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Second Laplacian eigenvalue on real projective space

Hanna N. Kim

Abstract

In this paper, we prove an upper bound on the second non-zero Laplacian eigenvalue on $n$-dimensional real projective space. The sharp result for 2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric degenerates to that of the disjoint union of a round projective space and a sphere. That conjecture is open in higher dimensions, but this paper proves it up to a constant factor that tends to 1 as the dimension tends to infinity. Also, we introduce a topological argument that deals with the orthogonality conditions in a single step proof.

Second Laplacian eigenvalue on real projective space

Abstract

In this paper, we prove an upper bound on the second non-zero Laplacian eigenvalue on -dimensional real projective space. The sharp result for 2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric degenerates to that of the disjoint union of a round projective space and a sphere. That conjecture is open in higher dimensions, but this paper proves it up to a constant factor that tends to 1 as the dimension tends to infinity. Also, we introduce a topological argument that deals with the orthogonality conditions in a single step proof.
Paper Structure (14 sections, 16 theorems, 90 equations)

This paper contains 14 sections, 16 theorems, 90 equations.

Table of Contents

  1. Introduction and results
  2. Results on the second eigenvalue
  3. Overview of the proof
  4. Veronese embedding of projective space into a sphere
  5. Trial Functions and Orthogonality
  6. Möbius transformations
  7. Spherical caps, reflection, and folding
  8. Center of mass and trial functions
  9. Orthogonality of the trial functions by two-step proof
  10. Orthogonality of the trial functions by one-step proof
  11. Calculation of the degree of the self-maps
  12. Estimating the Rayleigh Quotient --- proof of \ref{['thm']}
  13. Remark about \ref{['conj']}
  14. Calculations

Key Result

Theorem 1

Assume that the metric $wg$ is normalized so that ${\text{Vol}}_n(w)={\text{Vol}}_n(1)$. Then the first nonzero eigenvalue of $-\Delta_{wg}$ satisfies,

Theorems & Definitions (29)

  • Theorem 1: Li and Yau LY82 for $n=2$ and El Soufi and Ilias EI86 for $n \geq 2$; first eigenvalue on ${\mathbb{RP}}^n$
  • Theorem 2: Nadirashvili and Penskoi NA18; second eigenvalue on ${\mathbb{RP}}^2$
  • Conjecture 1: Second eigenvalue on ${\mathbb{RP}}^n$, $n \geq 3$
  • Theorem 3: Second eigenvalue on ${\mathbb{RP}}^n$, $n \geq 3$
  • Definition 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 19 more