# Characterizing Classical Minimal Surfaces Via the Entropy Differential

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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.

### Keywords

Minimal surfaces Conservation laws Schwarzian derivative### Mathematics Subject Classification

53A10 70S10## Notes

### Acknowledgements

The authors would like to thank Rob Kusner and Daniel Fox for several stimulating discussions regarding the topics of this paper. The authors are also grateful to the anonymous referee for carefully reading the article and many useful suggestions. The first author was partially supported by the NSF Grants DMS-0902721 and DMS-1307953 and by the EPSRC Programme Grant “Singularities of Geometric Partial Differential Equations” number EP/K00865X/1. The second author was partially supported by the Mathematical Sciences Research Institute in Berkeley and by the Forschungsinstitut für Mathematik at ETH Zürich.

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