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Prescribed Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures

Weike Yu

TL;DR

This work investigates prescribing the Chern scalar curvature on complete noncompact Hermitian manifolds with nonpositive curvature, formulating a Chern-Yamabe-type equation and proving existence of a complete conformal metric with negative target curvature under geometric decay conditions. The author develops a monotone iteration framework anchored by lower-solution construction, leveraging Laplacian comparison and barrier techniques to handle the noncompact setting. A key contribution is establishing a precise necessary-and-sufficient condition for solvability when the target curvature is negative, and deriving explicit existence results under various growth/decay regimes for the background geometry (second Chern Ricci curvature and torsion) and the prescribed function S. The results extend Aviles–McOwen-type existence theory to higher-dimensional complete Hermitian manifolds and provide a flexible toolkit for creating complete metrics with prescribed negative Chern scalar curvature, including corollaries for constant targets like $-1$.

Abstract

In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures, and establish some existence results. In particular, we obtain some sufficient conditions for the existence of a constant negative Chern scalar curvature metric in the conformal class.

Prescribed Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures

TL;DR

This work investigates prescribing the Chern scalar curvature on complete noncompact Hermitian manifolds with nonpositive curvature, formulating a Chern-Yamabe-type equation and proving existence of a complete conformal metric with negative target curvature under geometric decay conditions. The author develops a monotone iteration framework anchored by lower-solution construction, leveraging Laplacian comparison and barrier techniques to handle the noncompact setting. A key contribution is establishing a precise necessary-and-sufficient condition for solvability when the target curvature is negative, and deriving explicit existence results under various growth/decay regimes for the background geometry (second Chern Ricci curvature and torsion) and the prescribed function S. The results extend Aviles–McOwen-type existence theory to higher-dimensional complete Hermitian manifolds and provide a flexible toolkit for creating complete metrics with prescribed negative Chern scalar curvature, including corollaries for constant targets like .

Abstract

In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures, and establish some existence results. In particular, we obtain some sufficient conditions for the existence of a constant negative Chern scalar curvature metric in the conformal class.
Paper Structure (4 sections, 17 theorems, 122 equations)

This paper contains 4 sections, 17 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A sufficient and necessary condition for the existence of solutions
  4. The construction of lower solutions and the proof of main theorems

Key Result

Theorem 1.1

Let $(M^n, \omega)$ be a complete noncompact Hermitian manifold with its second Chern Ricci curvature and the torsion where $X, Y\in T^{1,0}_{x}M$ satisfy $\|X\|=\|Y\|=1$, $r(x)$ is the Riemannian distance between a fixed point $x_0$ and $x$ in $(M, \omega)$, $C_1, C_2>0$ and $\alpha\geq 0$ are three constants. Suppose that the Chern scalar curvature of $(M, \omega)$ satisfies where $D$ is a co

Theorems & Definitions (18)

  • Theorem 1.1: =Theorem \ref{['thm4.5']}
  • Corollary 1.2
  • Theorem 1.3: =Theorem \ref{['theorem4.7']}
  • Corollary 1.4: [Yuan], [Yu5]
  • Lemma 2.1: cf. [Gau]
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • ...and 8 more