Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II: Morse Theory

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both pap ers the construction relies on a Lyapunov- Schmidt reduction, the difficulty being the M ̈obius degenera tion of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore func tional on exponentiated small Clifford tori degenerating, under the action of the M ̈obius group, to smal l geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient condition s, in terms of the ambient curvature tensors and Morse inequalities, for having existence/mult iplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (su fficiently small) area.