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Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. (English) Zbl 1207.35148

The author considers the problem \[ \Delta_g u + \lambda \left( \frac{Ke^u}{\int_M Ke^u \, dv_g} - \frac{1}{|M|} \right) = 4\pi \sum_{j=1}^N \beta_j \left( \delta_{p_j} - \frac{1}{|M|} \right) (*) \] on a closed surface \((M,g)\), where \(\Delta_g\) and \(dv_g\) denote respectively the Laplace-Beltrami operator and the volume element relative to the metric \(g\), \(\lambda>0\) and \(\beta_j \in (-1,+\infty)\) are parameters, \(p_1\dots,p_N\in M\), and \(K\) is a smooth positive function on \(M\). As usual \(\delta_p\) denotes the Dirac distribution on \((M,g)\) with pole at \(p\), and \(|M|\) is the surface area of \(M\) relative to \(g\).
The author describes a variety of problems leading to \((*)\), starting with the well known Nirenberg problem of finding a metric conformal to the standard metric on \(\mathbf{S}^2\) and having Gauss curvature \(K\), which can be recast in the form \((*)\) with all \(\beta_j=0\) and \(\lambda=8\pi\).
A more closely related problem arises if, in the conformal change of metric problem, we allow the conformal class to contain metrics that introduce conical type singularities on \(M\), as considered by M. Troyanov [Trans. Am. Math. Soc. 324, No.2, 793–821 (1991; Zbl 0724.53023)]. Given singular points \(p_1,\dots,p_N\in M\) with relative orders \(\beta_1,\dots,\beta_N\in (-1,+\infty)\), a version of the Gauss-Bonnet theorem that takes into account the Gauss curvature at the singular points leads to \((*)\), in the case \(\mu=\chi(M)+\sum_{j=1}^N \beta_j>0\) and for \(g\) equal to the constant curvature metric on \(M\).
The author’s primary motivation, however, is the connection of \((*)\) with vortex-like configurations that occur naturally in physical applications, such as the study of periodic electroweak vortices. In this case the problem reduces to the existence of doubly periodic solutions of a coupled pair of equations similar to \((*)\). A key observation is that one of these equations has many analytical features in common with \((*)\). Consequently, any strategy that yields solutions for \((*)\) has implications for the system describing periodic electroweak vortices.
The author discusses a large number of recent results for \((*)\), including existence, nonexistence, uniqueness and multiplicity results, depending on the topological and geometrical properties of \((M,g)\). The results are too numerous and technical to describe here.

MSC:

35J60 Nonlinear elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05 Elliptic equations on manifolds, general theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J93 Quasilinear elliptic equations with mean curvature operator

Citations:

Zbl 0724.53023
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