Existence of mean curvature flow singularities with bounded mean curvature (Q6046463)

The mean curvature flow is one of the most important curvature flows, and has many interesting applications in deriving isoperimetric inequalities. For example, \textit{P. Topping} [J. Reine Angew. Math. 503, 47--61 (1998; Zbl 0909.53044)] used the mean curvature flow tool to get an isoperimetric inequality for closed \(2\)-discs (equipped with some metric \(g\)) if the boundary convexity (of these discs) and some Gaussian curvature constraint were imposed. A classical result by \textit{G. Huisken} [J. Differ. Geom. 20, 237--266 (1984; Zbl 0556.53001)] tells us that for a smooth family of closed, embedded hypersurfaces \(\{\Sigma^{N-1}(t)\}_{t\in[0,T)}\) (in the Euclidean space \(\mathbb{R}^{N}\)) moving by the mean curvature flow, the second fundamental form \(A_{\Sigma(t)}\) blows up at the singularity time \(T<\infty\) in the sense that \[ \limsup_{t\nearrow T}\max_{x\in\Sigma(t)}|A_{\Sigma(t)}(x)|=\infty. \] \textit{W.-X. Shi} [J. Differ. Geom. 30, No. 1, 223--301 (1989; Zbl 0676.53044)] generalized Huisken's this result to noncompact hypersurfaces. One might naturally ask if the mean curvature \(H_{\Sigma(t)}\) necessarily blows up at a finite-time singularity, which is actually Problem 2.4.10 in [\textit{C. Mantegazza}, Lecture notes on mean curvature flow. Basel: Birkhäuser (2011; Zbl 1230.53002)]. The present author makes a remarkable and important progress on this open problem and obtains that for any dimension \(N\geq8\) there exists a smooth, properly embedded mean curvature flow of complete, noncompact hypersurfaces \(\{\Sigma^{N-1}(t)\subset\mathbb{R}^N\}_{t\in[0,T)}\) such that \[ \limsup_{t\nearrow T}\sup_{x\in\Sigma(t)}|A_{\Sigma(t)}(x)|=\infty, \quad \text{and} \quad \sup_{t\in[0,T)}\sup_{x\in\Sigma(t)}|H_{\Sigma(t)}(x)|<\infty. \] This means, roughly speaking, that for the mean curvature flow, in general, the mean curvature needs not blow up at a finite-time singularity.