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Stefano Trapani

Publications and source records attributed to Stefano Trapani.

At least 19 recordsLinked to original sources
arXiv2025

(CMC) 1-immersions of surfaces into hyperbolic 3-manifolds

Constant Mean Curvature (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds are natural and yet rather curious objects in hyperbolic geometry with interesting applications. Firstly, Bryant revealed surprising relations between (CMC) $1$-immersions of surfaces into $\mathbb H^3$ (Bryant surfaces) and (cousins) minimal immersions into $\mathbb E^3.$ In addition, the interest to (CMC) immersions of a surface $S$ (closed, orientable, with genus $\mathfrak{g} \geq2$) into hyperbolic 3-manifolds was motivated by Uhlenbeck in connection to irreducible representations of the fundamental group $\pi_{1}(S)$ into $PSL(2,\mathbb{C}).$ However a (CMC) 1-immersed compact surface is likely to develop singularities (punctures at finitely many points), and indeed in our analysis the prescribed value 1 of the mean curvature enters as a "critical" parameter. In fact, Huang-Lucia-Tarantello showed that (CMC) $c$-immersions of $S$ into hyperbolic 3-manifolds exist for $|c | <1$ and are parametrized by elements of the tangent bundle of the Teichmueller space of $S.$ More importantly, (CMC) $1$-immersions are attained only as "limits" for $|c| \to 1^-$ . In general the passage to the limit can be prevented by possible blow-up phenomena captured in terms of the Kodaira map and its suitable extension respectively for genus $\mathfrak{g}=2$ and $\mathfrak{g}=3.$ Here we handle the case of surfaces of any genus. In Theorem , we are able to encompass the blow up situation in terms of an appropriate "orthogonality" condition. Subsequently, we can provide the existence and uniqueness of (CMC) 1-immersions under an appropriate "generic" condition, see Theorem 2.

math.DG↗
arXiv2025

Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)

We introduce the notion of K\"ahler topologically hyperbolic manifold, as a"topological" generalization of K\"ahler [Gro91] and weakly K\"ahler [BDET24] hyperbolic manifolds. Analogously to [BCDT24], we show the birational invariance of this property and then that K\"ahler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact K\"ahler manifolds with trivial first real Chern class. Then, we prove spectral gap theorems for positive holomorphic Hermitian vector bundles on K\"ahler topologically hyperbolic manifolds, obtaining in particular effective non vanishing results \`a la Kawamata for adjoint line bundles. We finally explore the effects of K\"ahler topologically hyperbolicity on Ricci and scalar curvature of K\"ahler metrics. In the appendix, it is given an explicit description of degree~$2$ hyperbolic classes for finitely presented groups, and an algebro-geometric consequence for K\"ahler topologically hyperbolic surfaces: they are necessarily of general type.

math.CV↗
arXiv2024

Convexity of the Mabuchi functional in big cohomology classes

We study the Mabuchi functional associated to a big cohomology class. We define an invariant associated to transcendental Fujita approximations, whose vanishing is related to the Yau-Tian Donaldson conjecture. Assuming vanishing (finiteness) of this invariant we establish (almost) convexity along weak geodesics. As an application, we give an explicit expression of the distance $d_p$ in the big setting for finite entropy potentials.

math.DG↗
arXiv2024

On constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds

Motivated by the work of Bryant on constant mean curvature (CMC) $1$-immersions of surfaces into the hyperbolic space H^3 and after the results of Tarantello (2023), we pursue a possible parametrization for the moduli space of (CMC) 1-immersions of a surface S (closed, orientable and of genus >1) into hyperbolic 3-manifolds. Those immersions enter as "critical" object in our analysis. In fact, they can be attained only as limits of the (CMC) c-immersions (as c tends to 1), obtained in Huang-Lucia-Tarantello (2022), for |c|<1. However, such passage to the limit can be prevented by possible blow-up phenomena, so that the pullback metrics of the (CMC) c-immersions may yield (at the limit) to a singular metric with conical singularities at finitely many points (the blow-up points). In case of genus g=2, blow up can occur only at a single point, and in Tarantello (2023) it was shown how it could be prevented and the passage to the limit ensured in terms of the Kodaira map. In this note we sharpen this result and for genus g=2, we obtaina condition (we believe sharp) which involves only the Kodaira map on the six Weierstrass points. In addition we tackle the case of higher genus, where multiple blow-up points occur. In this case, we need to identify a suitable replacement of the Kodaira map, now defined on the space of non-zero effective divisors. More importantly, we need to improve in a substantial way the asymptotic analysis of Tarantello (2023) limited to the case of "blow-up" with minimal mass. In this direction we give a contribution which best applies to the case of genus g=3, but also provides a relevant step and a convincing indication on what should happen in the general case.

