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Free Boundary Hamiltonian Stationary Lagrangian Discs in C 2. c2中的自由边界哈密顿静止拉格朗日盘。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-04-04 DOI: 10.1007/s12220-025-01962-0
Filippo Gaia
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Free Boundary Hamiltonian Stationary Lagrangian Discs in <ns0:math> <ns0:msup><ns0:mrow><ns0:mi>C</ns0:mi></ns0:mrow> <ns0:mn>2</ns0:mn></ns0:msup></ns0:math>.","authors":"Filippo Gaia","doi":"10.1007/s12220-025-01962-0","DOIUrl":"https://doi.org/10.1007/s12220-025-01962-0","url":null,"abstract":"<p><p>Let <math><mrow><mi>Ω</mi> <mo>⊂</mo> <msup><mrow><mi>C</mi></mrow> <mn>2</mn></msup> </mrow> </math> be a smooth domain. We establish conditions under which a weakly conformal, branched <math><mi>Ω</mi></math> -free boundary Hamiltonian stationary Lagrangian immersion <i>u</i> of a disc in <math> <msup><mrow><mi>C</mi></mrow> <mn>2</mn></msup> </math> is a <math><mi>Ω</mi></math> -free boundary minimal immersion. We deduce that if <math><mi>u</mi></math> is a weakly conformal, branched <math> <mrow><msub><mi>B</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mn>0</mn> <mo>)</mo></mrow> </mrow> </math> -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then <math><mrow><mi>u</mi> <mo>(</mo> <msup><mi>D</mi> <mn>2</mn></msup> <mo>)</mo></mrow> </math> is a Lagrangian equatorial plane disc. Furthermore, we present examples of <math><mi>Ω</mi></math> -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 5","pages":"160"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971164/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797164","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
Geometric Bounds for Low Steklov Eigenvalues of Finite Volume Hyperbolic Surfaces. 有限体积双曲曲面的低Steklov特征值的几何界。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-04-04 DOI: 10.1007/s12220-025-01990-w
Asma Hassannezhad, Antoine Métras, Hélène Perrin
{"title":"Geometric Bounds for Low Steklov Eigenvalues of Finite Volume Hyperbolic Surfaces.","authors":"Asma Hassannezhad, Antoine Métras, Hélène Perrin","doi":"10.1007/s12220-025-01990-w","DOIUrl":"https://doi.org/10.1007/s12220-025-01990-w","url":null,"abstract":"<p><p>We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the corresponding Steklov eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 5","pages":"158"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971064/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143797170","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
On the Cheeger Inequality in Carnot-Carathéodory Spaces. carnot - carathacimodory空间中的Cheeger不等式。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-02-06 DOI: 10.1007/s12220-025-01912-w
Martijn Kluitenberg
{"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}
引用次数: 0
Lipschitz Stability of Travel Time Data. 旅行时间数据的Lipschitz稳定性。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-06-28 DOI: 10.1007/s12220-025-02084-3
Joonas Ilmavirta, Antti Kykkänen, Matti Lassas, Teemu Saksala, Andrew Shedlock
{"title":"Lipschitz Stability of Travel Time Data.","authors":"Joonas Ilmavirta, Antti Kykkänen, Matti Lassas, Teemu Saksala, Andrew Shedlock","doi":"10.1007/s12220-025-02084-3","DOIUrl":"10.1007/s12220-025-02084-3","url":null,"abstract":"<p><p>We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 8","pages":"244"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12206210/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144531301","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
On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian. 等周等径不等式及拉普拉斯函数特征值的最小化。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-01-04 DOI: 10.1007/s12220-024-01887-0
Sam Farrington
{"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}
引用次数: 0
Interpolating with generalized Assouad dimensions. 广义关联维插值。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-07-12 DOI: 10.1007/s12220-025-02099-w
Amlan Banaji, Alex Rutar, Sascha Troscheit
