{"title":"Equivariant CR Yamabe problem","authors":"Pak Tung Ho","doi":"10.1007/s10231-024-01484-6","DOIUrl":"https://doi.org/10.1007/s10231-024-01484-6","url":null,"abstract":"<p>As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup <i>G</i> of the isometry group, find a <i>G</i>-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we introduce the equivariant CR Yamabe problem and prove some related results.</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"87 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The characteristic group of locally conformally product structures","authors":"Brice Flamencourt","doi":"10.1007/s10231-024-01479-3","DOIUrl":"https://doi.org/10.1007/s10231-024-01479-3","url":null,"abstract":"<p>A compact manifold <i>M</i> together with a Riemannian metric <i>h</i> on its universal cover <span>(tilde{M})</span> for which <span>(pi _1(M))</span> acts by similarities is called a similarity structure. In the case where <span>(pi _1(M) not subset textrm{Isom}(tilde{M}, h))</span> and <span>((tilde{M}, h))</span> is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial <span>(mathbb {R}^p)</span>-principal bundles over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure.</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ignazio Longhi, Nadir Murru, Francesco M. Saettone
{"title":"Heights and transcendence of p-adic continued fractions","authors":"Ignazio Longhi, Nadir Murru, Francesco M. Saettone","doi":"10.1007/s10231-024-01476-6","DOIUrl":"https://doi.org/10.1007/s10231-024-01476-6","url":null,"abstract":"<p>Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous <i>p</i>–adic problem. More specifically, we deal with Browkin <i>p</i>–adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a <i>p</i>–adic Euclidean algorithm. Then, we focus on the heights of some <i>p</i>–adic numbers having a periodic <i>p</i>–adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with <i>p</i>–adic Roth-like results, in order to prove the transcendence of three families of <i>p</i>–adic continued fractions.\u0000</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"75 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups","authors":"Leonardo Biliotti, Oluwagbenga Joshua Windare","doi":"10.1007/s10231-024-01480-w","DOIUrl":"https://doi.org/10.1007/s10231-024-01480-w","url":null,"abstract":"<p>We study a Hilbert–Mumford criterion for polystablility associated with an action of a real reductive Lie group <i>G</i> on a real submanifold <i>X</i> of a Kähler manifold <i>Z</i>. Suppose the action of a compact Lie group with Lie algebra <span>(mathfrak {u})</span> extends holomorphically to an action of the complexified group <span>(U^{mathbb {C}})</span> and that the <i>U</i>-action on <i>Z</i> is Hamiltonian. If <span>(Gsubset U^{mathbb {C}})</span> is compatible, there is a corresponding gradient map <span>(mu _mathfrak {p}: Xrightarrow mathfrak {p})</span>, where <span>(mathfrak {g}= mathfrak {k}oplus mathfrak {p})</span> is a Cartan decomposition of the Lie algebra of <i>G</i>. Under some mild restrictions on the <i>G</i>-action on <i>X</i>, we characterize which <i>G</i>-orbits in <i>X</i> intersect <span>(mu _mathfrak {p}^{-1}(0))</span> in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity (<span>(partial _infty G/K)</span>) of the symmetric space <i>G</i>/<i>K</i>. We also establish the Hilbert–Mumford criterion for polystability of the action of <i>G</i> on measures.</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Partial separability and symplectic-Haantjes manifolds","authors":"Daniel Reyes, Piergiulio Tempesta, Giorgio Tondo","doi":"10.1007/s10231-024-01462-y","DOIUrl":"10.1007/s10231-024-01462-y","url":null,"abstract":"<div><p>A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) that allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra. We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized Stäckel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. The new systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"203 6","pages":"2677 - 2710"},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-024-01462-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasilinear elliptic problem in anisotropic Orlicz–Sobolev space on unbounded domain","authors":"Karol Wroński","doi":"10.1007/s10231-024-01477-5","DOIUrl":"https://doi.org/10.1007/s10231-024-01477-5","url":null,"abstract":"<p>We study a quasilinear elliptic problem <span>(-text {div} (nabla Phi (nabla u))+V(x)N'(u)=f(u))</span> with anisotropic convex function <span>(Phi )</span> on the whole <span>(mathbb {R}^n)</span>. To prove existence of a nontrivial weak solution we use the mountain pass theorem for a functional defined on anisotropic Orlicz–Sobolev space <span>({{{,mathrm{textbf{W}},}}^1}{{,mathrm{textbf{L}},}}^{{Phi }} (mathbb {R}^n))</span>. As the domain is unbounded we need to use Lions type lemma formulated for Young functions. Our assumptions broaden the class of considered functions <span>(Phi )</span> so our result generalizes earlier analogous results proved in isotropic setting.