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

Abstract 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> \mathbb{C}^{3} , there are no complete Calabi–Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li–Conlon–Rochon–Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> \mathbb{C}^{3} raised by Conlon–Rochon when the symmetry is spherical.