The surgery and level-set approaches to mean curvature Flow
Head, John · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2011
We study mean curvature flow (MCF) of smooth, closed hypersurfaces in Euclidean space. In the setting of two-convex surfaces of dimension at least three, Huisken and Sinestrari have recently developed a surgery-based approach to extending the classical evolution beyond the singular time. According to their program, one interrupts smooth MCF shortly before the singular time and manually excises appropriate high-curvature regions – so as to effect a fixed drop in the curvature of the surface – before resuming the smooth flow and repeating the procedure. This algorithm is controlled (across all surgeries) by a set of parameters which depend only on the initial data. The starting point for our work in this thesis is the observation that the choice of surgery times and locations – that is, the choice of surgery parameters – is not canonical. This motivates our central result, which can be thought of as a reconciliation between the flow with surgeries and the wellknown weak solution of the level-set flow. These are (heretofore) independent attempts at a geometrically reasonable model of mean curvature flow beyond the singular time. More specifically, we prove that the weak solution can be approximated (in an appropriate quantitative sense) by MCF with “small-scale” surgeries. This is a result of great utility as it can be used to establish new regularity properties of the weak solution; we additionally record some consequences of this nature. Our first object of study is classical MCF: we use estimates developed by Huisken and Sinestrari to bound certain Lp norms of the mean curvature under the smooth evolution. This result is of independent interest and boasts various novel features and consequences. It bounds, for example, the corresponding Lp norms of the second fundamental which play a central role in regularity theory for MCF – most notably in recent work by Ecker on the size of the singular set at the first singular time. We then discuss Huisken and Sinestrari’s MCF with surgeries. We prove that the aforementioned estimates for the classical evolution survive the explicit surgery procedure introduced by Huisken and Sinestrari. This leads to a new bound on the required number of surgeries which depends explicitly on the surgery parameters and which is essential for our main application. We finally consider a sequence of MCFs with surgeries along which we vary the surgery parameters such that the regions removed by surgery become smaller and smaller. The key ingredient in our analysis is a new geometric barrier construction controlling the relative positions of the weak evolution and the flow with surgeries; our approach makes use of familiar tools from the theory of weak solutions including Brakke’s “clearing out lemma”. In combination with the estimate on the required number of surgeries described above, our barrier construction dictates that the sequence of MCFs with surgeries converges (in an appropriate limit of the surgery parameters) to the weak solution of the level-set flow.