# Abstracts

### On the linear stability of higher dimensional Schwarzschild black hole

The linear stability of the four dimensional Schwarzschild black hole has been proved using a new approach that generalizes Chandrasekhar's and relies on spherical symmetry. I will discuss how this approach is applied to study the linear stability of higher dimensional Schwarzschild black hole. This is based on joint work with Pei-Ken Hung and Jordan Keller.

### Evaluating Quasi-local Angular Momentum and Center-of-Mass at Null Infinity

We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau, where the authors calculate the quasi-local energy and linear momentum at null infinity. Finiteness of the quasi-local center-of-mass requires that the spacetime be in the so-called center-of-mass frame, a mild assumption on the mass aspect function amounting to vanishing of linear momentum at null infinity. With this condition and the assumption that the mass aspect function is non-trivial, we obtain explicit expressions for the angular momentum and center-of-mass at future null infinity in terms of the observables appearing in the Bondi-Sachs expansion of the spacetime metric. This is joint work with Ye-Kai Wang and Shing-Tung Yau.

### Linear stability of Reissner-Nordstrom spacetime,

*We address the linear stability of the Reissner-Nordstrom family of charged black holes, subject to coupled gravitational and electromagnetic perturbations. Boundedness and decay of the gauge-invariant quantities verifying the Teukolsky equation is the first step towards controlling all the terms in the perturbation. Then, we should identify the residual gauge freedom and the Kerr-Newman parameters to obtain boundedness and decay of all the remaining gauge-dependent quantities. ***E.Giorgi**

### Boundedness and decay for the Teukolsky system in Reissner-Nordstrom,

*We prove boundedness and polynomial decay statements for solutions of the spin 2 generalized Teukolsky system on a Reissner-Nordstrom background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $\alpha$. The second equation, coupled with the first one, is a new equation for a gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of a generalization of the higher order quantities in previous works on linear and non-linear stability of Schwarzschild, as well as in the Teukolsky equation in Kerr. ***E.Giorgi**