GeCo GeDi Papershttps://gecogedi.dimai.unifi.it/papers/en-usSat, 25 Mar 2023 14:11:03 +0000Mini-Workshop: Almost Complex Geometryhttps://gecogedi.dimai.unifi.it/paper/481/D. Angella, J. Cirici, J. P. Demailly, S. Wilson.https://gecogedi.dimai.unifi.it/paper/481/On finite generation in magnitude (co)homology, and its torsionhttps://gecogedi.dimai.unifi.it/paper/480/L. Caputi, C. Collari.<p> The aim of this paper is to apply the framework, which was developed by Samand Snowden, to study structural properties of graph homologies, in the spiritof Ramos, Miyata and Proudfoot. Our main results concern the magnitude homologyof graphs introduced by Hepworth and Willerton. More precisely, for graphs ofbounded genus, we prove that magnitude cohomology, in each homological degree,has rank which grows at most polynomially in the number of vertices, and thatits torsion is bounded. As a consequence, we obtain analogous results for pathhomology of (undirected) graphs. We complement the work with a proof that thecategory of planar graphs of bounded genus and marked edges, with contractions,is quasi-Gr\"obner.</p>https://gecogedi.dimai.unifi.it/paper/480/Every complex H{é}non map satisfies the Central Limit Theoremhttps://gecogedi.dimai.unifi.it/paper/479/F. Bianchi, T. C. Dinh.<p> We consider a measurable dynamical system preserving a probability measure$\nu$. If the system is exponentially mixing of all orders for suitableobservables, we prove that these observables satisfy the Central Limit Theorem(CLT) with respect to $\nu$. We show that the measure of maximal entropy ofevery complex H{\'e}non map is exponentially mixing of all orders forH{\"o}lder observables. It follows that the CLT holds for all complex H{\'e}nonmaps and H{\"o}lder observables.</p>https://gecogedi.dimai.unifi.it/paper/479/Generic uniqueness for the Plateau problemhttps://gecogedi.dimai.unifi.it/paper/478/G. Caldini, A. Marchese, A. Merlo, S. Steinbruechel.<p>Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,\beta}$ and an oriented, closed submanifold $\Gamma \subset \mathcal M$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{h,\alpha}$-perturbations of $\Gamma$ inside $\mathcal{M}$, with $\alpha<\beta$, enjoying the following property. For the typical element $b\in\mathcal B$, in the sense of Baire categories, there exists a unique $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem and it has multiplicity one.</p><p>Comments are welcome!</p>https://gecogedi.dimai.unifi.it/paper/478/GKM actions on cohomogeneity one manifoldshttps://gecogedi.dimai.unifi.it/paper/477/O. Goertsches, E. Loiudice, G. Russo.<p>We consider compact manifolds $M$ with a cohomogeneity one action of acompact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For$T$ a maximal torus of $G$, we find necessary and sufficient conditions on thegroup diagram of $M$ such that the $T$-action on $M$ is of GKM type, anddescribe its GKM graph. The general results are illustrated on explicitexamples.</p>https://gecogedi.dimai.unifi.it/paper/477/Non-symmetric intrinsic Hopf-Lax semigroup vs. intrinsic Lagrangianhttps://gecogedi.dimai.unifi.it/paper/476/D. Di Donato.<p> In this paper, we analyze the 'symmetrized' of the intrinsic Hopf-Laxsemigroup introduced by the author in the context of the intrinsicallyLipschitz sections in the setting of metric spaces. Indeed, in the usual case,we have that $d(x,y) =d(y,x)$ for any point $x$ and $y$ belong to the metricspace $X$; on the other hand, in our intrinsic context, we have that$d(f(x),\pi^{-1} (y)) \ne d(f(y),\pi^{-1} (x)),$ for every $x,y \in X$.Therefore, it is not trivial that we get the same result obtained for the"classical" intrinsic Hopf-Lax semigroup, i.e., the 'symmetrized' Hopf-Laxsemigroup is a subsolution of Hamilton-Jacobi type equation. Here, an importantobservation is that $f$ is just a continuous section of a quotient map $\pi$and it can not intrinsic Lipschitz. However, following Evans, the main result of this note is to show that the"new" intrinsic Hopf-Lax semigroup satisfies a suitable variational problemwhere the functional contained an intrinsic Lagrangian. Hence, we also defineand prove some basic properties of the intrinsic Fenchel-Legendre transform ofthis intrinsic Lagrangian that depends on a continuous section of $\pi$.</p>https://gecogedi.dimai.unifi.it/paper/476/The intrinsic Hopf-Lax semigroup vs. The intrinsic slopehttps://gecogedi.dimai.unifi.it/paper/475/D. Di Donato.<p> In this note, we introduce a natural notion of intrinsic Hopf-Lax semigroupin the context of the so-called intrinsically Lipschitz sections. The main aimsare to prove the link between the intrinsic Hopf-Lax semigroup and theintrinsic slope and to show that the intrinsic Hopf-Lax semigroup is asubsolution of Hamilton-Jacobi type equality.</p>https://gecogedi.dimai.unifi.it/paper/475/Bounded Cohomology Classes of Exact Formshttps://gecogedi.dimai.unifi.it/paper/474/L. Battista, S. Francaviglia, M. Moraschini, F. Sarti, A. Savini.