Small eigenvalues of random 3-manifolds

We show that for every $g\geq 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{\rm vol}(M)^2$.Comment: 52 pages. Major revisio

Improved time-decay for a class of scaling critical electromagnetic Schr\"odinger flows

We consider a Schr\"odinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then $\||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$

On the cohomological dimension of the moduli space of Riemann surfaces

The moduli space of Riemann surfaces of genus $g\geq 2$ is (up to a finite \'etale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$. This expectation is verified in low genus and supported by Harer's computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this paper we prove that such dimension is at most $2g-2$. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most $g$. In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: the construction of such a function relies on some basic geometric properties of translation surfaces.Comment: 37 page

The wave equation on black rings and the linear stability of slowly rotating Kerr spacetimes

The existence of black holes is perhaps the most spectacular prediction of Einstein's classical theory of general relativity. Recent advances in both theoretical and experimental physics, such as the discovery of gravitational waves, have confirmed that black holes are stable, macroscopic objects playing a fundamental role in our universe. On the other hand, modern physical theories demand for a higher dimensional formulation of general relativity, and thus to understand the stability properties of black holes within a wider scenario than the one directly probed by the astrophysical observations. However, the mathematical question of whether black holes are stale as solutions to the vacuum Einstein equations Ric(g)=0, known as the black hole stability problem, remains, to large extent, open. The present thesis contributes to the black hole stability problem with two theorems. Chapter 2 of the thesis considers a family of higher dimensional black holes, known as black rings. The potential stability of this family, and the part that physical theories should reserve to them if unstable, have been largely investigated in the physics literature. The main theorem of the chapter is the first mathematically rigorous result suggesting that these black holes are unstable to gravitational perturbations. In particular, we establish a logarithmic lower bound for the uniform energy decay rate of scalar linear perturbations on black ring spacetimes. Chapter 3 of the thesis deals with the Kerr family of black holes, which is believed to characterise all the astrophysical stationary black holes. To agree with our physical expectation, the Kerr stability conjecture claims that these black holes are stable to gravitational perturbations. The content of the chapter represents the first part of work by the author providing the last missing ingredient towards a final proof of the conjecture for the slowly rotating members of the Kerr family. More precisely, we formulate the problem of linear stability of Kerr black holes to gravitational perturbations in a new geometric gauge.Open Acces