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Spectral Deformation Flow and Dimension Recovery: Invariant-Based Rigidity for Simply-Connected Closed Manifolds

Anton Alexa

Abstract

We study an effective spectral deformation flow for mode amplitudes $C_n(τ)$, governed by a second-order self-adjoint operator $\hat{C}$ on a compact interval. The flow is encoded in the multi-function $C(v,τ,n)$ and exhibits global stabilization toward a symmetric spectral attractor. To connect this dynamics with geometry, we introduce a deformation-spectrum encoding of compact Riemannian manifolds through a shifted Laplace--Beltrami spectrum. Within this framework, we analyze energy decay, entropy decay, and the bulk asymptotic spectral density of the encoded manifold spectrum, obtaining an information-theoretic and spectral route to dimension recovery. We further formulate a rigidity criterion showing that, when the deformation spectral invariants coincide with those of the round sphere, the spherical profile is the unique manifold-compatible asymptotic realization within the present framework. In dimension four, this yields a topological conclusion together with a spectral obstruction against exotic smooth structures that produce distinct invariants. The results position the spectral flow as an effective geometric model, rather than as a direct replacement for tensorial geometric flows on arbitrary manifolds.

Spectral Deformation Flow and Dimension Recovery: Invariant-Based Rigidity for Simply-Connected Closed Manifolds

Abstract

We study an effective spectral deformation flow for mode amplitudes , governed by a second-order self-adjoint operator on a compact interval. The flow is encoded in the multi-function and exhibits global stabilization toward a symmetric spectral attractor. To connect this dynamics with geometry, we introduce a deformation-spectrum encoding of compact Riemannian manifolds through a shifted Laplace--Beltrami spectrum. Within this framework, we analyze energy decay, entropy decay, and the bulk asymptotic spectral density of the encoded manifold spectrum, obtaining an information-theoretic and spectral route to dimension recovery. We further formulate a rigidity criterion showing that, when the deformation spectral invariants coincide with those of the round sphere, the spherical profile is the unique manifold-compatible asymptotic realization within the present framework. In dimension four, this yields a topological conclusion together with a spectral obstruction against exotic smooth structures that produce distinct invariants. The results position the spectral flow as an effective geometric model, rather than as a direct replacement for tensorial geometric flows on arbitrary manifolds.

Paper Structure

This paper contains 28 sections, 27 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Spectral Geometry and the Multifunction $C(v, \tau, n)$
  3. Spectral Operator and Eigenbasis
  4. Definition of the Multifunction
  5. Spectral Space and Geometric Interpretation
  6. Geometric Encoding of Compact Manifolds
  7. Variational Spectral Flow and Global Stability
  8. Action Principle and Flow Equation
  9. Energy Functional and Entropy Decay
  10. Cubic Interactions and Nonlinear Coupling
  11. Global Well-Posedness and Exponential Convergence
  12. Global Convergence under Nonlinear Interactions
  13. Canonical Structure and Quantization
  14. Spectral Rigidity for Simply-Connected Closed Manifolds
  15. Asymptotic Behavior and Spectral Stabilization
  16. ...and 13 more sections

Key Result

Lemma 1

For each $k \in \mathbb{N}$, there exists a constant $C_k > 0$ such that

Theorems & Definitions (77)

  • Lemma 1: Derivative Bounds
  • proof
  • Definition 1: Spectral Multifunction
  • Definition 2: Deformation Spectrum
  • Definition 3: Renormalized Spectral Representative
  • Proposition 1: Square-Summability of the Renormalized Deviation
  • proof
  • Theorem 1: Global Exponential Convergence of the Spectral Flow
  • proof
  • Theorem 2: Global Convergence of the Nonlinear Spectral Flow
  • ...and 67 more