Abstract:

We survey recent existence results, due to Conlon-Rochon, Y. Li and Szekelyhidi, of Calabi-Yau metrics on C^n with maximal volume growth. Such metrics are asymptotic to singular Calabi-Yau cones, thus disproving a conjecture of Tian, which says that every Calabi-Yau metric on C^n with maximal volume growth must be the Euclidean metric. We closely follow the construction of Szekelyhidi. If time permits, we will give a generalization of these results to constructing new families of such Calabi-Yau metrics, and discuss how to distinguish them.