The Morse property of limit functions appearing in mean field equations on surfaces with boundary (Q6536750)

Let \(\Sigma\) denote a smooth and compact surface with boundary \(\partial \Sigma\) of a Riemannian manifold with metric \(g\). Denote also by \(G^g\) the Green function of the Laplace-Beltrami operator with Neuman boundary condition.\N\NThis article discusses Morse properties of functions of the form \N\[\Nf_g(x)=\sum_{1\leq i\leq m} \sigma_i^2 R^g(x_i)+\sum_{1\leq i, j\leq m, i\neq j}\sigma_i \sigma_j G^g(x_i, x_j)+h(x_1, x_2, \dots, x_m), \N\]\Nwhere \(R^g\) is the corresponding Robin function of \(G^g\) and \(h\) is any \(C^2\) arbitrary function. The main result of the article establishes that for any Riemannian metric \(g\), there exists a metric \(\tilde g\) arbitrarily close to \(g\) and in the conformal class of with the property that \(f_{\tilde g}\) is a is a Morse function. As a byproduct, it is obtained that if all \(\sigma_i>0\), then the set of all Riemannian metrics \(g\) for which \(f_g\) is a Morse function becomes an open and dense set in the set of all Riemannian metrics.