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The Kähler-Ricci flow on Kähler manifolds with 2 traceless bisectional curvature operator

X. X. Chen, H. Li

TL;DR

The paper analyzes the Kähler-Ricci flow on compact Kähler manifolds through the lens of the traceless bisectional curvature operator. It proves that the 2-positive traceless condition is preserved in all dimensions and that it enforces nonnegativity of the orthogonal bisectional curvature, with scalar curvature controlling the full curvature. Under scalar-curvature bounds, the flow admits subsequence convergence to a Kähler-Ricci soliton, extending prior surface-level results to higher dimensions. The results hinge on a blend of Hamilton's maximum principle for tensors and careful analysis of the traceless curvature operator on Λ^{1,1}_0(X).

Abstract

It was proved by H. Chen earlier that the property of the sum of any two eigenvalues of the curvature operator is positive is preserved under the ricci flow in all dimensional. By a recent result of Phong-Sturm, a similar notion of positive 2-traceless bisectional curvature positive is preserved on complex surface. We prove that this holds in all dimensional K\"ahler manifold. Moreover, the scalar curvature controls full curvature for this type of metrics.

The Kähler-Ricci flow on Kähler manifolds with 2 traceless bisectional curvature operator

TL;DR

The paper analyzes the Kähler-Ricci flow on compact Kähler manifolds through the lens of the traceless bisectional curvature operator. It proves that the 2-positive traceless condition is preserved in all dimensions and that it enforces nonnegativity of the orthogonal bisectional curvature, with scalar curvature controlling the full curvature. Under scalar-curvature bounds, the flow admits subsequence convergence to a Kähler-Ricci soliton, extending prior surface-level results to higher dimensions. The results hinge on a blend of Hamilton's maximum principle for tensors and careful analysis of the traceless curvature operator on Λ^{1,1}_0(X).

Abstract

It was proved by H. Chen earlier that the property of the sum of any two eigenvalues of the curvature operator is positive is preserved under the ricci flow in all dimensional. By a recent result of Phong-Sturm, a similar notion of positive 2-traceless bisectional curvature positive is preserved on complex surface. We prove that this holds in all dimensional K\"ahler manifold. Moreover, the scalar curvature controls full curvature for this type of metrics.

Paper Structure

This paper contains 10 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Basic Kähler geometry
  3. Setup of notations
  4. The Kähler Ricci flow
  5. The traceless bisectional curvature operator
  6. Definition and the evolution equations
  7. Geometric properties of the traceless bisectional curvature operator
  8. Proof of Theorem 1.2
  9. Proof of Theorem 1.1
  10. Proof of Corollary 1.4

Key Result

Theorem 1.1

Let X be a compact Kähler manifold of dimension n\geq 2.\; Along the Kähler Ricci flow, we have Under either of these two conditions, the positivity of Ricci tensor is preserved under the Kähler Ricci flow.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 4.1
  • Lemma 5.1
  • Definition 6.1
  • ...and 1 more