Journal of Geometric Analysis最新文献

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Worm Domains are not Gromov Hyperbolic. Worm域不是Gromov双曲域。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-05-31 DOI: 10.1007/s12220-023-01320-y
Leandro Arosio, Gian Maria Dall'Ara, Matteo Fiacchi
{"title":"Worm Domains are not Gromov Hyperbolic.","authors":"Leandro Arosio,&nbsp;Gian Maria Dall'Ara,&nbsp;Matteo Fiacchi","doi":"10.1007/s12220-023-01320-y","DOIUrl":"10.1007/s12220-023-01320-y","url":null,"abstract":"<p><p>We show that Worm domains are not Gromov hyperbolic with respect to the Kobayashi distance.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 8","pages":"257"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10232651/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9578918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 1
On the Normal Stability of Triharmonic Hypersurfaces in Space Forms. 关于空间形式中三调和超曲面的正规稳定性。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-08-29 DOI: 10.1007/s12220-023-01414-7
Volker Branding
{"title":"On the Normal Stability of Triharmonic Hypersurfaces in Space Forms.","authors":"Volker Branding","doi":"10.1007/s12220-023-01414-7","DOIUrl":"10.1007/s12220-023-01414-7","url":null,"abstract":"<p><p>This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere <math><mrow><mi>ϕ</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mi>m</mi></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math> is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 11","pages":"355"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10465648/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10509996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Horizontally Affine Functions on Step-2 Carnot Algebras. Step-2卡诺代数上的水平仿射函数。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-09-09 DOI: 10.1007/s12220-023-01360-4
Enrico Le Donne, Daniele Morbidelli, Séverine Rigot
{"title":"Horizontally Affine Functions on Step-2 Carnot Algebras.","authors":"Enrico Le Donne,&nbsp;Daniele Morbidelli,&nbsp;Séverine Rigot","doi":"10.1007/s12220-023-01360-4","DOIUrl":"10.1007/s12220-023-01360-4","url":null,"abstract":"<p><p>In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-<i>n</i> Carnot algebra is isomorphic to the exterior algebra of <math><msup><mrow><mi>R</mi></mrow><mi>n</mi></msup></math>. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 11","pages":"359"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10492776/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10589130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 2
Multicomplexes on Carnot Groups and Their Associated Spectral Sequence. 卡诺群上的多重配合物及其相关谱序列。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2023-01-01 DOI: 10.1007/s12220-023-01259-0
Antonio Lerario, Francesca Tripaldi
{"title":"Multicomplexes on Carnot Groups and Their Associated Spectral Sequence.","authors":"Antonio Lerario,&nbsp;Francesca Tripaldi","doi":"10.1007/s12220-023-01259-0","DOIUrl":"https://doi.org/10.1007/s12220-023-01259-0","url":null,"abstract":"<p><p>The aim of this paper is to give a thorough insight into the relationship between the Rumin complex on Carnot groups and the spectral sequence obtained from the filtration on forms by homogeneous weights that computes the de Rham cohomology of the underlying group.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 7","pages":"199"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10119276/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9389909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 3
The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation. 拓扑状态导数:拓扑优化的最优控制视角。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-05-15 DOI: 10.1007/s12220-023-01295-w
Phillip Baumann, Idriss Mazari-Fouquer, Kevin Sturm
{"title":"The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation.","authors":"Phillip Baumann,&nbsp;Idriss Mazari-Fouquer,&nbsp;Kevin Sturm","doi":"10.1007/s12220-023-01295-w","DOIUrl":"10.1007/s12220-023-01295-w","url":null,"abstract":"<p><p>In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22-47, 2022; J Convex Anal 25(3):957-982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 8","pages":"243"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10185627/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9544949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds. 完全严格伪凸CR流形上的塞格格核渐近性。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-08-18 DOI: 10.1007/s12220-022-00990-4
Chin-Yu Hsiao, George Marinescu, Huan Wang
{"title":"Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds.","authors":"Chin-Yu Hsiao,&nbsp;George Marinescu,&nbsp;Huan Wang","doi":"10.1007/s12220-022-00990-4","DOIUrl":"https://doi.org/10.1007/s12220-022-00990-4","url":null,"abstract":"<p><p>We prove a Bochner-Kodaira-Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR <math><mi>R</mi></math> -action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 11","pages":"266"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9387902/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40437983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 1
