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

In the first part of this article, we complete the program announced in the preliminary note [8] by proving a conjecture presented in [9] that states the equivalence of contractibility and p_{1}-stability for generalized spaces of formal barycenters 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 these model spaces 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 $i\in{1,...,m}$ the value $α_{i}$ approaches -1. This disproves a conjecture made in Section 3 of [19] 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 maximum/comparison principle and of a Pohozaev type identity and applies for arbitrary choice both of the underlying metric g of $Σ$ and of the datum h.

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