arXiv subjects

Leonardo Enrique Abbrescia

Publications and source records attributed to Leonardo Enrique Abbrescia.

3 recordsLinked to original sources

arXiv2021

Global stability of some totally geodesic wave maps

We prove that wave maps that factor as $\mathbb{R}^{1+d} \overset{φ_{\text{S}}}{\to} \mathbb{R} \overset{φ_{\text{I}}}{\to} M$, subject to a sign condition, are globally nonlinear stable under small compactly supported perturbations when $M$ is a space-form. The main innovation is our assumption on $φ_{\text{S}}$, namely that it be a semi-Riemannian submersion. This implies that the background solution has infinite total energy, making this, to the best of our knowledge, the first stability result for factored wave maps with infinite energy backgrounds. We prove that the equations of motion for the perturbation decouple into a nonlinear wave--Klein-Gordon system. We prove global existence for this system and improve on the known regularity assumptions for equations of this type.

math.AP↗

arXiv2020

Geometric analysis of 1+1 dimensional quasilinear wave equations

We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time.

math.AP↗

arXiv2016

Reeb Dynamics of the Link of the $A_n$ Singularity

The link of the $A_n$ singularity, $L_{A_n} \subset \mathbb{C}^3$ admits a natural contact structure $ξ_0$ coming from the set of complex tangencies. The canonical contact form $α_0$ associated to $ξ_0$ is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to $ξ_0$ that has two isolated simple periodic Reeb orbits. We compute the Conley-Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive $S^1$-equivariant symplectic homology groups for $\left(L_{A_n}, ξ_0 \right)$. In addition, we prove that $\left(L_{A_n}, ξ_0 \right)$ is contactomorphic to the Lens space $L(n+1,n)$, equipped with its canonical contact structure $ξ_{std}$.

math.SG↗