{"title":"Interpolating with generalized Assouad dimensions.","authors":"Amlan Banaji, Alex Rutar, Sascha Troscheit","doi":"10.1007/s12220-025-02099-w","DOIUrl":"10.1007/s12220-025-02099-w","url":null,"abstract":"<p><p>The <math><mi>ϕ</mi></math> -Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to \"phase-transition\" phenomena in sets. In this article we establish a number of key properties of the <math><mi>ϕ</mi></math> -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space <i>F</i> and <math><mrow><mi>α</mi> <mo>∈</mo> <mi>R</mi></mrow> </math> satisfying <math> <mrow> <msub><mover><mtext>dim</mtext> <mo>¯</mo></mover> <mtext>B</mtext></msub> <mi>F</mi> <mo><</mo> <mi>α</mi> <mo>≤</mo> <msub><mtext>dim</mtext> <mtext>A</mtext></msub> <mi>F</mi></mrow> </math> that there is a function <math><mi>ϕ</mi></math> so that the <math><mi>ϕ</mi></math> -Assouad dimension of <i>F</i> is equal to <math><mi>α</mi></math> . We further show that the \"upper\" variant of the dimension is fully determined by the <math><mi>ϕ</mi></math> -Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the <math><mi>ϕ</mi></math> -Assouad dimensions for the boundary of Galton-Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. The proof of this result combines a sharp large deviations theorem for Galton-Watson processes with bounded offspring distribution and a general Borel-Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the <math><mi>ϕ</mi></math> -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 9","pages":"270"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12255623/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144638777","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
Width Stability of Rotationally Symmetric Metrics. 旋转对称度量的宽度稳定性。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-06-24 DOI: 10.1007/s12220-025-02020-5
Hunter Stufflebeam, Paul Sweeney
{"title":"Width Stability of Rotationally Symmetric Metrics.","authors":"Hunter Stufflebeam, Paul Sweeney","doi":"10.1007/s12220-025-02020-5","DOIUrl":"10.1007/s12220-025-02020-5","url":null,"abstract":"<p><p>In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the validity of this conjecture under the additional assumption of rotational symmetry. Furthermore, we obtain a rigidity theorem in dimensions at least three for rotationally symmetric manifolds, which is analogous to the width rigidity theorem of Marques and Neves. We also prove a volume preserving intrinsic flat stability result for this rigidity theorem. Lastly, we study variants of Marques and Neves' stability conjecture. In the first, we show Gromov-Hausdorff convergence outside of certain \"bad\" sets. In the second, we assume non-negative Ricci curvature and show Gromov-Hausdorff stability.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 8","pages":"238"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12187838/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144509587","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
A Unifying Framework for Complex-Valued Eigenfunctions via The Cartan Embedding. 基于Cartan嵌入的复值特征函数统一框架。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-07-07 DOI: 10.1007/s12220-025-02090-5
Sigmundur Gudmundsson, Adam Lindström
{"title":"A Unifying Framework for Complex-Valued Eigenfunctions via The Cartan Embedding.","authors":"Sigmundur Gudmundsson, Adam Lindström","doi":"10.1007/s12220-025-02090-5","DOIUrl":"https://doi.org/10.1007/s12220-025-02090-5","url":null,"abstract":"<p><p>In this work we find a unifying scheme for the known explicit complex-valued eigenfunctions on the classical compact Riemannian symmetric spaces. For this we employ the well-known Cartan embedding for those spaces. This also leads to the construction of new eigenfunctions on the quaternionic Grassmannians.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"35 9","pages":"251"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12234617/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144602377","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
More Weakly Biharmonic Maps from the Ball to the Sphere. 从球到球的更多弱比谐映射
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2024-11-22 DOI: 10.1007/s12220-024-01852-x
Volker Branding
{"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}
引用次数: 0
On Isolated Singularities and Generic Regularity of Min-Max CMC Hypersurfaces. 最小-最大CMC超曲面的孤立奇异性和一般正则性。
IF 1.2 2区 数学
Journal of Geometric Analysis Pub Date : 2025-01-01 Epub Date: 2025-03-18 DOI: 10.1007/s12220-025-01956-y
Costante Bellettini, Kobe Marshall-Stevens
{"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}
引用次数: 0
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