On Mañé's critical value for the two-component Hunter-Saxton system

In this talk, we will introduce Mañé's critical value for a Hamiltonian PDE, the two-component Hunter-Saxton system. We will introduce the magnetic two-component Hunter-Saxton system (M2HS), which is a magnetic geodesic equation on an infinite-dimensional Lie group. We prove that this magnetic system is magnetic isomorphic to a magnetic system on an infinite-dimensional sphere. Surprisingly each magnetic geodesic is tangent to the 3-sphere obtained by intersecting the ambient sphere with a complex plane. We use this geometric description of the (M2HS) to give explicit criteria for blow-ups and prove the existence of global weak solutions.