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Multiple solutions for Schrödinger equations on Riemannian manifolds via (nabla )-theorems 黎曼流形上Schrödinger方程的多解及其$$nabla$$Ş-定理
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-01-24 DOI: 10.1007/s10455-023-09885-1
Luigi Appolloni, Giovanni Molica Bisci, Simone Secchi
{"title":"Multiple solutions for Schrödinger equations on Riemannian manifolds via (nabla )-theorems","authors":"Luigi Appolloni,&nbsp;Giovanni Molica Bisci,&nbsp;Simone Secchi","doi":"10.1007/s10455-023-09885-1","DOIUrl":"10.1007/s10455-023-09885-1","url":null,"abstract":"<div><p>We consider a smooth, complete and non-compact Riemannian manifold <span>((mathcal {M},g))</span> of dimension <span>(d ge 3)</span>, and we look for solutions to the semilinear elliptic equation </p><div><div><span>$$begin{aligned} -varDelta _g w + V(sigma ) w = alpha (sigma ) f(w) + lambda w quad hbox {in }mathcal {M}. end{aligned}$$</span></div></div><p>The potential <span>(V :mathcal {M} rightarrow mathbb {R})</span> is a continuous function which is coercive in a suitable sense, while the nonlinearity <i>f</i> has a subcritical growth in the sense of Sobolev embeddings. By means of <span>(nabla )</span>-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter <span>(lambda )</span> is sufficiently close to an eigenvalue of the operator <span>(-varDelta _g)</span>.\u0000</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49143954","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}
引用次数: 0
On the Steklov spectrum of covering spaces and total spaces 关于覆盖空间和全空间的Steklov谱
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-01-17 DOI: 10.1007/s10455-023-09884-2
Panagiotis Polymerakis
{"title":"On the Steklov spectrum of covering spaces and total spaces","authors":"Panagiotis Polymerakis","doi":"10.1007/s10455-023-09884-2","DOIUrl":"10.1007/s10455-023-09884-2","url":null,"abstract":"<div><p>We show the existence of a natural Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the <span>(L^2)</span>-space of the boundary admits Friedrichs extension. We focus on the spectrum of this operator on covering spaces and total spaces of Riemannian principal bundles over compact manifolds.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09884-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41604374","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}
引用次数: 0
Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces 二维和三维双曲空间上齐次向量丛截面的Delorme交织条件
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-12-05 DOI: 10.1007/s10455-022-09882-w
Martin Olbrich, Guendalina Palmirotta
{"title":"Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces","authors":"Martin Olbrich,&nbsp;Guendalina Palmirotta","doi":"10.1007/s10455-022-09882-w","DOIUrl":"10.1007/s10455-022-09882-w","url":null,"abstract":"<div><p>The description of the Paley–Wiener space for compactly supported smooth functions <span>(C^infty _c(G))</span> on a semi-simple Lie group <i>G</i> involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for <span>(G=textbf{SL}(2,mathbb {R})^d)</span> (<span>(din mathbb {N})</span>) and <span>(G=textbf{SL}(2,mathbb {C}))</span>. Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49570414","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}
引用次数: 1
On the eigenforms of compact stratified spaces 关于紧致分层空间的本征形式
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-11-24 DOI: 10.1007/s10455-022-09883-9
Luobin Fang
{"title":"On the eigenforms of compact stratified spaces","authors":"Luobin Fang","doi":"10.1007/s10455-022-09883-9","DOIUrl":"10.1007/s10455-022-09883-9","url":null,"abstract":"<div><p>Let <i>X</i> be a compact Thom–Mather stratified pseudomanifold, and let <i>M</i> be the regular part of <i>X</i> endowed with an iterated metric. In this paper, we prove that if the curvature operator of <i>M</i> is bounded, then the <span>(L^2)</span> harmonic space of <i>M</i> is finite dimensional. Next we consider the absolute eigenvalue problems of the Hodge Laplacian of a sequence of compact domains <span>(Omega _j)</span> converging to <i>M</i>. We prove that when the curvature operator of <i>M</i> is bounded, the eigenvalues of <span>(Omega _j)</span> converge to eigenvalues of <i>M</i>, and the eigenforms of <span>(Omega _j)</span> converge to eigenforms of <i>M</i> in the Sobolev norm. This generalizes Chavel and Feldman’s theorem in Chavel and Feldman (J Funct Anal 30:198-222, 1978) from compact manifolds to compact pseudomanifolds and from functions to differential forms. Then, we apply our results to <span>(L^2)</span>-chomology. We will give a correspondence between boundary cohomology and <span>(L^2)</span>-cohomology.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45120432","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}
引用次数: 0
The effect of pinching conditions in prescribing ( Q )-curvature on standard spheres 规定$$ Q $$ -曲率时捏紧条件对标准球体的影响
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-11-07 DOI: 10.1007/s10455-022-09878-6
Mohamed Ben Ayed, Khalil El Mehdi
