Lectures on Geodesics in the Space of Kaehler Metrics, Lecture 3: Hrma, Hcma and Toric Kaehler Manifolds

It is very hard to find computable examples of geodesics in the space of Kaehler metrics. It seems that they come in two types: (i) geodesics of toric Kaehler metrics, the topic of Lecture 2; (iii) geodesics coming from special test configurations and Hele-Shaw flows. Besides Donaldson’s construction (reviewed in §1) there are not many examples of geodesic rays. Almost the only other explicit examples are toric cases or ‘test configurations’. This lecture is devoted to the HRMA and to toric Kahler manifolds where the toric geodesic equation reduces to the HRMA. The only (embedded) toric Kaehler manifolds of complex dimension one are the Riemannian metrics on S = CP which are invariant under rotations around the third axis. Thus we are interested in “ one parameter families of surfaces of revolution which are geodesics in the Mabuchi-Semmes-Donaldson metric.’ It is not obvious, but surfaces of revolution (i.e. toric Kaehler metrics on S) form a totally geodesic submanifold HTm of Hω. The reason that one can explicitly solve the MSD geodesic equation in the toric case is that the HCMA may be linearized by the Legendre transform. Roughly speaking, we transfer the problem from Kaehler potentials to symplectic potentials, where the equation for geodesics becomes linear and solvable. All of the difficulty lies in Legendre transforming back and dealing with the singularities that arise from this transform. The torus invarince in complex dimension one is S invariance and it reduces the HCMA to the HRMA. A toy model for the HRMA is the equation det Hessf = 0 for a function f(x, t) on the upper half plane t > 0. The initial value problem is briefly reviewed in §2.