A density function and the structure of singularities of the mean curvature flow
The author studies the formation of singularities in the mean curvature flow \({d\over dt} F(p, t)= H(p, t) \nu(p, t)\), where \(F(\cdot, t)\) is a family of smooth embeddings of a closed hypersurface in \(\mathbb{R}^{n+ 1}\), \(\nu(\cdot, t)\) the inward pointing unit normal and \(H(\cdot, t)\) the corresponding mean curvature at ``time'' \(t\). The results of this paper extend and refine previous ones of G. Huisken. Denoting by \(A\) the second fundamental form of a family of hypersurfaces as above one considers the first time \(T\) when \(\max_p|A|^2(p, t)\) becomes infinite. By the results of Huisken, such finite \(T\) exists and \(\max_p|A|^2(p, t)\geq {1\over 2}(T- t)^{- 1}\). As a fundamental hypothesis the converse estimate \(\max_p|A|^2(p, t)\leq {1\over 2} C_0(T- t)^{- 1}\) for some constant \(C_0\) is assumed throughout the paper; it is known to hold in many, though not all cases. For flow starting from a hypersurface with \(H\geq 0\) (and therefore staying in this class) the author's first result reads as follows: Whenever \(p_k\to p\) and \(t_k\uparrow T\) \((k\to \infty)\) such that \(|A|^2(p_k, t_k)\geq \delta(T- t_k)^{- 1}\) for some \(\delta> 0\) then already \(\limsup_{t\to T} (T- t)|A|^2(p, t)> 0\), which puts one in a situation analyzed before by G. Huisken: From any sequence \(t_k\to T\) one can extract a convergent subsequence (again denoted by \(t_k\)) such that the images of \((2(T- t_k))^{- 1/2}(F(\cdot, t_k)- \widehat p)\), \(\widehat p:= \lim_{t\to T} F(p, T)\), converge to a cylinder with cross section some \(m\)-dimensional round sphere, \(1\leq m\leq n\). A further result of the present author then states that the various blow up limits with respect to a fixed singular point \(\widehat p\) can only differ by a rotation of \(\mathbb{R}^{n+ 1}\). Furthermore, the author addresses the question whether blow up of \(|A|^2\) can occur with a slower rate than considered above and he proves curvature estimates which show e.g. that if \(|A|^2\) grows no faster than \(\varepsilon(T- t)^{- 1}\) on some fixed ball around the limit point \(\widehat p\), \(0< \varepsilon< {1/3}\), then \(|A|^2(p, t)\) actually stays bounded for all \(t< T\). These estimates under some additional restrictions also hold for surfaces with boundary. The paper concludes with an extension of Huisken's result on the shrinking of convex initial surfaces to ``round'' points under the mean curvature flow by allowing initial surfaces which may deviate from convex shape on portions of very small surface measure. The technical key point for the proofs is the analysis of density functions obtained by integrating the backward heat kernel with singularity at blow up time \(T\) over the hypersurface \(F(\cdot, t)\).
scientific article; zbMATH DE number 720499