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First eigenvalue estimates on complete balanced Hermitian manifolds

Liangdi Zhang

TL;DR

The paper develops sharp lower bounds for the first eigenvalue $\lambda_1$ of the Laplace–de Rham operator on complete balanced Hermitian manifolds by exploiting curvature of the Strominger–Bismut connection. It extends classical Lichnerowicz–Obata, Li–Yau, and Zhong–Yang estimates from Riemannian/Kähler settings to the Hermitian balanced context, using Bochner-type identities and a carefully constructed integral formula. Central tools include the complexified holomorphic Ricci curvature $\mathcal{R}ic^{SB,\mathbb{C}}$ and the SB torsion, with results showing that $\lambda_1$ is controlled by holomorphic curvature bounds and diameter, and in some cases yields rigidity (e.g., isometry to $\mathbb{CP}^1$ under Kählerity). The integral identity in Theorem 1.4 provides a versatile framework to relate spectral data to geometric torsion and curvature, enabling further corollaries and potential extensions. Overall, the work broadens spectral geometry on complex manifolds by integrating Hermitian connections beyond the Kähler paradigm.

Abstract

In analogy with classical results in Riemannian geometry, we establish estimates for the first eigenvalue of the Laplace-de Rham operator on complete balanced Hermitian manifolds in terms of either the holomorphic Ricci curvature or the holomorphic sectional curvature associated with the Strominger-Bismut connection.

First eigenvalue estimates on complete balanced Hermitian manifolds

TL;DR

The paper develops sharp lower bounds for the first eigenvalue of the Laplace–de Rham operator on complete balanced Hermitian manifolds by exploiting curvature of the Strominger–Bismut connection. It extends classical Lichnerowicz–Obata, Li–Yau, and Zhong–Yang estimates from Riemannian/Kähler settings to the Hermitian balanced context, using Bochner-type identities and a carefully constructed integral formula. Central tools include the complexified holomorphic Ricci curvature and the SB torsion, with results showing that is controlled by holomorphic curvature bounds and diameter, and in some cases yields rigidity (e.g., isometry to under Kählerity). The integral identity in Theorem 1.4 provides a versatile framework to relate spectral data to geometric torsion and curvature, enabling further corollaries and potential extensions. Overall, the work broadens spectral geometry on complex manifolds by integrating Hermitian connections beyond the Kähler paradigm.

Abstract

In analogy with classical results in Riemannian geometry, we establish estimates for the first eigenvalue of the Laplace-de Rham operator on complete balanced Hermitian manifolds in terms of either the holomorphic Ricci curvature or the holomorphic sectional curvature associated with the Strominger-Bismut connection.

Paper Structure

This paper contains 9 sections, 25 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Chern connection
  4. The Strominger--Bismut connection.
  5. Some operators and identities
  6. Lichnerowicz--Obata type estimates
  7. Li--Yau type estimates
  8. Zhong--Yang type estimates
  9. An integral formula and the proof of Theorem \ref{['thm1.4']}

Key Result

Theorem 1.1

Let $(M,\omega)$ be a complete balanced Hermitian manifold of complex dimension $n$. If there exists some constant $K>0$ such that the holomorphic Ricci curvature of the Strominger--Bismut connection satisfies then When equality is achieved in 1.2x, one has Furthermore, if in addition $\omega$ is Kähler, then $(M,\omega)$ is, up to a scaling of the metric, isometric to the complex projective li

Theorems & Definitions (39)

  • Theorem 1.1
  • Example 1.2
  • Conjecture 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 29 more