Mean curvature flow with surgery

Abstract

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a priori estimates for a more general class of flows in a local and flexible setting

Mean Curvature Flow of Mean Convex Hypersurfaces

“…Apart from streamlining the proofs, our results are local and depend only on the value of the Andrews constant. We will exploit this local and universal character in our forthcoming paper [17] to give a new and general construction of mean curvature flow with surgery; in particular, our construction also works in the case of mean convex surfaces in R 3 .…”

“…Note added in May 2015. Since the first version of this paper was posted on arxiv in April 2013, the estimates have been used to construct mean convex flow with surgery in R 3 by Brendle and Huisken [5] in September 2013 and in a paper by the authors [17] in April 2014.…”

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

“…An understanding of the finite-time singularities of these flows has been shown to be very useful in devising protocols for surgery along the flows, which in turn are crucial for using the flows to study the relationship between topological and geometric aspects of manifolds and hypersurfaces. Perelman's proof of the Poincaré and Geometrization Conjectures [21,22] using surgery provides the prime example of this for Ricci flow, while the analysis of two-convex hypersurfaces by Huisken and Sinestrari [20] and the analysis of mean-convex surfaces by Brendle and Huisken [8] provide prime examples of this for MCF (see also the work on MCF with surgery by Haslhofer and Kleiner [17]). …”

Low Complexity Solutions of the Allen–Cahn Equation on Three-spheres

“…By a recent result from Haslhofer-Ketover [HK,Thm. 1.4], which has been established using combined efforts from mean curvature flow with surgery [BH,HK17,BHH] and min-max theory [KMN], there also exists an embedded minimal two-sphere Σ 2 ⊂ (S 3 , g) which has index two. Arguing as above, for ε small enough the associated Pacard-Ritore solutions satisfy (2.4) ind(±u 1 ε ) = ind(Σ 1 ) = 1, and (2.5) ind(±u 2 ε ) = ind(Σ 2 ) = 2.…”