Sharp asymptotics of the first eigenvalue on some degenerating surfaces

TL;DR

This paper investigates how attaching collapsing handles or cross caps to a Riemannian surface affects the first eigenvalue normalized by area, revealing conditions under which it increases or decreases, with implications for eigenvalue optimization.

Contribution

It provides sharp asymptotic analysis of the first eigenvalue on degenerating surfaces and identifies symmetry conditions influencing eigenvalue behavior.

Findings

First eigenvalue normalized by area can be increased by attaching handles with symmetry.

Without symmetry, the normalized first eigenvalue decreases under similar degenerations.

Results motivate further monotonicity studies without symmetry assumptions.

Abstract

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in \cite{MS3}, where we show a monotonicity result for the normalized first eigenvalue without any symmetry assumptions.