A positive-definite energy functional for the axisymmetric perturbations of Kerr-Newman black hole spacetimes.
The mathematical problem of stability of black hole solutions of Einstein’s equations is important to establish the astrophysical significance of these solutions. The problem concerns the stability of the Kerr(-Newman) family of solutions of the Einstein(-Maxwell) equations. In terms of mathematical technique, an important obstacle is that the energy of waves propagating through such spacetimes is not necessarily positive-definite due to the existence of the ergoregion for non-zero angular momentum. In this talk, we shall discuss the proof that there exists an energy functional for axially symmetric linear perturbations of Kerr-Newman that is positive-definite and strictly conserved. The proof is based on a Hamiltonian approach to the Einstein equations and holds for the full sub-extremal range. This is joint work with Vincent Moncrief.
Consider a small spherical tube around a compact submanifold M in Euclidean space. In 1939 Weyl showed that the volume of such a tube only depends on the radius of the tube and the intrinsic curvature of M. What happens for tubes with more complicated cross sections D? We will see that under sufficiently strong symmetry assumptions on D the tube volume turns out to be still intrinsic. Joint work with Gert Heckman.
Comparison results for three-dimensional manifolds.
Typical comparison results in geometry require information on Ricci curvature. In dimension three, we show that this can sometimes be relaxed to scalar curvature. An example is a sharp upper bound for the bottom spectrum of the Laplacian on noncompact three-dimensional manifolds.
Manifold submetries are a geometric generalization of isometric group actions, as well as of singular Riemannian foliations. For the round sphere, we show that manifold submetries are in one-one correspondence with a certain class of algebras of polynomials intimately related to the Laplace operator. Surprisingly, this broader point of view yields interesting new results in Classical Invariant Theory. (Joint work with Marco Radeschi)