arXiv research

Round spheres are uniquely characterized by half-geodesics.

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length $l(γ)/2$. For each nonnegative integer $n$, we construct Riemannian manifolds diffeomorphic to $S^2$ admitting exactly $n$ half-geodesics. Additionally, we construct a sequence of Riemannian manifolds, each of which…

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…

Proves intersection properties of minimal hypersurfaces in various spaces.

We study the topological dynamics of the horocycle flow $h_\mathbb{R}$ on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for $h_\mathbb{R}$ in T^1 S. Suppose that the half-geodesic $u(\mathbb{R}^+)$ is almost minimizing and that the injectivity radius along $u(\mathbb{R}^+)$ has a finite …