Constant $k$th-mixed curvature

TL;DR

This work introduces the kth-mixed curvature $\mathcal{C}^{(k)}_{\alpha,\beta}$ for Hermitian manifolds, blending the $k$-th Chern Ricci curvature with holomorphic sectional curvature under $\beta\neq0$. Using averaging techniques, Weyl tensor analysis, and conformal-change arguments, the authors prove that compact Hermitian surfaces with constant $\mathcal{C}^{(k)}_{\alpha,\beta}$ are self-dual, and that constant 2nd-mixed curvature forces the metric to be Kähler; they also obtain partial results in higher dimensions for certain $(\alpha,\beta)$ relationships. The results extend understanding of curvature constraints beyond the Kähler setting and connect curvature constancy to global geometric structures like self-duality, Kählerity, and balanced metrics. These findings contribute to the broader program of classifying non-Kähler Hermitian manifolds under curvature constraints and provide concrete rigidity phenomena in low dimensions with implications for higher-dimensional geometry.

Abstract

In this paper, we consider general $k$th-mixed curvature $\mathcal{C}^{(k)}_{α,β}$ ($β\neq0$) for Hermitian manifolds, which is a convex combination of the $k$th Chern Ricci curvature and holomorphic sectional curvature. We prove that any compact Hermitian surface with constant $k$th-mixed curvature is self-dual. Furthermore, we show that if a compact Hermitian surface has constant 2th-mixed curvature $c$, then the Hermitian metric must be Kähler. For the higher-dimensional case, when the parameters $α$ and $β$ satisfy certain conditions, we can also obtain partial results.

Theorems & Definitions (12)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Lemma 2.1: ADM
• Lemma 3.1
• Lemma 3.2