Gromov’s filling area conjecture

The conjecture asks whether every compact orientable Riemannian surface filling a circle of circumference $2\pi$ has area at least $2\pi$. The full statement remains unresolved.

Known results

• The conjecture is known for disk fillings and genus-$1$ fillings.
• A discrete analogue yields $\operatorname{Area}(M)\ge \frac{\sqrt{3}}{4}\pi^2$ for arbitrary compact Riemannian fillings, without an orientability assumption.
• Systolic inequalities establish the conjectured bound for certain sufficiently high-genus fillings; these are restricted results, not a general proof.

September 2026 Fourier lower bound

A September 8, 2026 report describes a Fourier-based method claiming $\operatorname{Area}(M)\ge 14\zeta(3)/\pi\approx 5.35677$ generally and $\operatorname{Area}(M)>5.40154$ for orientable fillings of the circle of length $2\pi$. This is a claimed advance, not a complete resolution.

Current status (as of September 2026): The conjecture remains open; disk and genus-$1$ cases and other restricted bounds are known, while the new Fourier estimates are claimed progress but are not independently verified here.