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Choi-Wang inequality for affine connections

Yasuaki Fujitani

TL;DR

The paper extends the Choi-Wang lower bound for the first nonzero eigenvalue of the Laplacian on minimal hypersurfaces from manifolds with positive Ricci curvature to the Li-Xia type affine-connection setting, assuming $\mathrm{Ric}^D \ge K e^{(\alpha-\beta)u} g$. It develops a statistical manifold framework for the Li-Xia connection, establishes a Bochner-type inequality and equiaffine structure, and then uses a Reilly-type formula for the $D$-Laplacian to prove a Choi-Wang type inequality: $\lambda_1(\Delta^D_\Sigma) \ge \tfrac{K}{2}$. The work also proves a Bochner-type theorem implying Betti-number bounds in this setting and discusses limitations related to the finiteness of the fundamental group and Bishop-Gromov-type comparison, outlining future directions. By connecting affine-connection geometry with weighted curvature notions, the paper broadens spectral-geometry results to a unified framework involving substatic and 1-weighted curvature concepts.

Abstract

Choi-Wang established a lower bound for the first non-zero eigenvalue of the Laplacian on minimal hypersurfaces in manifolds with positive Ricci curvature. We extend this Choi-Wang type inequality to the setting of positive Ricci curvature with respect to the Li-Xia type affine connection.

Choi-Wang inequality for affine connections

TL;DR

The paper extends the Choi-Wang lower bound for the first nonzero eigenvalue of the Laplacian on minimal hypersurfaces from manifolds with positive Ricci curvature to the Li-Xia type affine-connection setting, assuming . It develops a statistical manifold framework for the Li-Xia connection, establishes a Bochner-type inequality and equiaffine structure, and then uses a Reilly-type formula for the -Laplacian to prove a Choi-Wang type inequality: . The work also proves a Bochner-type theorem implying Betti-number bounds in this setting and discusses limitations related to the finiteness of the fundamental group and Bishop-Gromov-type comparison, outlining future directions. By connecting affine-connection geometry with weighted curvature notions, the paper broadens spectral-geometry results to a unified framework involving substatic and 1-weighted curvature concepts.

Abstract

Choi-Wang established a lower bound for the first non-zero eigenvalue of the Laplacian on minimal hypersurfaces in manifolds with positive Ricci curvature. We extend this Choi-Wang type inequality to the setting of positive Ricci curvature with respect to the Li-Xia type affine connection.

Paper Structure

This paper contains 3 sections, 4 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Bochner type theorem
  3. Proof of the Choi-Wang inequality

Key Result

Theorem 1.1

Let $(M,g)$ be a compact orientable Riemannian manifold, $u\in C^{\infty}(M)$ and $\alpha,\beta \in \mathbb{R}$. We set $D:= \nabla^{u,\alpha,\beta}$. For $K > 0$, we assume Let $\Sigma$ be a compact orientable embedded $D$-minimal hypersurface in $M$. Then the first non-zero eigenvalue $\lambda_{1}(\Delta^D_\Sigma)$ of the $D$-Laplacian $\Delta^D_{\Sigma}$ on $\Sigma$ satisfies

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 4 more