K-stable valuations and Calabi–Yau metrics on affine spherical varieties

After providing an explicit K-stability condition for a Q-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete K-invariant Calabi-Yau metric in the trivial class of an affine G-spherical manifold, K being the maximal compact subgroup of G. Next, we prove that the valuation induced by K-invariant Calabi-Yau metrics on affine G-spherical manifolds is in fact G-invariant. As an application, we point out an affine smoothing of a Calabi-Yau cone that does not admit any K-invariant Calabi-Yau metrics asymptotic to the cone. Another corollary is that on C³, there are no other complete Calabi-Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li-Conlon-Rochon-Sz\'ekelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on C^3 raised by Conlon-Rochon when the symmetry is spherical.