ECH Weyl law for a blown-up ball

Prove the ECH Weyl law for the blown-up ball B^4(1)#\overline{CP^2}(λ), namely establish that its ECH capacities satisfy the volume asymptotic with the precise leading term.

Background

The ECH Weyl law is known for Liouville domains whose ECH capacities are finite, but the paper points out that finiteness alone does not ensure that the law has been established in a particular example. For the one-point blow-up of a ball, only a range for the limiting normalized capacities is currently available.

The paper proves the corresponding precise Weyl-law statement for the alternative ECH capacities by embedding the blown-up ball into a closed symplectic 4-manifold with b_2+=1. This does not resolve the analogous question for the ordinary ECH capacities.

References

Even when the ECH capacities are finite, it might not be known that the Weyl law eq:Weyl holds. For example, the ECH Weyl law is not known for a blown-up ball $(X,) = B4(1)# \overline{C P2}(\lambda)$; one has at best an approximate Weyl law:

eq:Weyl:

ckAlt(X,)=4vol(X,)k+O(1)(k→∞).c_k^{Alt}(X,) = \sqrt{4\mathrm{vol}(X,)k} +O(1) \quad (k\to \infty).

— The ECH and alternative ECH capacities of closed symplectic 4-manifolds  (2608.19042 - Beiner, 19 Aug 2026) in Section 1, subsection “Results on the alternative ECH capacities”