Preprint: arXiv:2609.08251 -- Le Chen, Xiaolong Li and Yimin Zhong, A Fourier approach to Gromov's filling area conjecture (20 pages)
Abstract: Let \(M\) be a compact, connected Riemannian surface whose boundary is a circle of length \(2\pi\). Assume that the intrinsic distance in \(M\) between any two boundary points agrees with their distance along the circle. Gromov's filling area conjecture asserts that \(\operatorname{Area}(M) \geq 2\pi\), with equality attained by the round hemisphere. The conjecture is known for the disk and for orientable fillings of genus one, but remains open in higher genus. In this talk, I will describe a new Fourier approach that gives lower bounds \(\operatorname{Area}(M) \geq 14\zeta(3)/\pi \approx 5.35677\) for every filling, regardless of orientability or topological types. This is joint work with Le Chen and Yimin Zhong. The proof is AI-assisted.