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

Mohammadjavad Habibivostakolaei

TL;DR

The paper analyzes how the Dirichlet spectrum of an embedded annulus evolves when its boundary undergoes curve shortening flow and the ambient metric evolves under the 2D Ricci flow. By combining variational formulas, Rellich-type identities, and harmonic capacity theory, it relates eigenvalue changes to the boundary modulus and a deficit functional, establishing a geometric stability result: for small deficit D, the annulus is conformally close to a flat cylinder and its first Dirichlet eigenvalue satisfies λ(A) ≥ λ_cyl(h) + C√D, where h is the modulus. The approach yields a quantitative spectral gap controlled by D and yields a precise link between conformal geometry and spectral stability under coupled geometric flows, complemented by numerical experiments that corroborate the δλ ∝ √D scaling in model geometries. This work highlights how modulus control and deficit quantify spectral sensitivity under evolving domains and metrics, with implications for stability analyses in geometric analysis and spectral geometry.

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.

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

TL;DR

The paper analyzes how the Dirichlet spectrum of an embedded annulus evolves when its boundary undergoes curve shortening flow and the ambient metric evolves under the 2D Ricci flow. By combining variational formulas, Rellich-type identities, and harmonic capacity theory, it relates eigenvalue changes to the boundary modulus and a deficit functional, establishing a geometric stability result: for small deficit D, the annulus is conformally close to a flat cylinder and its first Dirichlet eigenvalue satisfies λ(A) ≥ λ_cyl(h) + C√D, where h is the modulus. The approach yields a quantitative spectral gap controlled by D and yields a precise link between conformal geometry and spectral stability under coupled geometric flows, complemented by numerical experiments that corroborate the δλ ∝ √D scaling in model geometries. This work highlights how modulus control and deficit quantify spectral sensitivity under evolving domains and metrics, with implications for stability analyses in geometric analysis and spectral geometry.

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.
Paper Structure (8 sections, 4 theorems, 158 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 158 equations, 4 figures.

Table of Contents

  1. Introduction and Motivation
  2. Preliminaries
  3. Background and Related Works
  4. Variational Formulas
  5. Spectral Comparison of an Annulus and a Cylinder
  6. Rellich--type Formula and the Deficit Boundary Term
  7. Geometric Stability and Spectral Gap
  8. Examples and Numerical Results

Key Result

Lemma 2.1

Let $\left(M, g\left(t\right)\right)$ be a smooth one--parameter family of Riemannian surfaces and let $A\left(t\right) \subset M$ be a smooth family of bounded domains diffeomorphic to an annulus with smooth boundary $\partial A\left(t\right) = \Gamma\left(t\right)$. If for each $t$, $\lambda\left( where $V$ is the normal speed of the boundary in the outward normal direction $\nu$ and here $\part

Figures (4)

  • Figure 1: Comparison between eigenvalues and deficit
  • Figure 2: Comparison between eigenvalues and outer radius $b$
  • Figure 3: Small deficit
  • Figure 4: Eigenvalues and epsilon parameter

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Remark 2.2: Special Cases
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5: Some specialisations
  • Remark 3.1: Few Points on Extension $X$
  • Theorem 3.2
  • Remark 3.3
  • ...and 6 more