Minimal surfaces and geometric flows

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The PI proposes, jointly with Toby Colding, to continue ourinvestigations on minimal surfaces and related areas of geometricanalysis, including geometric evolution equations such as the meancurvature (MCF) and Ricci flow. Our lamination theorem and one-sided curvature estimateplayed a key role in a number of results on minimal surfaces including theMeeks-Rosenberg proof of uniqueness of the helicoid, our proofof the Calabi-Yau Conjectures, and the Meeks-Perez-Ros proof ofthe uniqueness of the Riemann examples. Minimal surfaces with uniform curvature (or area) bounds have beenwell understood and the regularity theory is complete, yetessentially nothing was known without such bounds. The study isdivided into three cases depending on the topology of the surface. Our understanding is relatively complete in the first case when the surface is a disk.The main problems are to get a finer understanding of compactness in the higher genus cases, to understand singularities of potential limit laminations (especially generic properties), and to understand moduli spaces like the space of genus one helicoids of M. Weber, D. Hoffman and M. Wolf (cf. with Hoffman and B. White). The second broad area of the proposal centers on the study of singularities in MCF, including estimates for the size of the singular sets, a compactness theorem for the possible types of singularities, and a classification of generic singularities of the flow. Minimal surfaces are surfaces that locally minimize their surface area and, thus, model soap films where surface tension is in perfect equilibrium and the film does not change other time.They have been studied at least since Lagrange's 1762 memoir, but recent years have seenbreakthroughs on many long--standing problems in the theory ofminimal surfaces, with important contributions from manymathematicians. There is a time-varying analog of this where a surface (which is not in equilibrium) evolves to minimize its surface area as quickly as possible; this is called mean curvature flow or MCF. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Clearly, minimal surfaces remain static under the MCF. MCF and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth.While key foundational results have been obtained, several of the most basic questions remain unanswered.