Calabi-Yau metrics on rank two symmetric spaces with horospherical tangent cone at infinity

Overview & Synopsis

We show that on every non-$G_2$ complex symmetric space of rank two there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine Calabi-Yau smoothings of singular and irregular tangent cone. As a corollary, we obtain infinitely many examples of Calabi-Yau manifolds degenerating to the tangent cone in a single step, supporting a recent conjecture by Sun-Zhang, which was only proved for asymptotically conical Calabi-Yau manifolds in the sense of Conlon-Hein.