arXiv subjects

Ian Adelstein

Publications and source records attributed to Ian Adelstein.

21 records · Page 2Linked to original sources

arXiv2017

Morse theory for the uniform energy

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us to identify gradient-like vectors at those points where the function is not differentiable. These vectors are used to restart the standard negative gradient flow of the Morse energy at its critical points. We illustrate this procedure on the flat torus and demonstrate that the restarted flow improves the minimizing properties of the associated closed geodesics.

math.DG↗

arXiv2015

Existence and Non-existence of Half-Geodesics on $S^2$

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 is diffeomorphic to $S^2$ and admits no half-geodesics, yet which converge in the Gromov-Hausdorff sense to a limit space with infinitely many half-geodesics.

math.DG↗

arXiv2014

Minimizing Closed Geodesics via Critical Points of the Uniform Energy

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 persist under the Gromov-Hausdorff convergence of Riemannian manifolds. Additionally, we relate half-geodesics (1/2-geodesics) to the Grove-Shiohama critical points of the distance function. This relationship affords us the ability to study the behavior of half-geodesics via the well developed field of critical point theory. Along the way we provide a complete characterization of the differentiability of the Riemannian distance function.

math.DG↗