, finding unique ones for small knots.

The study proves the existence of geodesics on reversible Finsler spheres.

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a geodesic "superflow" on the cotangent bundle of the supermanifold. Integral curves…

Extends Newton's minimal resistance problem to Riemannian surfaces.

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.

Unique geodesics selected by energy minimization in Teichmüller space.

We derive the 2-component Camassa-Holm equation and corresponding N=1 super generalization as geodesic flows with respect to the $H^1$ metric on the extended Bott-Virasoro and superconformal groups, respectively.

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setti…

Extends K-energy to complexified Kähler classes for scalar curvature study.

The paper extends two-step homogeneous geodesics to homogeneous Finsler spaces.

New research proves uniqueness of maximal spacetime boundaries under certain conditions.

Study geodesic distances and convexity in contact sets.

Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.

Describes geodesic scattering on hyperboloids using quadrics results.

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second …

Zeta functions extended to nonorientable surfaces, order of vanishing computed.

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents…

All inextendible null geodesics in four dimensional de Sitter space dS^4 are complete and globally achronal. This achronality is related to the fact that all observer horizons in dS^4 are eternal, i.e. extend from future infinity scri^+ all the way back to past infinity scri^-. We show that the property of having a nul…

We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.

We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous…

New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of t…

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of…