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

Abstract

After providing an explicit K-stability condition for a $\mathbb{Q}$-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 $\mathbb{C}^3$, 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 $\mathbb{C}^{3}$ raised by Conlon-Rochon when the symmetry is spherical.