Journal of Geometric Analysis最新文献

{"title":"On the Cheeger Inequality in Carnot-Carathéodory Spaces.","authors":"Martijn Kluitenberg","doi":"10.1007/s12220-025-01912-w","DOIUrl":"https://doi.org/10.1007/s12220-025-01912-w","url":null,"abstract":"<p><p>We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carathéodory spaces and we describe a concrete method to lower bound the Cheeger constant. The proof is geometric, and works for Dirichlet, Neumann and mixed boundary conditions. One of the main technical tools in the proof is a generalization of Courant's nodal domain theorem, which is proven from scratch for Neumann and mixed boundary conditions. Carnot groups and the Baouendi-Grushin cylinder are treated as examples.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 3","pages":"82"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11880090/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143574622","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":"On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian.","authors":"Sam Farrington","doi":"10.1007/s12220-024-01887-0","DOIUrl":"https://doi.org/10.1007/s12220-024-01887-0","url":null,"abstract":"<p><p>We consider the problem of minimising the <i>k</i>-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension <math><mrow><mi>d</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as <math><mrow><mi>k</mi> <mo>→</mo> <mo>+</mo> <mi>∞</mi></mrow> </math> . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension <math><mrow><mi>d</mi> <mo>=</mo> <mn>2</mn></mrow> </math> . We also consider these problems for Robin eigenvalues and mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 2","pages":"62"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11811466/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143410705","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":"More Weakly Biharmonic Maps from the Ball to the Sphere.","authors":"Volker Branding","doi":"10.1007/s12220-024-01852-x","DOIUrl":"10.1007/s12220-024-01852-x","url":null,"abstract":"<p><p>In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable critical points of the bienergy.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 1","pages":"23"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11584471/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142711340","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":"On Isolated Singularities and Generic Regularity of Min-Max CMC Hypersurfaces.","authors":"Costante Bellettini, Kobe Marshall-Stevens","doi":"10.1007/s12220-025-01956-y","DOIUrl":"https://doi.org/10.1007/s12220-025-01956-y","url":null,"abstract":"<p><p>In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure in Bellettini and Wickramasekera (arXiv:2010.05847, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real <math><mi>λ</mi></math> we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature <math><mi>λ</mi></math> ; the minimal case ( <math><mrow><mi>λ</mi> <mo>=</mo> <mn>0</mn></mrow> </math> ) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 4","pages":"126"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11920008/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143671740","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":"Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves.","authors":"Martin Bauer, Patrick Heslin, Cy Maor","doi":"10.1007/s12220-024-01652-3","DOIUrl":"https://doi.org/10.1007/s12220-024-01652-3","url":null,"abstract":"<p><p>We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order <math><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math>. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if <math><mrow><mi>q</mi><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if <math><mrow><mi>q</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math>, whereas if <math><mrow><mi>q</mi><mo><</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math> then finite-time blowup may occur. The geodesic completeness for <math><mrow><mi>q</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></math> is obtained by proving metric completeness of the space of <math><msup><mi>H</mi><mi>q</mi></msup></math>-immersed curves with the distance induced by the Riemannian metric.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 7","pages":"214"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11068588/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140857102","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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The <ns0:math><ns0:msub><ns0:mi>A</ns0:mi><ns0:mi>∞</ns0:mi></ns0:msub></ns0:math> Condition, <ns0:math><ns0:mi>ε</ns0:mi></ns0:math>-Approximators, and Varopoulos Extensions in Uniform Domains.","authors":"S Bortz, B Poggi, O Tapiola, X Tolsa","doi":"10.1007/s12220-024-01666-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01666-x","url":null,"abstract":"<p><p>Suppose that <math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math>, <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>, is a uniform domain with <i>n</i>-Ahlfors regular boundary and <i>L</i> is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in <math><mi>Ω</mi></math>. We show that the corresponding elliptic measure <math><msub><mi>ω</mi><mi>L</mi></msub></math> is quantitatively absolutely continuous with respect to surface measure of <math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math> in the sense that <math><mrow><msub><mi>ω</mi><mi>L</mi></msub><mo>∈</mo><msub><mi>A</mi><mi>∞</mi></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math> if and only if any bounded solution <i>u</i> to <math><mrow><mi>L</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math> in <math><mi>Ω</mi></math> is <math><mi>ε</mi></math>-approximable for any <math><mrow><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math>. By <math><mi>ε</mi></math>-approximability of <i>u</i> we mean that there exists a function <math><mrow><mi>Φ</mi><mo>=</mo><msup><mi>Φ</mi><mi>ε</mi></msup></mrow></math> such that <math><mrow><msub><mrow><mo>‖</mo><mi>u</mi><mo>-</mo><mi>Φ</mi><mo>‖</mo></mrow><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><mi>ε</mi><msub><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub></mrow></math> and the measure <math><msub><mover><mi>μ</mi><mo>~</mo></mover><mi>Φ</mi></msub></math> with <math><mrow><mi>d</mi><mover><mi>μ</mi><mo>~</mo></mover><mo>=</mo><mrow><mo>|</mo><mi>∇</mi><mi>Φ</mi><mrow><mo>(</mo><mi>Y</mi><mo>)</mo></mrow><mo>|</mo></mrow><mspace></mspace><mi>d</mi><mi>Y</mi></mrow></math> is a Carleson measure with <math><msup><mi>L</mi><mi>∞</mi></msup></math> control over the Carleson norm. As a consequence of this approximability result, we show that boundary <math><mrow><mspace></mspace><mtext>BMO</mtext><mspace></mspace></mrow></math> functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy <math><msup><mi>L</mi><mn>1</mn></msup></math>-type Carleson measure estimates with <math><mrow><mspace></mspace><mtext>BMO</mtext><mspace></mspace></mrow></math> control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 7","pages":"218"},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11087277/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140913353","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":"On the Dimension of the Singular Set of Perimeter Minimizers in Spaces with a Two-Sided Bound on the Ricci Curvature.","authors":"Alessandro Cucinotta, Francesco Fiorani","doi":"10.1007/s12220-024-01784-6","DOIUrl":"https://doi.org/10.1007/s12220-024-01784-6","url":null,"abstract":"<p><p>We show that the Hausdorff dimension of the singular set of perimeter minimizers in noncollapsed limits of manifolds with two-sided bounds on the Ricci curvature is at most <math><mrow><mi>n</mi> <mo>-</mo> <mn>5</mn></mrow> </math> , where <i>n</i> is the dimension of the ambient space. The estimate is sharp.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"34 12","pages":"381"},"PeriodicalIF":1.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11845540/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143484951","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":"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}

{"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}

{"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}