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Compact four-manifolds with pinched self-dual Weyl curvature

Inyoung Kim

TL;DR

The paper investigates compact oriented four-manifolds with harmonic self-dual Weyl curvature under curvature pinching, extending anti-self-duality results via Guan’s method. By analyzing W^{±} eigenvalues and holomorphic sectional curvature on Kähler–Einstein surfaces, it derives conditions under which curvatures force constant holomorphic sectional curvature and leads to classifications like CP^{2} or CP^{1}×CP^{1}. It also shows nonpositive biorthogonal curvature implies nonnegative signature and, in zero-signature cases, local product structure, using Weitzenböck formulas and curvature identities. Additionally, the work unifies known results for δW^{+}=0 with constant |W^{+}|, establishing when the manifold is conformal to a Kahler metric with positive scalar curvature or anti-self-dual, under det W^{+} constraints and pinching assumptions.

Abstract

We consider compact oriented four-manifolds with harmonic self-dual Weyl curvature in addition to a pinching condition.

Compact four-manifolds with pinched self-dual Weyl curvature

TL;DR

The paper investigates compact oriented four-manifolds with harmonic self-dual Weyl curvature under curvature pinching, extending anti-self-duality results via Guan’s method. By analyzing W^{±} eigenvalues and holomorphic sectional curvature on Kähler–Einstein surfaces, it derives conditions under which curvatures force constant holomorphic sectional curvature and leads to classifications like CP^{2} or CP^{1}×CP^{1}. It also shows nonpositive biorthogonal curvature implies nonnegative signature and, in zero-signature cases, local product structure, using Weitzenböck formulas and curvature identities. Additionally, the work unifies known results for δW^{+}=0 with constant |W^{+}|, establishing when the manifold is conformal to a Kahler metric with positive scalar curvature or anti-self-dual, under det W^{+} constraints and pinching assumptions.

Abstract

We consider compact oriented four-manifolds with harmonic self-dual Weyl curvature in addition to a pinching condition.

Paper Structure

This paper contains 5 sections, 31 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Kähler-Einstein surfaces and holomorphic sectional curvature
  3. Kähler surfaces with nonpositive biorthogonal curvature
  4. Harmonic self-dual Weyl curvature with constant length
  5. Pinched self-dual Weyl curvature and nonpositive scalar curvature

Key Result

Lemma 1

Let $(M, g, J)$ be an almost-Hermitian four-manifolds. Then given an anti-self-dual 2-form $\phi$ with length $\sqrt{2}$, there exists an orthonormal basis $\{X, JX, Y, JY\}$ such that $\phi=X\wedge JX-Y\wedge JY$.

Theorems & Definitions (56)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • ...and 46 more