Remarks on the Extension of the Ricci Flow

Abstract

Abstract. We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.

Mixed Integral Norms for Ricci Flow

“…2 . His result was extended by Ma and Cheng to complete manifolds in [13]; similar results were then obtained, for Ricci and Mean Curvature flow, see [6,[9][10][11][12][22][23][24]. In particular, we remark that Theorem 1.6 in [10] considers mixed integral norms for the Mean Curvature flow case.…”

“…boundedness of the L 1 -norm of the maximum of the Ricci curvature is sufficient to extend the flow, as shown in [21] and subsequently in [6].…”

Notes on the extension of the mean curvature flow

“…Analogous extension theorems for the Ricci flow have been proved recently [Wang 2012;He 2014]. Some general regularity results have been obtained by Cheeger, Haslhofer and Naber [Cheeger et al 2013] and Ecker [2013], among others.…”

“…Some general regularity results have been obtained by Cheeger, Haslhofer and Naber [Cheeger et al 2013] and Ecker [2013], among others. To prove our theorems we combine the ideas in [Cooper 2011] and [He 2014]. First, by a suitable blow-up argument we get a minimal submanifold in Euclidean space.…”

Untitled

“…More precisely, we prove the following theorem. Analogous extension theorems for the Ricci flow have been proved recently [Wang 2012;He 2014]. Some general regularity results have been obtained by Cheeger, Haslhofer and Naber [Cheeger et al 2013] and Ecker [2013], among others.…”