Formation of singularities evolving by mean curvature (Q2717348)

The flow by mean curvature is defined for a regular \(d\)-submanifold \({\mathcal M}\) in \(\mathbb{R}^{d+1}\) and for \(t\) belonging to some maximal interval \([0,T[\), by \({\mathcal M}_t= F({\mathcal M},t)\), where \(F:{\mathcal M}\times [0,T[\to \mathbb{R}^{d+1}\) solves \(\frac{\partial F}{\partial t}=-H\nu\), \(F(\cdot,0)= \text{Id}\); \(H(\cdot,t)\) and \(\nu(\cdot,t)\) denoting respectively the mean curvature and the normal vector field of \({\mathcal M}_t\). For example, if \({\mathcal M}\) is a sphere then \({\mathcal M}_t\) shrinks to a point at finite time \(T\). The same is true if \({\mathcal M}\) bounds a convex compact. Moreover in this case the shape of \({\mathcal M}_t\) tends to become spherical. NEWLINENEWLINENEWLINEIn general, \(T\) is the time of appearance of a singularity. It can be infinite, but is finite for compact \({\mathcal M}\)'s. If \(H\geq 0\) holds for \(t=0\), then it holds for \(t\in [0,T[\). To investigate the shape of the singularity, for \({\mathcal M}\) compact with \(H\geq 0\) henceforth, a general way of rescaling \({\mathcal M}_t\), near its singularity time \(T\), into some \(\widetilde{\mathcal M}_t\), is defined, such that \(\widetilde{\mathcal M}_t\) also evolves by mean curvature flow, and bounds a convex. The author finally gives the following alternative for the formation of singularity: either \(\widetilde{\mathcal M}_t\) shrinks by dilatations, or \(\widetilde{\mathcal M}_t\) moves by constant speed translation.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00026].