Minimal Surfaces and Mean Curvature Flow

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AbstractAward: DMS 1404282, Principal Investigator: Brian WhiteA minimal surface is the mathematical counterpart to a soap film spanning a curved wire. The elegant appearance of a film of soap is associated to the soap film's property of spanning the wire in the most efficient way possible, where efficiency here means using as little surface area as possible. The physical elegance of the soap film is reflected in the mathematical description of its efficiency, which was discovered in the 18th century and continues to motivate progress in geometry and differential equations. Among the topics to be pursued under the support of this grant are the properties of helicoid-like minimal surfaces, which spiral in space like the surface of a screw or a multi-story parking ramp; the classical helicoid was discovered in the 1770s to be a minimal surfaces, but a number of other examples that closely resemble helicoids in the large but are more complicated near the origin were discovered only in the last ten years.The principal investigator plans to study helicoid-like minimal surfaces, densities of minimal cones, the dependence of a minimal surface on the total curvature of its boundary, the branching behavior of minimal surfaces, and properties of mean curvature flow, particularly singularity formation and the non-uniqueness known as "fattening." The methods to be employed are a combination of classical minimal surface theory, geometric measure theory, and partial differential equations.