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Triple Products of Eigenfunctions and Spectral Geometry

Joe Schaefer

TL;DR

The paper addresses the inverse spectral problem for closed Riemannian manifolds by introducing the triple-product invariants $M^{i,j,k}$ of Laplace-Beltrami eigenfunctions as a discrete global geometric signature. It proves that the map $(M,g,\{e^i\})\mapsto \{\lambda_i, M^{i,j,k}\}$ is injective up to isometry, with sufficiency in the multiplicity-one case, thereby enabling isometry determination from spectral data augmented by the $M^{i,j,k}$ set. The main contributions include a concrete framework linking harmonic analysis of pointwise multiplication to geometric isometry classifications, a connection to Abelian $C^*$-algebras and conformal-field-theory-inspired structures, and a detailed flat-torus example illustrating the method. This work advances the understanding of when spectral data suffices to determine geometry by providing a discrete, computable invariant that complements the spectrum.

Abstract

Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^*$ Algebras, we uncover a novel, yet familiar, global geometric invariant -- namely the indexed set of integrals of triple products of eigenfunctions of the Laplace-Beltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.

Triple Products of Eigenfunctions and Spectral Geometry

TL;DR

The paper addresses the inverse spectral problem for closed Riemannian manifolds by introducing the triple-product invariants of Laplace-Beltrami eigenfunctions as a discrete global geometric signature. It proves that the map is injective up to isometry, with sufficiency in the multiplicity-one case, thereby enabling isometry determination from spectral data augmented by the set. The main contributions include a concrete framework linking harmonic analysis of pointwise multiplication to geometric isometry classifications, a connection to Abelian -algebras and conformal-field-theory-inspired structures, and a detailed flat-torus example illustrating the method. This work advances the understanding of when spectral data suffices to determine geometry by providing a discrete, computable invariant that complements the spectrum.

Abstract

Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian Algebras, we uncover a novel, yet familiar, global geometric invariant -- namely the indexed set of integrals of triple products of eigenfunctions of the Laplace-Beltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.
Paper Structure (7 sections, 2 theorems, 15 equations)

This paper contains 7 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Proof of Theorem \ref{['thm']}
  5. Discussion of Conjecture \ref{['conj']}
  6. Example
  7. Acknowledgements

Key Result

Theorem 1

Given a (non-decreasing on the eigenvalues) orthonormal basis of eigenfunctions $\set{e^i}_{i=0}^{\infty}$ for the (non-negative) Laplacian $\Delta_M$ on $L^2(M,g)$ associated with a closed Riemannian manifold $(M,g)$, define To be isometric to $(M,g)$, it is a necessary and sufficient condition for another isospectral closed Riemannian manifold to have an orthonormal basis of eigenfunctions (for

Theorems & Definitions (5)

  • Theorem 1
  • Conjecture 2
  • proof
  • Lemma 3
  • proof : Proof of Lemma \ref{['lma']}