Axially Symmetric Perturbations of Kerr Black Holes II: Boundary Behaviour of the Dynamics in the Orbit Space

In a previous work, we constructed a positive-definite total energy functional for the axially symmetric linear perturbative theory of Kerr black hole spacetimes. That work is based on the dimensional reduction of dynamical axisymmetric spacetimes into 2+1 Einstein-wave map system. In the construction of the positive-definite energy, various dynamical terms, at the boundary of the orbit space, critically occur. In this work, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time, by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of Kerr black holes