On the $\kappa-$solutions of the Ricci flow on noncompact 3-manifolds

TL;DR

This paper proves that certain noncompact 3-manifolds with positive curvature cannot develop finite-time singularities under Ricci flow, supporting a conjecture by Perelman.

Contribution

It establishes a nonexistence result for $ppa$-solutions with specific curvature blow-up conditions, partially confirming Perelman's conjecture.

Findings

No $ppa$-solutions with positive sectional curvature blow up at finite time under given conditions.

Supports Perelman's conjecture on Ricci flow singularities.

Advances understanding of Ricci flow behavior on noncompact 3-manifolds.

Abstract

In this paper we prove that there is no $\kappa$-solution of Ricci flow on 3-dimensional noncompact manifold with strictly positive sectional curvature and blow up at some finite time $T$ satisfying $\int^T_0 \sqrt{T-t} R(p_0,t)dt< \infty$ for some point $p_0$. This partially confirms a conjecture of Perelman.