{"title":"The effect of pinching conditions in prescribing ( Q )-curvature on standard spheres","authors":"Mohamed Ben Ayed,&nbsp;Khalil El Mehdi","doi":"10.1007/s10455-022-09878-6","DOIUrl":"10.1007/s10455-022-09878-6","url":null,"abstract":"<div><p>In this paper, we study the problem of prescribing a fourth-order conformal invariant on standard spheres. This problem is variational but it is noncompact due to the presence of nonconverging orbits of the gradient flow, the so called <i>critical points at infinity</i>. Following the method advised by Bahri we determine all such critical points at infinity and compute their contribution to the difference of topology between the level sets of the associated Euler–Lagrange functional. We then derive some existence results under pinching conditions.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42363648","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}
引用次数: 0
Compact geodesic orbit spaces with a simple isotropy group 具有简单各向同性群的紧致测地线轨道空间
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-11-07 DOI: 10.1007/s10455-022-09877-7
Z. Chen, Y. Nikolayevsky, Yu Nikonorov
{"title":"Compact geodesic orbit spaces with a simple isotropy group","authors":"Z. Chen,&nbsp;Y. Nikolayevsky,&nbsp;Yu Nikonorov","doi":"10.1007/s10455-022-09877-7","DOIUrl":"10.1007/s10455-022-09877-7","url":null,"abstract":"<div><p>Let <span>(M=G/H)</span> be a compact, simply connected, Riemannian homogeneous space, where <i>G</i> is (almost) effective and <i>H</i> is a <i>simple</i> Lie group. In this paper, we first classify all <i>G</i>-naturally reductive metrics on <i>M</i>, and then all <i>G</i>-geodesic orbit metrics on <i>M</i>.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-022-09877-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48223102","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}
引用次数: 6
Complexes, residues and obstructions for log-symplectic manifolds 对数辛流形的复形、残数和阻挡
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-11-07 DOI: 10.1007/s10455-022-09881-x
Ziv Ran
{"title":"Complexes, residues and obstructions for log-symplectic manifolds","authors":"Ziv Ran","doi":"10.1007/s10455-022-09881-x","DOIUrl":"10.1007/s10455-022-09881-x","url":null,"abstract":"<div><p>We consider compact Kählerian manifolds <i>X</i> of even dimension 4 or more, endowed with a log-symplectic structure <span>(Phi )</span>, a generically nondegenerate closed 2-form with simple poles on a divisor <i>D</i> with local normal crossings. A simple linear inequality involving the iterated Poincaré residues of <span>(Phi )</span> at components of the double locus of <i>D</i> ensures that the pair <span>((X, Phi ))</span> has unobstructed deformations and that <i>D</i> deforms locally trivially.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-022-09881-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48828458","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}
引用次数: 0
Higher-power harmonic maps and sections 高功率谐波图和截面
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-11-07 DOI: 10.1007/s10455-022-09875-9
A. Ramachandran, C. M. Wood
{"title":"Higher-power harmonic maps and sections","authors":"A. Ramachandran,&nbsp;C. M. Wood","doi":"10.1007/s10455-022-09875-9","DOIUrl":"10.1007/s10455-022-09875-9","url":null,"abstract":"<div><p>The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on three-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-022-09875-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45179907","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}
引用次数: 0
Umehara algebra and complex submanifolds of indefinite complex space forms 梅原代数与不定复空间形式的复子流形
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-10-27 DOI: 10.1007/s10455-022-09876-8
Xu Zhang, Donghai Ji
{"title":"Umehara algebra and complex submanifolds of indefinite complex space forms","authors":"Xu Zhang,&nbsp;Donghai Ji","doi":"10.1007/s10455-022-09876-8","DOIUrl":"10.1007/s10455-022-09876-8","url":null,"abstract":"<div><p>The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, and the symmetrized polydisc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44567395","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}
引用次数: 0
Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime GRW时空中类空间平均曲率流孤子的不存在性和刚度
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2022-10-24 DOI: 10.1007/s10455-022-09879-5
Allan Freitas, Henrique F. de Lima, Márcio S. Santos, Joyce S. Sindeaux
{"title":"Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime","authors":"Allan Freitas,&nbsp;Henrique F. de Lima,&nbsp;Márcio S. Santos,&nbsp;Joyce S. Sindeaux","doi":"10.1007/s10455-022-09879-5","DOIUrl":"10.1007/s10455-022-09879-5","url":null,"abstract":"<div><p>We study the nonexistence and rigidity of an important class of particular cases of trapped submanifolds, more precisely, <i>n</i>-dimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field <span>(mathcal K=f(t)partial _t)</span> (<span>(tin Isubset mathbb R)</span>) which is globally defined on an <span>((n+p+1))</span>-dimensional generalized Robertson–Walker (GRW) spacetime <span>(-Itimes _fM^{n+p})</span> with warping function <span>(fin C^infty (I))</span> and Riemannian fiber <span>(M^{n+p})</span>, via applications of suitable generalized maximum principles and under certain constraints on <i>f</i> and on the curvatures of <span>(M^{n+p})</span>. In codimension 1, we also obtain new Calabi–Bernstein-type results concerning the spacelike mean curvature flow soliton equation in a GRW spacetime.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44719079","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}
引用次数: 0
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