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 $\mathbb{C} \subset \mathbb{C}^n$, $\lambda_{\mathbb{C}}^n \geq 2$, $\lambda_{\mathbb{C}}^2 \geq 3$ 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 $\leq \frac{3\pi}{2\sqrt{2}} Area(\Sigma)$. Another corollary is that a closed curve of length $l$ in $\mathbb{C}^n$ bounds an isotropic surface of area $\leq \frac{3l^2}{8\sqrt{2}}$. 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) \leq I(D) \leq \lambda_{\mathbb{C}}^n \cdot Area(D)$. Moreover the first inequality becomes an equality for $D=\mathbb{C}P^1$.