Self-focal points of ellipsoids of dimension $\geq 3$

TL;DR

This paper investigates the existence of self-focal points on high-dimensional ellipsoids, proving their absence in certain cases and implications for Laplace eigenfunction behavior.

Contribution

It establishes new results on the non-existence of self-focal points in high-dimensional ellipsoids with at least four axes, and explores their presence in other cases.

Findings

Ellipsoids with ≥4 axes have no self-focal points.

Certain ellipsoids with 3 axes have non-polar self-focal points.

Ellipsoids with ≤2 axes always have self-focal points.

Abstract

A self-focal point of a Riemannian manifold $(M,g)$ is a point $p$ so that every geodesic starting from $p$ returns to $p$ at some positive time. It is called a pole if all geodesics through $p$ are closed, and a non-polar self-focal point if all geodesics loop back but not all are smoothly closed. Umbilic points of two dimensional tri-axial ellipsoids are non-polar self-focal points. Little is known about existence of self-focal points for Riemannian manifolds of dimension $\geq 3$. We prove that ellipsoids of dimension $\geq 3$ with at least 4 distinct axes have no self-focal points. Certain ellipsoids of dimension $\geq 3$ with three distinct axes do have non-polar self-focal points. Ellipsoids with $\leq 2$ distinct axes always have self-focal points. Self-focal points play an important role in the study of $L^{\infty}$ norms of Laplace eigenfunctions. Our results imply that Laplace eigenfunctions on ellipsoids of dimension $\geq 3$ with at least $4$ distinct axes never achieve maximal sup-norm growth.