Gromov’s filling area conjecture in higher genus

Prove Gromov’s filling area conjecture for every orientable Riemannian isometric filling of a circle of length 2π, in particular for fillings of genus at least two, by establishing the lower bound Area(M) ≥ 2π.

Background

Gromov’s filling area conjecture asserts that every orientable Riemannian isometric filling of a circle of length 2π has area at least 2π, with equality attained by the unit round hemisphere. The paper notes that the conjecture is known for disk fillings and orientable genus-one fillings, but not in general for higher-genus fillings.

The paper establishes substantially weaker universal lower bounds, including 14ζ(3)/π for fillings of arbitrary topology and a stronger bound for orientable fillings. These results do not resolve the conjectured 2π bound for genus at least two.

References

Despite the partial results and related works in , , , and , the conjecture remains open in general for genus at least two.

— A Fourier approach to Gromov's filling area conjecture  (2609.08251 - Chen et al., 8 Sep 2026) in Section 1, Introduction

It remains open whether their discrete problem is equivalent to Gromov's filling area conjecture.

— A Fourier approach to Gromov's filling area conjecture  (2609.08251 - Chen et al., 8 Sep 2026) in Section 1, Introduction

We leave open the question of improving the bounds by other nonlinear couplings or by using the topology of the filling.

— A Fourier approach to Gromov's filling area conjecture  (2609.08251 - Chen et al., 8 Sep 2026) in Section 1, Introduction

It remains open whether additional resonant perturbations can produce a substantially stronger estimate or even approach the conjectured bound $2\pi$.

— A Fourier approach to Gromov's filling area conjecture  (2609.08251 - Chen et al., 8 Sep 2026) in Remark following the proof of Theorem 3.1, Section 4