The Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface Σ with boundary ∂Σ. Given a Riemannian metric g on Σ we consider functions of the form \[ f_g(x) := ∑ᵢ₌₁^mσ_i^2R^g(x_i)+∑i,j=1\≠ j^mσ_iσ_jG^g(x_i,x_j)+h(x_1,…,x_m), \] where σᵢ ≠ 0 for i=1,…,m, Gᵍ is the Green function of the Laplace-Beltrami operator on (Σ,g) with Neumann boundary conditions, Rᵍ is the corresponding Robin function, and h ∈ C²(Σᵐ,R) is arbitrary. We prove that for any Riemannian metric g, there exists a metric g which is arbitrarily close to g and in the conformal class of g such that fg is a Morse function. Furthermore we show that, if all σᵢ>0, then the set of Riemannian metrics for which fg is a Morse function is open and dense in the set of all Riemannian metrics.