Finiteness of the ECH capacities of prequantization bundles

Determine whether all ECH capacities of the prequantization contact manifold over a genus-g surface with Euler parameter e are finite whenever e>2g−2.

Background

The paper proves that the first ECH capacity of a prequantization contact manifold is infinite when the Euler parameter satisfies e≤2g−2. The complementary range e>2g−2 is relevant to the proof of the conjectured sharpness criterion for embedded Liouville domains.

Finiteness is known for prequantization bundles over the sphere and torus, and the paper notes that subsequent work addresses the remaining cases. The quoted passage explicitly records the unresolved expectation at the point where the problem is introduced.