Isoperimetric gaps in CAT(0) spaces

A metric space $X$ satisfies a Euclidean isoperimetric inequality for $n$-spheres, if every $n$-sphere $S\subset X$ bounds a ball $B\subset X$ with $\operatorname{vol}_{n+1}(B)\leq C\cdot \operatorname{vol}_n(S)^\frac{n+1}{n}$. Every CAT(0) space $X$ satisfies Euclidean isoperimetric inequalities for $1$-spheres with the sharp constant $C=1/4\pi$. Moreover, if such inequalities hold with a constant strictly smaller than $1/4\pi$, then $X$ has to be Gromov hyperbolic. In particular, a sharp isoperimetric gap appears. In the talk I will focus on the case $n=2$, namely fillings of 2-spheres by 3-balls.

This is based on joint work with Drutu, Lang and Papasoglu.