Curvature tensor under the complete non-compact Ricci Flow

TL;DR

This paper proves that under the Ricci flow on a complete non-compact manifold with bounded Ricci curvature, the full curvature tensor remains uniformly bounded up to finite time T.

Contribution

It establishes the boundedness of the curvature tensor under Ricci flow given bounded Ricci curvature on complete non-compact manifolds, extending previous results.

Findings

Curvature tensor remains bounded under specified conditions.

Results apply to finite-time Ricci flow solutions.

Provides new insights into curvature behavior on non-compact manifolds.

Abstract

We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\times [0,T)$. Some other results are also presented.