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arXiv · 1301.1663

Characterizing classical minimal surfaces via the entropy differential

Jacob Bernstein·Thomas Mettler
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Abstract

We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of particular interest is the realization of its real part as a conservation law for a natural geometric functional -- which is, essentially, the entropy of the Gauss curvature. We characterize several classical surfaces -- including Enneper's surface, the catenoid and the helicoid -- in terms of $P$. As an application, we prove a novel curvature estimate for embedded minimal surfaces with small entropy differential and an associated compactness theorem.

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Jacob Bernstein, Thomas Mettler. 2013-01-08. Characterizing classical minimal surfaces via the entropy differential. https://doi.org/10.1007/s12220-017-9759-6

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