Journal of Geometric Analysis最新文献

{"title":"Fatou Components of Attracting Skew-Products.","authors":"Han Peters,&nbsp;Iris Marjan Smit","doi":"10.1007/s12220-017-9811-6","DOIUrl":"https://doi.org/10.1007/s12220-017-9811-6","url":null,"abstract":"<p><p>We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 1","pages":"84-110"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9811-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36866346","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}

{"title":"Littlewood-Paley Theory for Triangle Buildings.","authors":"Tim Steger,&nbsp;Bartosz Trojan","doi":"10.1007/s12220-017-9856-6","DOIUrl":"https://doi.org/10.1007/s12220-017-9856-6","url":null,"abstract":"<p><p>For the natural two-parameter filtration <math> <mfenced><msub><mi>F</mi> <mi>λ</mi></msub> <mo>:</mo> <mrow><mi>λ</mi> <mo>∈</mo> <mi>P</mi></mrow> </mfenced> </math> on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on <math> <mrow><msup><mi>L</mi> <mi>p</mi></msup> <mrow><mo>(</mo> <msub><mi>Ω</mi> <mn>0</mn></msub> <mo>)</mo></mrow> </mrow> </math> for <math><mrow><mi>p</mi> <mo>∈</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo></mrow> </math> . At the end, we consider <math> <mrow><msup><mi>L</mi> <mi>p</mi></msup> <mrow><mo>(</mo> <msub><mi>Ω</mi> <mn>0</mn></msub> <mo>)</mo></mrow> </mrow> </math> boundedness of martingale transforms. If the building is of <math><mrow><mtext>GL</mtext> <mo>(</mo> <mn>3</mn> <mo>,</mo> <msub><mi>Q</mi> <mi>p</mi></msub> <mo>)</mo></mrow> </math> , then <math><msub><mi>Ω</mi> <mn>0</mn></msub> </math> can be identified with <i>p</i>-adic Heisenberg group.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 2","pages":"1122-1150"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9856-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37028502","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}

{"title":"A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.","authors":"Saskia Roos","doi":"10.1007/s12220-017-9930-0","DOIUrl":"https://doi.org/10.1007/s12220-017-9930-0","url":null,"abstract":"<p><p>Let <math><mrow><mi>M</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo></mrow> </math> be the space of closed <i>n</i>-dimensional Riemannian manifolds (<i>M</i>, <i>g</i>) with <math><mrow><mi>diam</mi> <mo>(</mo> <mi>M</mi> <mo>)</mo> <mo>≤</mo> <mi>D</mi></mrow> </math> and <math> <mrow><mrow><mo>|</mo></mrow> <msup><mo>sec</mo> <mi>M</mi></msup> <mrow><mo>|</mo> <mo>≤</mo> <mn>1</mn></mrow> </mrow> </math> . In this paper we consider sequences <math><mrow><mo>(</mo> <msub><mi>M</mi> <mi>i</mi></msub> <mo>,</mo> <msub><mi>g</mi> <mi>i</mi></msub> <mo>)</mo></mrow> </math> in <math><mrow><mi>M</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo></mrow> </math> converging in the Gromov-Hausdorff topology to a compact metric space <i>Y</i>. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number <i>r</i> such that the quotient <math> <mfrac><mrow><mi>vol</mi> <mo>(</mo> <msubsup><mi>B</mi> <mi>r</mi> <msub><mi>M</mi> <mi>i</mi></msub> </msubsup> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> <mrow> <msup><mrow><mi>inj</mi></mrow> <msub><mi>M</mi> <mi>i</mi></msub> </msup> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow> </mfrac> </math> can be uniformly bounded from below by a positive constant <i>C</i>(<i>n</i>, <i>r</i>, <i>Y</i>) for all points <math><mrow><mi>x</mi> <mo>∈</mo> <msub><mi>M</mi> <mi>i</mi></msub> </mrow> </math> . On the other hand, we show that if the limit space has at most codimension one then for all positive <i>r</i> there is a positive constant <i>C</i>(<i>n</i>, <i>r</i>, <i>Y</i>) bounding the quotient <math> <mfrac><mrow><mi>vol</mi> <mo>(</mo> <msubsup><mi>B</mi> <mi>r</mi> <msub><mi>M</mi> <mi>i</mi></msub> </msubsup> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> <mrow> <msup><mrow><mi>inj</mi></mrow> <msub><mi>M</mi> <mi>i</mi></msub> </msup> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow> </mfrac> </math> uniformly from below for all <math><mrow><mi>x</mi> <mo>∈</mo> <msub><mi>M</mi> <mi>i</mi></msub> </mrow> </math> . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in <math><mrow><mi>M</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo></mrow> </math> with <math><mrow><mi>C</mi> <mo>≤</mo> <mfrac><mrow><mi>vol</mi> <mo>(</mo> <mi>M</mi> <mo>)</mo></mrow> <mrow><mi>inj</mi> <mo>(</mo> <mi>M</mi> <mo>)</mo></mrow> </mfrac> </mrow> </math> .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 3","pages":"2707-2724"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9930-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37028499","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}

