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arXiv · 2304.11129

The symmetric (log-)epiperimetric inequality and a decay-growth estimate

Nick Edelen·Luca Spolaor·Bozhidar Velichkov
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Abstract

We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: as the radius increases energy decays while negative and grows while positive. One can view the symmetric epiperimetric inequality as giving a log-convexity of energy, analogous to the 3-annulus lemma or frequency formula. We establish the symmetric epiperimetric inequality for some free-boundary problems and almost-minimizing currents, and give some applications including a ``propagation of graphicality'' estimate, uniqueness of blow-downs at infinity, and a local Liouville-type theorem.

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Nick Edelen, Luca Spolaor, Bozhidar Velichkov. 2023-04-21. The symmetric (log-)epiperimetric inequality and a decay-growth estimate. https://arxiv.org/abs/2304.11129

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