On the isoperimetric inequality on a minimal surface
An explicit formula for the isoperimetric defect \(L^2-4\pi A\) of an arbitrary minimal surface \(\Sigma^2\subset\mathbb{R}^n\) of area \(A\), with smooth boundary \(\partial\Sigma\) of length \(L\), is derived -- in terms of a double surface integral of certain geometric quantities (Lemma 5.1). As a by-product the author shows that the best known universal isoperimetric estimate, that \(L^2\geq 2\pi A\) for any minimal surface \(\Sigma^2\subset\mathbb{R}^n\) (due to L. Simon), may be improved by the factor \(\sqrt 2\) to the universal estimate \(L^2\geq 2\pi A\sqrt 2\).
scientific article; zbMATH DE number 1973281