A Fourier approach to Gromov's filling area conjecture

Abstract

We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\mathrm{Area}(M)\ge \frac{14\zeta (3)}{\pi }\approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary points. For orientable fillings, we use a cubic resonant perturbation to obtain $\mathrm{Area}(M)>5.40154$.

20 pages; comments are welcome