arXiv research

Minimal surfaces can't have certain epitrochoid geodesics.

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

Improved lower bound for geodesics on manifolds.

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…

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

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

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number ex…

In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the $n$-sphere. We also bound the length of certain minimizing geodesic nets.

Minimal geodesics on hyperbolic surfaces are long.

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

The paper proves properties of curves in Riemannian manifolds.

Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

Totally geodesic submanifolds in hyperbolic space up to codimension two.

Lightlike hypersurfaces in cone structures minimize time.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological ent…

Constructs minimal annuli with free boundary in hyperbolic 3-space.

In hyperbolic space $H^n$ we set a geodesic ball of radius $ρ$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic $k$ dimensional submanifold passing through…

Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.

Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.

Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.

Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.

Extends Newton's minimal resistance problem to Riemannian surfaces.

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…

In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first ideas as to the solutions.

Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the Riemannian metric.

We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.

Improved bounds on geodesic intersections on hyperbolic surfaces.

The paper proves geodesics and conic sections are length-minimizing under specific metrics.

New inequality linking geodesic length and volume in complex projective plane.

Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a…

Solves Dirichlet problem for harmonic maps to give geodesic insights.

New para-Kähler structure found in geodesic space.

Local minimizers are convex and close to Wulff shapes.

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds $M_+,M_-$ introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…

This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.

Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geod…

Minimal surfaces in hyperbolic space have a renormalized area criterion.

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(γ)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy pe…

New proof shows nonholonomic motions are geodesics, minimizing distance.

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

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

Study shows not all smooth paths are optimal in certain geometric structures.

Study finds new minimal surfaces in Schwarzschild space.