Analytic Torsion of Generic Rank Two Distributions in Dimension Five. 五维一般二阶分布的解析扭转。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-07-26 DOI: 10.1007/s12220-022-00987-z
Stefan Haller
{"title":"Analytic Torsion of Generic Rank Two Distributions in Dimension Five.","authors":"Stefan Haller","doi":"10.1007/s12220-022-00987-z","DOIUrl":"10.1007/s12220-022-00987-z","url":null,"abstract":"<p><p>We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 10","pages":"248"},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9325871/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40574965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Hermitian-Yang-Mills Connections on Collapsing Elliptically Fibered K3 Surfaces. 塌缩椭圆纤维K3曲面上的Hermitian-Yang-Mills连接。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-01-07 DOI: 10.1007/s12220-021-00808-9
Ved Datar, Adam Jacob
{"title":"Hermitian-Yang-Mills Connections on Collapsing Elliptically Fibered <i>K</i>3 Surfaces.","authors":"Ved Datar,&nbsp;Adam Jacob","doi":"10.1007/s12220-021-00808-9","DOIUrl":"https://doi.org/10.1007/s12220-021-00808-9","url":null,"abstract":"<p><p>Let <math><mrow><mi>X</mi> <mo>→</mo> <msup><mrow><mi>P</mi></mrow> <mn>1</mn></msup> </mrow> </math> be an elliptically fibered <i>K</i>3 surface, admitting a sequence <math><msub><mi>ω</mi> <mi>i</mi></msub> </math> of Ricci-flat metrics collapsing the fibers. Let <i>V</i> be a holomorphic <i>SU</i>(<i>n</i>) bundle over <i>X</i>, stable with respect to <math><msub><mi>ω</mi> <mi>i</mi></msub> </math> . Given the corresponding sequence <math><msub><mi>Ξ</mi> <mi>i</mi></msub> </math> of Hermitian-Yang-Mills connections on <i>V</i>, we prove that, if <i>E</i> is a generic fiber, the restricted sequence <math> <mrow><msub><mi>Ξ</mi> <mi>i</mi></msub> <msub><mrow><mo>|</mo></mrow> <mi>E</mi></msub> </mrow> </math> converges to a flat connection <math><msub><mi>A</mi> <mn>0</mn></msub> </math> . Furthermore, if the restriction <math> <msub><mrow><mi>V</mi> <mo>|</mo></mrow> <mi>E</mi></msub> </math> is of the form <math> <mrow><msubsup><mo>⊕</mo> <mrow><mi>j</mi> <mo>=</mo> <mn>1</mn></mrow> <mi>n</mi></msubsup> <msub><mi>O</mi> <mi>E</mi></msub> <mrow><mo>(</mo> <msub><mi>q</mi> <mi>j</mi></msub> <mo>-</mo> <mn>0</mn> <mo>)</mo></mrow> </mrow> </math> for <i>n</i> distinct points <math> <mrow><msub><mi>q</mi> <mi>j</mi></msub> <mo>∈</mo> <mi>E</mi></mrow> </math> , then these points uniquely determine <math><msub><mi>A</mi> <mn>0</mn></msub> </math> .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 2","pages":"69"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8741718/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39882092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 2
On Rectifiable Measures in Carnot Groups: Existence of Density. 卡诺群中的可校正测度:密度的存在性。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-07-18 DOI: 10.1007/s12220-022-00971-7
Gioacchino Antonelli, Andrea Merlo
{"title":"On Rectifiable Measures in Carnot Groups: Existence of Density.","authors":"Gioacchino Antonelli,&nbsp;Andrea Merlo","doi":"10.1007/s12220-022-00971-7","DOIUrl":"https://doi.org/10.1007/s12220-022-00971-7","url":null,"abstract":"<p><p>In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable, for <math><mrow><mi>h</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> , if it has positive <i>h</i>-lower density and finite <i>h</i>-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measures. Namely, we prove that the support of a <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a <math><msub><mi>P</mi> <mi>h</mi></msub> </math> -rectifiable measure has almost everywhere positive and finite <i>h</i>-density whenever the tangents admit at least one complementary subgroup.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 9","pages":"239"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9293879/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40534631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 9
Exception Sets of Intrinsic and Piecewise Lipschitz Functions. 本征函数和片状 Lipschitz 函数的例外集。
IF 1.1 2区 数学
Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-02-01 DOI: 10.1007/s12220-021-00860-5
Gunther Leobacher, Alexander Steinicke
{"title":"Exception Sets of Intrinsic and Piecewise Lipschitz Functions.","authors":"Gunther Leobacher, Alexander Steinicke","doi":"10.1007/s12220-021-00860-5","DOIUrl":"10.1007/s12220-021-00860-5","url":null,"abstract":"<p><p>We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , which include Lipschitz submanifolds.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 4","pages":"118"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8807473/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39914097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
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