math.DG↗
arXiv2024

Weak K\"ahler hyperbolicity is birational

We show that a compact K\"ahler manifold bimeromorphic to a weakly K\"ahler hyperbolic manifold is weakly K\"ahler hyperbolic, providing an answer to a problem raised by J. Koll\'ar in his 1995 book "Shafarevic maps and automorphic forms"

math.AG↗
arXiv2024

Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, θ, ϕ)$, for a model potential $ϕ$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, θ, ϕ)$ (provided $n>1$), while they have the same singularities of $ϕ$ when $n=1$.

math.CV↗
arXiv2023

Weakly Kähler hyperbolic manifolds and the Green--Griffiths--Lang conjecture

We introduce the notion of weakly Kähler hyperbolic manifold which generalizes that of Kähler hyperbolic manifold given in the early '90s by M. Gromov, and establish its basic features. We then investigate its spectral properties and show a spectral gap result (on a suitable modification). As applications, we prove that weakly Kähler hyperbolic manifolds are of general type and we study the geometry of their subvarieties and entire curves, verifying -- among other things -- various aspects of the Lang and the Green--Griffiths conjectures for this class of manifolds.

math.CV↗
arXiv2023

Exchangeability and irreducible rotational invariance

In this note we prove that a finite family $\{X_1,\dots,X_d\}$ of real r.v.'s that is exchangeable and such that $(X_1,\dots,X_d)$ is invariant with respect to a subgroup of $SO(d)$ acting irreducibly, is actually invariant with respect to the action of the full group $SO(d)$. Three immediate consequences are deduced: a characterization of isotropic spherical random eigenfunctions whose Fourier coefficients are exchangeable, an extension of Bernstein's characterization of the Gaussian and a characterization of the Lebesgue measure on the sphere.

math.PR↗
arXiv2023

The Regularity of Envelopes

Let $X$ be a compact complex manifold of complex dimension $n$ and $α$ be a smooth closed real form on $X$ such that its cohomology class $\{ α\}\in H^{1,1}(X, \mathbb{R})$ is big. In this paper we prove that, given a bounded function $f$ with bounded distributional laplacian in $X,$ the $α$-psh envelope $P(f)$ is also locally bounded with locally bounded distributional laplacian on the ample locus of $\{α\}.$

math.CV↗
arXiv2020

The pluricomplex Poisson kernel for strongly pseudoconvex domains

In this paper we introduce, via a Phragmen-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the {\sl pluricomplex Poisson kernel} because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new "intrinsic" version of the classical Julia's Lemma and Julia-Wolff-Carathéodory Theorem.

math.CV↗
arXiv2019

Monge-Ampère measures on contact sets

Let $(X, ω)$ be a compact Kähler manifold of complex dimension n and $θ$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ θ\}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $φ$ be a $θ$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $ω$ such that $φ\leq f. $ Then the non-pluripolar measure $θ_φ^n:= (θ+ dd^c φ)^n$ satisfies the equality: $$ {\bf{1}}_{\{ φ= f \}} \ θ_φ^n = {\bf{1}}_{\{ φ= f \}} \ θ_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ θ_{P_θ(f)}^n= { \bf {1}}_{\{P_θ(f) = f\}} \ θ_f^n\qquad {\rm and }\qquad θ_{P_θ[φ](f)}^n = { \bf {1}}_{\{P_θ[φ](f) = f \}} \ θ_f^n. \]

math.CV↗
arXiv2017

The equality case in Wu-Yau inequalities

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.

math.DG↗
arXiv2015

Divisorial Zariski Decomposition and some properties of full mass currents

Let $α$ be a big class on a compact Kähler manifold. We prove that a decomposition $α=α_1+α_2$ into the sum of a modified nef class $α_1$ and a pseudoeffective class $α_2$ is the divisorial Zariski decomposition of $α$ if and only if $\operatorname{vol}(α)=\operatorname{vol}(α_1)$. We deduce from this result some properties of full mass currents.

math.AG↗
arXiv2013

Fourier coefficients of invariant random fields on homogeneous spaces of compact groups

Let $T$ be a random field invariant under the action of a compact group $G$. In the line of previous work we investigate properties of the Fourier coefficients as orthogonality and Gaussianity. In particular we give conditions ensuring that independence of the random Fourier coefficients implies Gaussianity. As a consequence, in general, it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients.

math.PR↗
arXiv2010

A classification of taut, Stein surfaces with a proper $\R$-action

We present a classification of 2-dimensional, taut, Stein manifolds with a proper $\R$-action. For such manifolds the globalization with respect to the induced local $\C$-action turns out to be Stein. As an application we determine all 2-dimensional taut, non-complete, Hartogs domains over a Riemann surface.

math.CV↗