Smooth compactness of self-shrinkers (Q424813)

A surface \(\Sigma \subset \mathbb{R}^3\) is said to be a ``self-shrinker'' if it satisfies NEWLINE\[NEWLINE H = \frac{ \langle x , N \rangle }{2} \, , NEWLINE\]NEWLINE where \(H = \text{div} \, N\) is the mean curvature, \(x\) is the position vector, and \(N\) is the unit normal vector. Since self-shrinkers describe all possible blow ups at a given singularity of a mean curvature flow, they model singularities in mean curvature flow.NEWLINENEWLINEIn this paper, the authors prove a smooth compactness theorem for embedded self-shrinkers in \(\mathbb{R}^3\). In the theorem, surfaces are assumed to be homeomorphic to closed surfaces with finitely many disjoint disks removed and the genus of the surface is defined to be the genus of the corresponding closed surface. The result is the following: Given an integer \(g \geq 0\) and a constant \(\rho>0\), the space of smooth complete embedded self-shrinkers \(\Sigma \subset \mathbb{R}^3\) satisfying the following conditions is compact:NEWLINENEWLINE(1) the genus is at most \(g\),NEWLINENEWLINE(2) \(\partial \Sigma = \emptyset\), andNEWLINENEWLINE(3) \(\text{Area} \, \left( B_{r}(x_0) \cap \Sigma \right) \leq \rho \, r^2\) for all \(x_0 \in \mathbb{R}^3\) and all \(r > 0\).NEWLINENEWLINEIn other words, any sequence of such surfaces has a subsequence that converges in the topology of \(C^m\) convergence on compact subsets for any \(m \geq 2\).NEWLINENEWLINEThis compactness theorem plays a key role in understanding the generic mean curvature flow [the authors, Ann. Math. (2) 175, No. 2, 755--833 (2012; Zbl 1239.53084)]. To obtain the result, the authors first use a well-known local singular compactness proposition for embedded minimal surfaces in any Riemannian \(3\)-manifold in order to prove a global singular compactness theorem for self-shrinkers. Then, they show that if the convergence is not smooth, the limiting self-shrinker is \(L\)-stable, where \(L\) is the second order operator given by linearization of the self-shrinker equation [loc. cit.] NEWLINE\[NEWLINE L \, = \Delta \, + |A|^2 \, - \frac{1}{2} \, \langle x , \nabla (\cdot) \rangle + \frac{1}{2} \, \, . NEWLINE\]NEWLINE Finally, they use the following proposition from [loc. cit.]: There are no \(L\)-stable smooth complete self-shrinkers without boundary and with polynomial volume growth in \(\mathbb{R}^{n+1}\).