Two-dimensional solutions of a mean field equation on flat tori

We study the mean field equation on the flat torus T-sigma := C/(Z + Z sigma) Delta u + rho(e(u)/integral(T sigma)e(u) - 1/vertical bar T-sigma vertical bar) = 0, where rho is a real parameter. For a general flat torus, we obtain the existence of two-dimensional solutions bifurcating from the trivial solution at each eigenvalue (up to a multiplicative constant vertical bar T-sigma vertical bar) of Laplace operator on the torus in the space of even symmetric functions. We further characterize the subset of all eigenvalues through which only one bifurcating curve passes. Finally local convexity near bifurcating points of the solution curves are obtained. (C) 2020 Elsevier Inc. All rights reserved.