Abstract. We prove a Skoda-type division theorem via a degeneration argument. The proof is inspired by Berndtsson and Lempert’s approach to the L² extension theorem and is based on positivity of direct image bundles. The same tools are then used to slightly simplify the proof of the L² extension theorem given by Berndtsson and Lempert.

Abstract. We prove the existence of a family of non-trivial solutions of the Liouville equation in dimensions two and four with infinite volume. These solutions are perturbations of a finite-volume solution of the same equation in one dimension less. In particular, they are periodic in one variable and decay linearly to \(-\infty\) in the other variables. In dimension two, we also prove that the periods are arbitrarily close to \(\pi k, k \in \mathbb{N}\) (from the positive side). The main tool we employ is bifurcation theory in weighted Hölder spaces.

Degeneration techniques in complex geometry

Abstract. In 2009, B. Berndtsson proved a theorem on the positivity of direct image bundles of positive line bundles. Berndtsson’s theorem has been successfully used to give radically new proofs of some fundamental theorems in the part of complex geometry often referred to as L² methods. Among these is a proof of the L² extension theorem with sharp estimates. This thesis is a step towards determining how much of the classical L² theory can be recovered by this technique. The main contribution is a new proof of a Skoda-type L² division theorem.

Solutions of Liouville equations with non-trivial profile

Thesis for the Galilean School of Higher Education

This was a continuation of my Master’s Thesis. The results of these two theses had been summarized and completed in my 2021 paper Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4.

Solutions of Liouville equations with non-trivial profile in dimension 4

Abstract. Liouville equations have been widely studied for more than a century. In particular, the interest in this class of PDEs renewed during the last three decades, after the introduction of the so-called Q-curvature and the discovery that they are intimately related to several fundamental concepts both in Analysis and in Geometry. In this work, we will show the existence of a class of non-trivial solutions of the 2D Liouville equation with infinite volume, employing basic tools of bifurcation theory. Using some more advanced techniques of bifurcation theory and Morse theory, we will also lay the groundwork for the study of the same problem in dimension 4.

Rappresentazioni di gruppi di Lie compatti ed equazioni alle derivate parziali (in Italian)

Abstract. This work will study representation theory on compact Lie groups and the decomposition of the space of square-integrable functions with respect to the Haar measure. After proving that this decomposition is the spectral decomposition of the Laplace operator on the group SU(2), this work will use these results in order to study the heat equation on the group. In particular, it will solve the Cauchy problem with a Dirac delta function as the initial data.