On the solvability of singular Liouville equations on compact surfaces of arbitrary genus

Abstract:

In the first part of this article, we complete the program announced in the preliminary note by the author and Malchiodi by proving a conjecture presented in a previous paper that states the equivalence of contractibility and $p_{1}-$stability for generalized spaces of formal barycenters $\Sigma _{\rho ,\underline {\alpha }}$ and hence we get purely algebraic conditions for the solvability of the singular Liouville equation on Riemann surfaces. This relies on a structure decomposition theorem for $\Sigma _{\rho ,\underline {\alpha }}$ in terms of maximal strata, and on elementary combinatorial arguments based on the selection rules that define such spaces. Moreover, we also show that these solvability conditions on the parameters are not only sufficient, but also necessary, at least when for some $i\in \left \{1,\ldots ,m\right \}$ the value $\alpha _{i}$ approaches $-1$. This disproves a conjecture made in Section 3 of a paper by Tarantello and gives the first non-existence result for this class of PDEs without any genus restriction. The argument we present is based on a combined use of the maximum/comparison principle and of a Pohozaev type identity and applies for an arbitrary choice both of the underlying metric $g$ of $\Sigma$ and of the datum $h$.