Renormalized energy for Ginzburg-Landau vortices on closed surfaces (Q1358234)

In their book ``Ginzburg-Landau vortices'' (1994; Zbl 0802.35142) \textit{F. Bethuel, H. Brezis} and \textit{F. Hélein} defined the notion of renormalized energy associated to a prescribed configuration of singularities with given multiplicities and to a Dirichlet boundary data. One of their main results in this monograph is that the configuration of vortices is a minimum point of the renormalized energy. In this paper it is transferred the study of Bethuel-Brezis-Hélein to the case of a hermitian line bundle of prescribed degree \(d\) over a closed Riemann surface \(M\). It is obtained the expression of the renormalized energy, which depends only on the given metric of \(M\) and the degree \(d\) and it is also shown that the configuration of the limits of vortices of minimizers is a minimum point for this functional. After proving the existence of minimizers of the Ginzburg-Landau energy in the considered case the author establishes some asymptotic properties of these minimizers. One of the basic technical tools is a local Pohozaev type inequality, which is the analogue of Theorem III.2 from the cited monograph by Bethuel-Brezis-Hélein. These auxiliary results are useful in the study of the moduli space of solutions of prescribed isolated singularities and for finding the expression of the renormalized energy. Another generalization of the classical results is the fact that the space of \(S^1\) harmonic maps from \(M\) with prescribed singularities \(\{p_k,d_k\}\) is non-empty if and only if \(\sum d_k=0\). The paper is interesting and applies modern geometric and analytic methods for the study of Yang-Mills-Higgs functionals.