Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy.

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we focus primarily on the case of stationary algebro-geometric CH solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.