Mathematical Research Letters

Volume 13, Issue 3, May 2006 pp. 333-342.

Area comparison results for isotropic surfaces

Authors: Edward Goldstein
Author institution: Brandeis University

Summary: We prove an area comparison theorem for non-orientable isotropic surfaces with the same boundary as holomorphic curves: There exists a positive constant $\lambda$, $3 \leq \lambda \leq \frac{3 \pi}{2 \sqrt{2}}$ such that the following hold:\\ 1) Consider $\CC^1 \subset \CC^2$ and let $D$ be a bounded region in $\CC^1$ with piecewise smooth boundary. Let $L(D)$ be the infimum of areas of all Lagrangian surfaces in $\CC^2$ with the same boundary as $D$. Then $L(D)= \lambda \cdot Area(D)$.\\ 2) Consider $\CC P^1 \subset \CC P^n$ and let $D$ be a region in $\CC P^1$ with piecewise smooth boundary. Let $I(D)$ be the infimum of areas of all isotropic surfaces in $\CC P^n$ with the same boundary as $D$ representing the same relative homology class mod $2$ as $D$. Then $ 2 \cdot Area(D) \leq I(D) \leq \lambda \cdot Area(D)$. Moreover the first inequality becomes an equality for $D=\CC P^1$.

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