Minimizing and min-max Yamabe metrics on conical manifolds

We discuss the existence of Yamabe metrics on conical manifolds with Ricci-flat tangent cones at singular points. We prove an analogue of Aubin’s classical result, obtaining solutions as minimizers of the Yamabe quotient. In contrast to the smooth case, when this condition fails, minimizers may not exist. In dimension four, and in the presence of at least two Z/2Z-orbifold points, we still obtain solutions via a min-max variational scheme.

Based on joint works with Andrea Malchiodi (SNS) and Francesco Malizia (SNS).