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Vittorio Martino

Publications and source records attributed to Vittorio Martino.

At least 19 recordsLinked to original sources
arXiv2026

A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the Aubib-type inequality corresponding to the problem is always strict, unless $(M,g)$ is conformal to the round sphere. In particular, this result provides an existence result for a ground state to the conformal Dirac-Einstein problem in dimension four. We point out that aside from some perturbative or special cases, this presents the first general existence result for the conformal Dirac-Einstein equations in dimension four.

math.DG↗
arXiv2025

Conformally Invariant Dirac Equation with Non-Local Nonlinearity

We study a conformally invariant equation involving the Dirac operator and a non-linearity of convolution type. This non-linearity is inspired from the conformal Einstein-Dirac problem in dimension 4. We first investigate the compactness, bubbling and energy quantization of the associated energy functional then we characterize the ground state solutions of the problem on the standard sphere. As a consequence, we prove an Aubin-type inequality that assures the existence of solutions to our problem and in particular the conformal Einstein-Dirac problem in dimension 4. Moreover, we investigate the effect of a linear perturbation to our problem, leading us to a Brezis-Nirenberg type result.

math.DG↗
arXiv2024

Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere

In this paper we investigate the existence of singular solutions to the conformal Dirac-Einstein system. Because of its conformal invariance, there are many similarities with the classical construction of singular solutions for the Yamabe problem. We construct here a family of singular solutions, on the three-dimensional sphere, having exactly two singularities.

math.DG↗
arXiv2023

Overdetermined problems for gauge balls in the Heisenberg group

In this paper we aim at characterizing the gauge balls in the Heisenberg group $\mathbb{H}^n$ as the only domains where suitable overdetermined problems of Serrin type can be solved. We discuss a one parameter family of overdetermined problems where both the source functions and the Neumann-like data are non-constant and they are related to the geometry of the underlying setting. The uniqueness results are established in the class of domains in $\mathbb{H}^n$ having partial symmetries of cylindrical type for any $n\geq 1$, and they are sharper in the lowest dimensional cases of $\mathbb{H}^1$ and $\mathbb{H}^2$ where we can respectively treat domains with $S^1$ and $S^1\times S^1$ invariances.

math.AP↗
arXiv2022

Conformal Dirac-Einstein equations on manifolds with boundary

In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result.

math.DG↗
arXiv2022

A characterization of gauge balls in $\mathbb{H}^n$ by horizontal curvature

In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces

math.DG↗
arXiv2022

$\bar{Q}'$-curvature flow on Pseudo-Einstein CR manifolds

In this paper we consider the problem of prescribing the $\bar{Q}'$-curvature on three dimensional Pseudo-Einstein CR manifolds. We study the gradient flow generated by the related functional and we will prove its convergence to a limit function under suitable assumptions.

math.DG↗
arXiv2022

Compactness of Dirac-Einstein spin manifolds and horizontal deformations

In this paper we consider the Hilbert-Einstein-Dirac functional, whose critical points are pairs, metrics-spinors, that satisfy a system coupling the Riemannian and the spinorial part. Under some assumptions, on the sign of the scalar curvature and the diameter, we prove a compactness result for this class of pairs, in dimension three and four. This can be seen as the equivalent of the study of compactness of sequences of Einstein manifolds as in \cite{And0,N}. Indeed, we study the compactness of sequences of critical points of the Hilbert-Einstein-Dirac functional which is an extension of the Hilbert-Einstein functional having Einstein manifolds as critical points. Moreover we will study the second variation of the energy, characterizing the horizontal deformations for which the second variation vanishes. Finally we will exhibit some explicit examples.

math.DG↗
arXiv2019

Singular CR structures of constant Webster curvature and applications

We consider the sphere $\Sph^{2n+1}$ equipped with its standard CR structure. In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus \Sph^{2k+1}$, which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular the curvature is positive if $2k< n-2$. As main applications, we provide two perturbative results. In the first one we prove the existence of infinitely many contact structures on $\Sph^{2n+1}\setminus τ(\Sph^{1})$ conformal to the standard one and having constant Webster curvature, where $τ(\Sph^{1})$ is a small perturbation of $\Sph^1$. In the second application, we show that there exist infinitely many bifurcating branches of periodic solutions to the CR Yamabe problem on $\Sph^{2n+1}\setminus \Sph^{1}$ having constant Webster curvature.

math.DG↗
arXiv2018

Palais-Smale sequences for the fractional CR Yamabe functional and multiplicity results

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely many solutions to the related equation.

math.AP↗
arXiv2015

The Rabinowitz-Floer homology for a class of semilinear problems and applications

In this paper, we construct a Rabinowitz-Floer type homology for a class of non-linear problems having a \emph{starshaped} potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations.

math.AP↗
arXiv2013

The topology of a subspace of the Legendrian curves in a closed contact 3-manifold

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

math.DG↗
arXiv2010

A symmetry result on Reinhardt domains

We show the following symmetry property of a bounded Reinhardt domain $Ω$ in $\mathbb{C}^{n+1}$: let $M=\partialΩ$ be the smooth boundary of $Ω$ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In Appendix we state the result from an hamiltonian point of view.

math.DG↗
arXiv2010

On the Characteristic Curvature Operator

We introduce the Characteristic Curvature as the curvature of the trajectories of the hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces and by using the Second Fundamental Form we relate it to the Classical and Levi Mean Curvature. Then we prove existence and uniqueness of viscosity solutions for the related Dirichlet problem and we show the Lipschitz regularity of the solutions under suitable hypotheses. Moreover we prove a non existence result on the balls when the prescribed curvature is a positive constant. At the end we show that neither Strong Comparison Principle nor Hopf Lemma do hold for the Characteristic Curvature Operator.

math.AP↗