We study the mean curvature evolution of smooth, closed, twoconvex hypersurfaces in Rn+1 for n ≥ 3.Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in [HS3]—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain Lp-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

We prove that a constrained Willmore immersion of a 2–torus into the conformal 4–sphere S4 is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in R4 = S4\{1} for some point 1 2 S4 at infinity. This implies that all constrained Willmore tori in S4 can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin’s approach [19] to the study of harmonic tori in S3.

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