Calculus of Variations and Geometric Measure Theory
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Isoperimetric Problems Between Analysis and Geometry

Almgren’s unfinished manuscript “Isoperimetric inequalities for anisotropic surface energie”

Jean Taylor (Rutgers)

created by paolini on 02 Jun 2014

16 jun 2014 -- 10:00   [open in google calendar]


Fred Almgren’s list of publications at his death in 1997 included the unfinished manuscript of the title of this talk. My objective is to state and explain the theorem and give an outline of Almgren’s method of proof. I would hope that this workshop can determine whether this result has been proved by someone else during the intervening 17 years. If so, I will have given a summary of a result that people should know. If not, I hope to enlist one or more participants in helping me to complete the manuscript for publication. Briefly, the usual isoperimetric inequality states that given an oriented rectifiable closed curve C in $\mathbb R^3$, there exists an oriented rectifiable surface S with boundary C such that the area of S is no greater than $1/(4π)$ times the square of the length of C. A much more useful version applies to every S with boundary C, with the integral over S of its mean curvature being added to the length of C. This sum of terms is in fact |δV |(R3), the first variation measure of $\mathbb R^3$ for the varifold V associated to S. More generally, Allard’s “On the first variation of a varifold” gives this version of the isoperimetric inequality for rectifiable varifolds. Almgren’s manuscript seeks to prove it for any anisotropic surface energy (parametric integrand) Φ belonging to a class that he identifies, so that the mean curvature is replaced by the weighted mean curvature appropriate to Φ.

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