</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141496166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-radial ground state solutions for fractional Schrödinger–Poisson systems in (mathbb {R}^{2})","authors":"Guofeng Che, Juntao Sun, Tsung-Fang Wu","doi":"10.1007/s10231-024-01470-y","DOIUrl":"10.1007/s10231-024-01470-y","url":null,"abstract":"<div><p>In this paper, we study the fractional Schrödinger–Poisson system with a general nonlinearity as follows: </p><div><div><span>$$begin{aligned} left{ begin{array}{ll} (-Delta )^{s}u+u+ l(x)phi u=f(u) &{} text { in }mathbb {R}^{2}, (-Delta )^{t}phi =l(x)u^{2} &{} text { in }mathbb {R}^{2}, end{array} right. end{aligned}$$</span></div></div><p>where <span>(frac{1}{2}<tle s<1)</span>, the potential <span>(lin C(mathbb {R}^{2},mathbb {R}^{+}))</span> and <span>(fin C(mathbb {R},mathbb {R}))</span> does not require the classical (AR)-condition. When <span>(l(x)equiv mu >0)</span> is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of <span>(mu )</span>. As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on <i>l</i>(<i>x</i>). When <i>l</i>(<i>x</i>) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on <i>l</i>(<i>x</i>), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"203 6","pages":"2863 - 2888"},"PeriodicalIF":1.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stratified ocean gyres with Stuart-type vortices","authors":"Qixing Ding, Luigi Roberti","doi":"10.1007/s10231-024-01469-5","DOIUrl":"10.1007/s10231-024-01469-5","url":null,"abstract":"<div><p>In the setting of the thin-shell approximation of the Euler equations in spherical coordinates for oceanic flows with variable density on the spinning Earth, we study a vorticity equation for a pseudo stream function <span>(psi )</span>, whereby the assumption of incompressibility allows us to express the density as a function of <span>(psi )</span>. Via an elliptic comparison argument, we show that, under certain assumptions, the (explicit) solution in the case of zero rate of rotation (i.e., on a fixed sphere) in a bounded region with smooth boundary contained either in the Northern or in the Southern Hemisphere is an approximation, in a suitable sense, of the corresponding solution of the equation with positive rate of rotation in the same region. This provides new insight into the dynamics of ocean gyres.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"203 6","pages":"2847 - 2862"},"PeriodicalIF":1.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-024-01469-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141382607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Gehring–Hayman type theorem on pseudoconvex domains of finite type in (mathbb {C}^2)","authors":"Haichou Li, Xingsi Pu, Hongyu Wang","doi":"10.1007/s10231-024-01466-8","DOIUrl":"10.1007/s10231-024-01466-8","url":null,"abstract":"<div><p>In this paper, we obtain the Gehring–Hayman type theorem on smoothly bounded pseudoconvex domains of finite type in <span>(mathbb {C}^2)</span>. As an application, we provide a quantitative comparison between global and local Kobayashi distances near a boundary point for these domains. \u0000</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"203 6","pages":"2785 - 2799"},"PeriodicalIF":1.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-024-01466-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141383485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extensions of extremal Kähler submanifolds of complex projective spaces","authors":"Chao Li","doi":"10.1007/s10231-024-01468-6","DOIUrl":"10.1007/s10231-024-01468-6","url":null,"abstract":"<div><p>In this paper we show that every connected extremal Kähler submanifold of a complex projective space has a natural extension which is a complete Kähler manifold and admits a holomorphic isometric immersion into the same ambient space. We also give an application to study the scalar curvatures of extremal Hypersurfaces of complex projective spaces.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"203 6","pages":"2825 - 2845"},"PeriodicalIF":1.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-024-01468-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142452929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}