Weyls's law for Compact Rank One Symmetric Spaces

TL;DR

This paper investigates Weyl's law for eigenvalues of the Laplacian on Compact Rank One Symmetric Spaces (CROSSes), establishing the sharpness of the error term for CROSSes and polynomial improvements for their products, supporting existing conjectures.

Contribution

It extends known results on Weyl's law error terms to CROSSes and their products, confirming sharpness and polynomial improvements respectively.

Findings

Error term is sharp for CROSSes.

Polynomial improvement of error term for products of CROSSes.

Supports conjecture on general products of CROSSes.

Abstract

Weyls law is a fundamental result governing the asymptotic behaviour of the eigenvalues of teh Laplacian. It states that for a compact d dimensional manifold M (without boundary), the eigenvalue counting function has an asymptotic growth, whose leading term is of the order of d and the error term is no worse than order d-1. A natural question is: when is the error term sharp and when can it be improved? It has been known for a long time that the error term is sharp for the round sphere (since 1968). In contrast, it has only recently been shown (in 2019) by Iosevich and Wyman that for the product of spheres, the error term can be polynomially improved. They conjecture that a polynomial improvement should be true for products in general. In this paper we extend both these results to Compact Rank One Symmetric Spaces (CROSSes). We show that for CROSSes, the error term is sharp. Furthermore, we show that for a product of CROSSes, the error term can be polynomially improved. This gives further evidence to the conjecture made by Iosevich and Wyman.