On the isoperimetric inequality on a minimal surface

Andrew Stone
August 2003
Calculus of Variations & Partial Differential Equations;Aug2003, Vol. 17 Issue 4, p369
Academic Journal
We derive an explicit formula for the isoperimetric defect $L�2 - 4\pi A$ of an arbitrary minimal surface $\Sigma�2 \subset {\bf R}�n$,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the "correct sign". As a by-product of these computations we show that the best known universal isoperimetric estimate, that $L�2 \geq 2\pi A$ for any minimal surface $\Sigma�2 \subset {\bf R}�n$ (due to L. Simon), may be improved to the universal estimate $L�2 \geq 2\sqrt{2} \pi A$.


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