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Nam Q. Le

Publications and source records attributed to Nam Q. Le.

57 records · Page 4Linked to original sources
arXiv2010

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example $\int_{0}^{t} \int_{M_{s}} \frac{\abs{A}^{n + 2}}{log (2 + \abs{A})} dμds,$ blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.

math.DG↗
arXiv2009

On the extension of the mean curvature flow

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. In \cite{Cooper} Cooper has recently proved that the mean curvature blows up at the singular time $T$. We show that if the second fundamental form stays bounded from below all the way to $T$, then the scaling invariant mean curvature integral bound is enough to extend the flow past time $T$, and this integral bound is optimal in some sense explained below.

math.DG↗