The Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary
Zhengni Hu, Thomas Bartsch · Journal of Geometric Analysis · 2024
Abstract In this paper, we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $$\Sigma $$ Σ with boundary $$\partial \Sigma $$ ∂ Σ . Given a Riemannian metric g on $$\Sigma $$ Σ we consider functions of the form "Equation missing" where $$\sigma _i e 0$$ σ i ≠ 0 for $$i=1,\ldots ,m$$ i = 1 , … , m , $$G^g$$ G g is the Green function of the Laplace-Beltrami operator on $$(\Sigma ,g)$$ ( Σ , g ) with Neumann boundary conditions, $$R^g$$ R g is the corresponding Robin function, and $$h \in {{\mathcal {C}}}^{2}(\Sigma ^m,\mathbb {R})$$ h ∈ C 2 ( Σ m , R ) is arbitrary. We prove that for any Riemannian metric g , there exists a metric $$\widetilde{g}$$ g ~ which is arbitrarily close to g and in the conformal class of g such that $$f_{\widetilde{g}}$$ f g ~ is a Morse function. Furthermore we show that, if all $$\sigma _i>0$$ σ i > 0 , then the set of Riemannian metrics for which $$f_g$$ f g is a Morse function is open and dense in the set of all Riemannian metrics.