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The curvature operator of the second kind in dimension three

Harry Fluck, Xiaolong Li

Abstract

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that $\a$-positive/$\a$-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all $\a \in [1,5]$.

The curvature operator of the second kind in dimension three

Abstract

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that -positive/-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all .
Paper Structure (5 sections, 14 theorems, 60 equations)

This paper contains 5 sections, 14 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Diagonalization
  4. Proof of Theorem \ref{['thm eigenvalue 3D']}
  5. preserving $\alpha$-nonnegativity of $\mathring{R}$

Key Result

Theorem 1.1

Let $R\in S^2_B(\wedge^2 \mathbb{R}^3)$ be an algebraic curvature operator on $\mathbb{R}^3$ and denote by $a\leq b \leq c$ the eigenvalues of the curvature operator $\hat{R}$. Then the eigenvalues of the curvature operator of the second kind $\mathring{R}$ are given by where

Theorems & Definitions (29)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['thm diagonilization']}
  • ...and 19 more