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This is the second of a series of two papers where we construct embeddedWillmore tori with small area constraint in Riemannian three-manifolds. In bothpapers the construction relies on a Lyapunov-Schmidt reduction, the difficultybeing the M\"obius degeneration of the tori. In the first paper theconstruction was performed via minimization, here by Morse Theory; to this aimwe establish new geometric expansions of the derivative of the Willmorefunctional on exponentiated small Clifford tori degenerating, under the actionof the M\"obius group, to small geodesic spheres with a small handle. By usingthese sharp asymptotics we give sufficient conditions, in terms of the ambientcurvature tensors and Morse inequalities, for having existence/multiplicity ofembedded tori stationary for the Willmore functional under the constraint ofprescribed (sufficiently small) area.

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