Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and $2$D Ricci Flows

TL;DR

This paper investigates how the spectrum of an embedded annulus changes under coupled curve shortening and Ricci flows, providing quantitative bounds and stability estimates relating spectral and geometric properties.

Contribution

It introduces a novel analysis linking spectral stability to geometric deficits for annuli evolving under coupled geometric flows.

Findings

Established bounds comparing spectra of evolving annuli and flat cylinders.

Derived a spectral gap estimate controlled by geometric deficit.

Demonstrated geometric stability under coupled flows.

Abstract

We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich--type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.