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Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

Weike Yu

TL;DR

The paper addresses the prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree using a flow that preserves the conformal class. It proves global existence for arbitrary smooth target functions $f$ and, under a balanced metric assumption together with the existence of a supersolution $u^*$, proves convergence of the flow to a conformal metric with $S^{Ch} = f$. It derives corollaries for the cases $f \le 0$ and for small-sign-changing perturbations, aided by an explicit supersolution construction. The analysis combines Krylov-Safonov parabolic estimates, energy monotonicity under a balanced Gauduchon metric, and a Simon-type compactness argument to obtain convergence to a stationary solution.

Abstract

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we give some sufficient conditions on the candidate curvature function $f$ which guaranties the convergence of the flow to a conformal Hermitian metric whose Chern scalar curvature is $f$.

Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

TL;DR

The paper addresses the prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree using a flow that preserves the conformal class. It proves global existence for arbitrary smooth target functions and, under a balanced metric assumption together with the existence of a supersolution , proves convergence of the flow to a conformal metric with . It derives corollaries for the cases and for small-sign-changing perturbations, aided by an explicit supersolution construction. The analysis combines Krylov-Safonov parabolic estimates, energy monotonicity under a balanced Gauduchon metric, and a Simon-type compactness argument to obtain convergence to a stationary solution.

Abstract

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we give some sufficient conditions on the candidate curvature function which guaranties the convergence of the flow to a conformal Hermitian metric whose Chern scalar curvature is .
Paper Structure (5 sections, 11 theorems, 70 equations)

This paper contains 5 sections, 11 theorems, 70 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Global existence of the flow
  4. Long-time behavior of the flow
  5. The construction of super-solutions

Key Result

Theorem 1.1

Let $(M^n, \omega_0)$ be a compact Hermitian manifold with Gauduchon degree $\Gamma(\{\omega_0\})<0$. Then for any function $f\in C^\infty(M)$ and any $u_0\in C^\infty(M)$, the flow 1.2 has a unique global smooth solution $u\in C^\infty(M\times [0,\infty))$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Lemma 2.1: cf. [Gau]
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 3.1
  • ...and 2 more