[Paper Review] A disc maximizes Laplace eigenvalues among isoperimetric surfaces of revolution

Sinan Ariturk|arXiv (Cornell University)|Oct 7, 2014

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper proves that among all compact, connected, immersed surfaces of revolution in ℝ³ with a fixed boundary circle, the flat disc uniquely maximizes all Dirichlet eigenvalues of the Laplace-Beltrami operator. Using variational methods and spectral analysis on radial curves, the author shows that any deviation from the disc's flat geometry strictly decreases eigenvalues, extending classical eigenvalue inequalities to the class of surfaces of revolution with fixed boundary.

ABSTRACT

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary.

Motivation & Objective

• To establish that the flat disc maximizes all Dirichlet eigenvalues among surfaces of revolution in ℝ³ with a fixed boundary circle.
• To extend the Rayleigh-Faber-Krahn inequality to the geometric class of surfaces of revolution with fixed boundary, not just planar domains.
• To resolve a geometric spectral optimization problem by proving strict eigenvalue maximization for the disc under isoperimetric constraints.
• To analyze the spectral behavior of surfaces of revolution through radial parametrization and eigenvalue functionals depending on the radial profile.
• To demonstrate that any non-flat surface of revolution yields strictly smaller eigenvalues than the disc, even for higher eigenvalues.

Proposed method

• Parametrize the meridian curve of a surface of revolution using arc-length in the half-plane ℝ²₊, with the boundary at (R,0) and axis of symmetry along x=0.
• Define eigenvalue functionals λₖ,ₙ(α) as Rayleigh quotients over weighted L² spaces involving the radial function Fₐ and angular momentum terms k²/Fₐ.
• Use variational characterization of eigenvalues via min-max over n-dimensional subspaces of C₀¹(0,L) functions vanishing at the origin.
• Construct a continuous deformation path from any non-flat curve α to the disc’s radial profile ω via a one-parameter family of curves ωₛ, preserving boundary and radial structure.
• Prove continuity and upper semicontinuity of eigenvalue functionals along the deformation path using convergence of integrals and uniform convergence of weight functions.
• Establish monotonicity of eigenvalues along the deformation path by showing the lower left Dini derivative is non-negative, with strict decrease unless the curve is identical to ω.

Experimental results

Research questions

• RQ1 Does the flat disc uniquely maximize the Dirichlet eigenvalues among all surfaces of revolution in ℝ³ with a fixed boundary circle?
• RQ2 Can the classical Rayleigh-Faber-Krahn inequality be extended to surfaces of revolution that are not planar?
• RQ3 How do the eigenvalues of the Laplace-Beltrami operator behave under geometric perturbations of a surface of revolution away from the flat disc?
• RQ4 Is there a strict spectral gap between the disc and any other surface of revolution with the same boundary?
• RQ5 What role does the radial profile Fₐ play in determining the spectrum of the Laplace-Beltrami operator on surfaces of revolution?

Key findings

• For any surface of revolution Σ ≠ D with the same boundary radius R, all Dirichlet eigenvalues satisfy λⱼ(Σ) < λⱼ(D) for all j ≥ 1.
• The eigenvalue functionals λₖ,ₙ(α) for a meridian curve α are strictly increasing along a continuous deformation path from any non-flat α to the disc’s profile ω.
• The eigenvalue λₖ,ₙ(α) is maximized uniquely when the meridian curve α coincides with the radial profile ω(t) = (R−t, 0), corresponding to the flat disc.
• The proof relies on showing that the lower left Dini derivative of the eigenvalue functional along the deformation path is non-negative, with strict negativity at some point if α ≠ ω.
• The eigenvalue functional is continuous and upper semicontinuous, ensuring that the maximum is attained and uniquely achieved at the disc.
• The result holds for all eigenvalues, not just the first, and applies to all surfaces of revolution with one smooth boundary component and fixed boundary radius.