Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear potential theory and Ricci-pinched 3-manifolds

Luca Benatti, Ariadna León Quirós, Francesca Oronzio, Alessandra Pluda

Abstract

Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold. In this short note, we give an alternative proof, based on the nonlinear potential theory, of the fact that if $(M,g)$ satisfies the Ricci-pinching condition and superquadratic volume growth, then it is flat. This result is one of the building blocks of the proof of Hamilton's pinching conjecture.

Nonlinear potential theory and Ricci-pinched 3-manifolds

Abstract

Let be a complete, connected, noncompact Riemannian -manifold. In this short note, we give an alternative proof, based on the nonlinear potential theory, of the fact that if satisfies the Ricci-pinching condition and superquadratic volume growth, then it is flat. This result is one of the building blocks of the proof of Hamilton's pinching conjecture.
Paper Structure (9 sections, 9 theorems, 26 equations)

This paper contains 9 sections, 9 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Basic results
  5. Monotone quantities
  6. Proof of the main theorem
  7. Weak Gauss-Bonnet theorem
  8. Asymptotic behavior of $\mathscr{F}_p$ and $\mathscr{G}_p$
  9. Proof of theorem \ref{['main']}

Key Result

Theorem 1

Let $(M, g)$ be a complete, connected Riemannian $3$-manifold. Suppose that $M$ is Ricci-pinched. Then, $M$ is flat or compact.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Proposition 9