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On the isoperimetric inequality on a minimal surface

Authors:	Stone Andrew

Institution:	(1) Economic Research Department, Reserve Bank of Australia, GPO Box 3947, NSW 2001 Sydney, Australia

Abstract:	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 $L^2 \geq 2\sqrt{2} \pi A$ .Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002

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