<p> On negatively curved compact manifolds, it is possible to associate to everyclosed form a bounded cocycle - hence a bounded cohomology class - viaintegration over straight simplices. The kernel of this map is contained in thespace of exact forms. We show that in degree 2 this kernel is trivial, incontrast with higher degree. In other words, exact non-zero 2-forms havenon-trivial bounded cohomology classes. This result is the higher dimensionalversion of a classical theorem by Barge and Ghys for surfaces. As aconsequence, one gets that the second bounded cohomology of negatively curvedmanifolds contains an infinite dimensional space, whose classes are explicitlydescribed by integration of forms. This also showcases that some recent resultsby Marasco (arXiv:2202.04419, arXiv:2209.00560) can be applied in higherdimension to obtain new non-trivial results on the vanishing of certain cupproducts and Massey products. Some other applications are discussed.</p>https://gecogedi.dimai.unifi.it/paper/474/Well-posedness properties of geometric variational problems: existence, regularity and uniqueness resultshttps://gecogedi.dimai.unifi.it/paper/473/G. Caldini.<p>This thesis is devoted to the study of well-posedness properties of some geometric variational problems: existence, regularity and uniqueness of solutions. We study two specific problems arising in the context of geometric calculus of variations and sharing strong analogies: the Plateau’s problem and the optimal branched transport problem. The first part of the thesis discusses the existence theory. Both problems are formulated in the language of Federer and Fleming’s theory of currents. After an exposition of the main results, we will present the core ideas of the (interior) regularity theory for area-minimizing currents and for optimal transport paths. The last part of the thesis contains two original results: the generic uniqueness of solutions both for the Plateau’s problem (in any dimension and codimension) and for the optimal branched transport problem.</p>https://gecogedi.dimai.unifi.it/paper/473/Geometrization of purely hyperbolic representations in $\text{PSL}_2\Bbb R$https://gecogedi.dimai.unifi.it/paper/472/G. Faraco.<p>Let $S$ be a surface of genus $g$ at least $2$. A representation$\rho:\pi_1S\longrightarrow \text{PSL}_2\Bbb R$ is said to be purely hyperbolicif its image consists only of hyperbolic elements other than the identity. Wemay wonder under which conditions such representations arise as holonomy of ahyperbolic cone-structure on $S$. In this work we will characterize themcompletely, giving necessary and sufficient conditions.</p>https://gecogedi.dimai.unifi.it/paper/472/Distances on the moduli space of complex projective structureshttps://gecogedi.dimai.unifi.it/paper/471/G. Faraco.<p>Let $S$ be a closed and oriented surface of genus $g$ at least $2$. In this(mostly expository) article, the object of study is the space $\mathcal{P}(S)$of marked isomorphism classes of projective structures on $S$. We show that$\mathcal{P}(S)$, endowed with the canonical complex structure, carries exotichermitian structures that extend the classical ones on the Teichm\"uller space$\mathcal{T}(S)$ of $S$. We shall notice also that the Kobayashi andCarath\'eodory pseudodistances, which can be defined for any complex manifold,can not be upgraded to a distance. We finally show that $\mathcal{P}(S)$ doesnot carry any Bergman pseudometric.</p>https://gecogedi.dimai.unifi.it/paper/471/Translation surfaces and periods of meromorphic differentialshttps://gecogedi.dimai.unifi.it/paper/470/S. Chenakkod, G. Faraco, S. Gupta.<p>Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods ofany meromorphic differential on $S$, with respect to a choice of complexstructure, determine a representation $\chi:\Gamma_{g,n} \to\mathbb C$ where$\Gamma_{g,n}$ is the first homology group of $S$. We characterize therepresentations that thus arise, that is, lie in the image of the period map$\textsf{Per}:\Omega\mathcal{M}_{g,n}\to\textsf{Hom}(\Gamma_{g,n},\mathbb{C})$. This generalizes a classical result ofHaupt in the holomorphic case. Moreover, we determine the image of this periodmap when restricted to any stratum of meromorphic differentials, havingprescribed orders of zeros and poles. Our proofs are geometric, as they aim toconstruct a translation structure on $S$ with the prescribed holonomy $\chi$.Along the way, we describe a connection with the Hurwitz problem concerning theexistence of branched covers with prescribed branching data.</p>https://gecogedi.dimai.unifi.it/paper/470/Modular orbits on the representation spaces of compact abelian Lie groupshttps://gecogedi.dimai.unifi.it/paper/469/Y. Bouilly, G. Faraco.<p>Let $S$ be a closed surface of genus $g$ greater than zero. In the presentpaper we study the topological-dynamical action of the mapping class group onthe $\Bbb T^n$-character variety giving necessary and sufficient conditions forMod$(S)$-orbits to be dense. As an application, such a characterisationprovides a dynamical proof of the Kronecker's Theorem concerning inhomogeneousdiophantine approximation.</p>https://gecogedi.dimai.unifi.it/paper/469/Monodromy of Schwarzian equations with regular singularitieshttps://gecogedi.dimai.unifi.it/paper/468/G. Faraco, S. Gupta.