3 problems

Gromov's filling area conjecture. The area of $M$ satisfies

Let $\mathcal{H}$ be a smooth Riemannian metric on the circle $\mathbb{S}^{1}$, and let $FillVol(\mathbb{S}^{1},\mathcal{H})$ denote the infimum of the areas of compact orientable…

Let a membrane have genus $p\ge 0$ and a prescribed boundary metric, and let an Ahlfors map mean the extremal conformal map used to compute its area through isothermic coordinates.…