Area comparison results for isotropic surfaces

Abstract

Consider a 2-plane $P\subset {\mathbb{C}}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in ${\mathbb{C}}^n$ with the same boundary as $D$. Then $I(D)={\lambda }_P^n\cdot Area(D)$. If $P$ is not complex, ${\lambda }_P^n<\frac{3\pi }{2\sqrt{2}}$. For a complex plane $\mathbb{C}\subset {\mathbb{C}}^n$, ${\lambda }_{\mathbb{C}}^n\ge 2$, ${\lambda }_{\mathbb{C}}^2\ge 3$ and also $\frac{3{\pi }^2}{2\sqrt{2}}$ is the area of an explicit Hamiltonian stationary isotropic Mobius band embedded in ${\mathbb{C}}^n$ whose boundary is a unit circle in $\mathbb{C}$. As a corollary, a compact surface $\Sigma$ (possibly with boundary) in a symplectic manifold can be approximated by isotropic surfaces of area $\le \frac{3\pi }{2\sqrt{2}}Area(\Sigma )$. Another corollary is that a closed curve of length $l$ in ${\mathbb{C}}^n$ bounds an isotropic surface of area $\le \frac{3l^2}{8\sqrt{2}}$. A related result is the following: consider $\mathbb{C}{P}^1\subset \mathbb{C}{P}^n$ and let $D$ be a region in $\mathbb{C}{P}^1$. Let $I(D)$ be the infimum of areas of all isotropic surfaces in $\mathbb{C}{P}^n$ with the same boundary as $D$ representing the same relative homology class mod 2 as $D$. Then $2\cdot Area(D)\le I(D)\le {\lambda }_{\mathbb{C}}^n\cdot Area(D)$. Moreover the first inequality becomes an equality for $D=\mathbb{C}{P}^1$.

10 pages