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 $2π$ satisfies $$\operatorname{Area}(M) \geq \frac{14ζ(3)}π \approx 5.35677$$, 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 $\operatorname{Area}(M)>5.40154$.