23.1.41 Sharp damping under unitary rotations5746F7AEAB40

The general damping estimate controls mass by the depth of a potential. On the standard Hopf threefold, unitary symmetry gives a stronger estimate: after integration along the elliptic fibers, a positive radial flux has total mass one. A semiconvexity estimate bounds the remaining radial coefficient by the square root of the depth. The resulting damping threshold is sharp, and at its endpoint finite measures can still concentrate on a pole divisor.

We retain the standard threefold $X$, form $\omega $, projection $\pi :X\to \mathbb P^2$, fiber $E$ and probability area $\mathrm dA_ E$ of this chapter. Let $\mathcal R$ be the class of $u\in \mathrm{PSH}(X,\omega )$ with $\sup _ Xu=0$ invariant under

This holomorphic action preserves $\omega $. The class includes bounded and unbounded potentials. For a positive $C^1$ nonincreasing function $W:[0,+\infty )\to (0,+\infty )$ tending to zero, write

The weighted measure gives zero mass to the pole set, as does the extended non-pluripolar measure. Capacity always means $\operatorname {Cap}_\omega $ of section 5.29.