{"title":"The Trace Theorem, the Luzin <i>N</i>- and Morse-Sard Properties for the Sharp Case of Sobolev-Lorentz Mappings.","authors":"Mikhail V Korobkov,&nbsp;Jan Kristensen","doi":"10.1007/s12220-017-9936-7","DOIUrl":"https://doi.org/10.1007/s12220-017-9936-7","url":null,"abstract":"<p><p>We prove Luzin <i>N</i>- and Morse-Sard properties for mappings <math><mrow><mi>v</mi> <mo>:</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> <mo>→</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> of the Sobolev-Lorentz class <math> <msubsup><mrow><mi>W</mi></mrow> <mrow><mi>p</mi> <mo>,</mo> <mn>1</mn></mrow> <mi>k</mi></msubsup> </math> , <math><mrow><mi>p</mi> <mo>=</mo> <mfrac><mi>n</mi> <mi>k</mi></mfrac> </mrow> </math> (this is the sharp case that guaranties the continuity of mappings). Our main tool is a new trace theorem for Riesz potentials of Lorentz functions for the limiting case <math><mrow><mi>q</mi> <mo>=</mo> <mi>p</mi></mrow> </math> . Using these results, we find also some very natural approximation and differentiability properties for functions in <math> <msubsup><mrow><mi>W</mi></mrow> <mrow><mi>p</mi> <mo>,</mo> <mn>1</mn></mrow> <mi>k</mi></msubsup> </math> with exceptional set of small Hausdorff content.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 3","pages":"2834-2856"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9936-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37030163","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}

{"title":"The Geometry of <i>m</i>-Hyperconvex Domains.","authors":"Per Åhag,&nbsp;Rafał Czyż,&nbsp;Lisa Hed","doi":"10.1007/s12220-017-9957-2","DOIUrl":"https://doi.org/10.1007/s12220-017-9957-2","url":null,"abstract":"<p><p>We study the geometry of <i>m</i>-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every <i>m</i>-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly <i>m</i>-subharmonic, and has bounded <i>m</i>-Hessian measure.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 4","pages":"3196-3222"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9957-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36822417","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}

{"title":"Morrey Spaces on Domains: Different Approaches and Growth Envelopes.","authors":"Dorothee D Haroske,&nbsp;Cornelia Schneider,&nbsp;Leszek Skrzypczak","doi":"10.1007/s12220-017-9843-y","DOIUrl":"https://doi.org/10.1007/s12220-017-9843-y","url":null,"abstract":"<p><p>We deal with Morrey spaces on bounded domains <math><mi>Ω</mi></math> obtained by different approaches. In particular, we consider three settings <math> <mrow><msub><mi>M</mi> <mrow><mi>u</mi> <mo>,</mo> <mi>p</mi></mrow> </msub> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> , <math> <mrow><msub><mi>M</mi> <mrow><mi>u</mi> <mo>,</mo> <mi>p</mi></mrow> </msub> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> and <math> <mrow><msub><mi>M</mi> <mrow><mi>u</mi> <mo>,</mo> <mi>p</mi></mrow> </msub> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </mrow> </math> , where <math><mrow><mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>u</mi> <mo><</mo> <mi>∞</mi></mrow> </math> , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes <math> <mrow><msub><mi>E</mi> <mi>G</mi></msub> <mrow><mo>(</mo> <msub><mi>M</mi> <mrow><mi>u</mi> <mo>,</mo> <mi>p</mi></mrow> </msub> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> </mrow> </math> as well as <math> <mrow><msub><mi>E</mi> <mi>G</mi></msub> <mrow><mo>(</mo> <msub><mi>M</mi> <mrow><mi>u</mi> <mo>,</mo> <mi>p</mi></mrow> </msub> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> </mrow> </math> , and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between <i>p</i> and <i>u</i> in the sense of whether <math> <mrow><mfrac><mi>n</mi> <mi>u</mi></mfrac> <mo>≥</mo> <mfrac><mn>1</mn> <mi>p</mi></mfrac> </mrow> </math> or <math> <mrow><mfrac><mi>n</mi> <mi>u</mi></mfrac> <mo><</mo> <mfrac><mn>1</mn> <mi>p</mi></mfrac> </mrow> </math> plays a decisive role when it comes to the behaviour of these spaces.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 2","pages":"817-841"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9843-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37203645","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}

{"title":"Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity.","authors":"Gui-Qiang G Chen,&nbsp;Siran Li","doi":"10.1007/s12220-017-9893-1","DOIUrl":"https://doi.org/10.1007/s12220-017-9893-1","url":null,"abstract":"<p><p>We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 3","pages":"1957-2007"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9893-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37030165","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}

{"title":"A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics","authors":"J. Jiménez-Garrido, J. Sanz, G. Schindl","doi":"10.1007/s12220-019-00203-5,","DOIUrl":"https://doi.org/10.1007/s12220-019-00203-5,","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"30 1","pages":"3458 - 3483"},"PeriodicalIF":1.1,"publicationDate":"2017-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41685299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Positive Scalar Curvature on Foliations:The Enlargeability","authors":"Weiping Zhang","doi":"10.1007/978-3-030-34953-0_22","DOIUrl":"https://doi.org/10.1007/978-3-030-34953-0_22","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"73 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80626439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotics of Partial Density Functions for Divisors.","authors":"Julius Ross,&nbsp;Michael Singer","doi":"10.1007/s12220-016-9741-8","DOIUrl":"https://doi.org/10.1007/s12220-016-9741-8","url":null,"abstract":"<p><p>We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor <i>Y</i>. Assuming the data in question is invariant under an <math><msup><mi>S</mi> <mn>1</mn></msup> </math> -action (locally around <i>Y</i>) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up. Moreover we recover the existence of the \"forbidden region\" <i>R</i> on which the density function is exponentially small, and prove that it has an \"error-function\" behaviour across the boundary <math><mrow><mi>∂</mi> <mi>R</mi></mrow> </math> . As an illustrative application, we use this to study a certain natural function that can be associated to a divisor in a Kähler manifold.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"27 3","pages":"1803-1854"},"PeriodicalIF":1.1,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-016-9741-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37030162","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}