Extend Mean Curvature Flow with Finite Integral Curvature

Abstract

In this note, we first prove that the solution of mean curvature flow on a finite time interval [0, T ) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval [0, T ) can be extended over time T if the space-time integration of the mean curvature is finite. Moreover, weshow that these conditions are optimal in some sense.

The mean curvature at the first singular time of the mean curvature flow

“…Note that the constant in Theorem 1.7 does not depend on the initial hypersurface, but only on a dimension. Finally, up to a constant depending on the dimension n and the initial hypersurface M 0 , our estimate (1.9) is implied as well by the estimate (27) in Theorem 14 of [15]. This can be seen using Hölder's inequality and volume bound during the mean curvature flow as in Lemma 1.4 in Ecker [8].…”

“…The precise estimate of the form (1.9) for the case of minimal submanifolds can be found in Shen-Zhu [20], Proposition 2.2 (see also [4]). Moreover, in [18,27,28], the authors showed that if the L n+2 norm in space-time of the second fundamental form (or the mean curvature but under various convexity assumptions) is finite then it is possible to extend the mean curvature flow beyond the time interval under consideration. Our theorem can be viewed as a local version of these results without imposing any convexity assumptions.…”

Remarks on the Extension of the Ricci Flow

“…In [21] B. Wang proved that if For the mean curvature flow, the same extension problem has also been studied. The supremum and certain scaling invariant space-time integrals of the norm of the second fundamental form are known to blow up at a finite singular time ( [11], [17], [18], [24]). Moreover, the surprising fact that a subcritical integral quantity has to blow up was proved by N. Le in [12]: Theorem 1.5 (N.Le).…”

The extension and convergence of mean curvature flow in higher codimension

“…On the other hand, Le-Šešum [12] and Xu-Ye-Zhao [25] obtained some integral conditions to extend the mean curvature flow of hypersurfaces in the Euclidean space independently. Later, Xu-Ye-Zhao [26] generalized these extension theorems to the case where the ambient space is a Riemannian manifold with bounded geometry.…”

“…Let A and H be the second fundamental form and the mean curvature vector of M , respectively. In this paper, we first generalize the extension theorems in [12,25,26] to the mean curvature flow of submanifolds in a Riemannian manifold with bounded geometry. LetÅ be the tracefree second fundamental form, which is defined byÅ = A − 1 n g ⊗ H. Denote by || · || p the L p -norm of a function or a tensor field.…”