<p> Let $S$ be a punctured surface of finite type and negative Eulercharacteristic. We determine all possible representations $\rho:\pi_1(S) \to\text{PSL}_2(\mathbb{C})$ that arise as the monodromy of the Schwarzianequation on $S$ with regular singularities at the punctures. Equivalently, wedetermine the holonomy representations of complex projective structures on $S$,whose Schwarzian derivatives (with respect to some uniformizing structure) havepoles of order at most two at the punctures. Following earlier work that dealtwith the case when there are no apparent singularities, our proof reduces tothe case of realizing a degenerate representation with apparent singularities.This mainly involves explicit constructions of complex affine structures onpunctured surfaces, with prescribed holonomy. As a corollary, we determine therepresentations that arise as the holonomy of spherical metrics on $S$ withcone-points at the punctures.</p>https://gecogedi.dimai.unifi.it/paper/468/On the automorphism groups of certain branched structures on surfaceshttps://gecogedi.dimai.unifi.it/paper/467/G. Faraco.<p> We consider translation surfaces with poles on surfaces. We shall prove thatany finite group appears as the automorphism group of some translation surfacewith poles. As a direct consequence we obtain the existence of structuresachieving the maximal possible number of automorphisms allowed by their genusand we finally extend the same results to branched projective structures.</p>https://gecogedi.dimai.unifi.it/paper/467/On the distality and expansivity of certain maps on sphereshttps://gecogedi.dimai.unifi.it/paper/466/Alok Kumar Yadav, M. Choudhuri, G. Faraco.<p> Any affine map on the (n+1)-dimensional Euclidean space gives rise to anatural map on the n-dimensional sphere whose dynamical aspects are not sowell-studied in the literature. We explore the dynamical aspects of these mapsby investigating about their distality and expansivity.</p>https://gecogedi.dimai.unifi.it/paper/466/Special cubulation of strict hyperbolizationhttps://gecogedi.dimai.unifi.it/paper/465/J. F. Lafont, L. Ruffoni.<p> We prove that the Gromov hyperbolic groups obtained by the stricthyperbolization procedure of Charney and Davis are virtually compact special,hence linear and residually finite, if the initial complex satisfies some minorconditions. Our strategy consists in constructing an action of a hyperbolizedgroup on a certain dual CAT(0) cubical complex. As a result, all the commonapplications of strict hyperbolization are shown to provide manifolds withvirtually compact special fundamental group. In particular, we obtain examplesof closed negatively curved Riemannian manifolds whose fundamental groups arelinear and virtually algebraically fiber.</p>https://gecogedi.dimai.unifi.it/paper/465/Manifolds without real projective or flat conformal structureshttps://gecogedi.dimai.unifi.it/paper/464/L. Ruffoni.<p> In any dimension at least five we construct examples of closed smoothmanifolds with the following properties: 1) they have neither real projectivenor flat conformal structures; 2) their fundamental group is a non-elementaryGromov hyperbolic group. These examples are obtained via relative stricthyperbolization.</p>https://gecogedi.dimai.unifi.it/paper/464/A Serrin-type symmetry result on model manifolds: an extension of the Weinberger argumenthttps://gecogedi.dimai.unifi.it/paper/463/A. Roncoroni.<p>We consider the classical "Serrin symmetry result" for the overdeterminedboundary value problem related to the equation $\Delta u=-1$ in a modelmanifold of non-negative Ricci curvature. Using an extension of the Weinbergerclassical argument we prove a Euclidean symmetry result under a suitable"compatibility" assumption between the solution and the geometry of the model.</p>https://gecogedi.dimai.unifi.it/paper/463/Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusion equationshttps://gecogedi.dimai.unifi.it/paper/462/M. Muratori, A. Roncoroni.<p>We investigate the validity, as well as the failure, of Sobolev-typeinequalities on Cartan-Hadamard manifolds under suitable bounds on thesectional and the Ricci curvatures. We prove that if the sectional curvaturesare bounded from above by a negative power of the distance from a fixed pole(times a negative constant), then all the $ L^p $ inequalities that interpolatebetween Poincar\'e and Sobolev hold for radial functions provided the powerlies in the interval $ (-2,0) $. The Poincar\'e inequality was established byH.P. McKean under a constant negative bound from above on the sectionalcurvatures. If the power is equal to the critical value $ -2 $ we show that $ p$ must necessarily be bounded away from $ 2 $. Upon assuming that the Riccicurvature vanishes at infinity, the nonradial version of such inequalitiesturns out to fail, except in the Sobolev case. Finally, we discuss applicationsof the here-established Sobolev-type inequalities to optimal smoothing effectsfor radial porous medium equations.</p>https://gecogedi.dimai.unifi.it/paper/462/