Differential Geometry and its Applications最新文献

{"title":"Natural parallel translation and connection associated to navigation data","authors":"A. Mezrag,&nbsp;Z. Muzsnay,&nbsp;Cs. Vincze","doi":"10.1016/j.difgeo.2025.102328","DOIUrl":"10.1016/j.difgeo.2025.102328","url":null,"abstract":"<div><div>In this paper, we consider the geometric setting of navigation data and introduce a natural parallel translation using the Riemannian parallelism. The geometry obtained in this way has some nice and natural features: the natural parallel translation is homogeneous (but in general nonlinear), preserves the Randers type Finslerian norm constituted by the navigation data, and the holonomy group is finite-dimensional.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102328"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symplectic Hodge theory on Lie algebroids","authors":"Rankin Shane","doi":"10.1016/j.difgeo.2025.102324","DOIUrl":"10.1016/j.difgeo.2025.102324","url":null,"abstract":"<div><div>We explore the natural analogues of the Brylinski condition, strong Lefschetz condition, and <em>dδ</em>-lemma in symplectic geometry originally explored by Brylinski, Mathieu, Yan, and Guillemin to the symplectic Lie algebroid case. The equivalence of the three conditions is re-established as a purely algebraic statement along with a primitive notion of the <em>dδ</em>-lemma shown established by Tseng, Yau, and Ho. We then show that the natural analogues of these in the Lie algebroid setting holds as well with examples given.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102324"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The group of symplectomorphisms of R2n and the Euler equations","authors":"Hasan İnci","doi":"10.1016/j.difgeo.2025.102320","DOIUrl":"10.1016/j.difgeo.2025.102320","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper we consider the “symplectic” version of the Euler equations studied by Ebin &lt;span&gt;&lt;span&gt;[7]&lt;/span&gt;&lt;/span&gt;. We show that these equations are globally well-posed on the Sobolev space &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. The mechanism underlying global well-posedness has similarities to the case of the 2D Euler equations. Moreover we consider the group of symplectomorphisms &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of Sobolev type &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; preserving the symplectic form &lt;span&gt;&lt;math&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. We show that &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a closed analytic submanifold of the full group &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of diffeomorphisms of Sobolev type &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; preserving the orientation. We prove that the symplectic version of the Euler equations has a Lagrangian formulation on &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; as an analytic second order ODE in the manner of the Euler-Arnold formalism &lt;span&gt;&lt;span&gt;[1]&lt;/span&gt;&lt;/span&gt;. In contrast to this “smooth” behavior in Lagrangian coordinates we show that it has a very “rough” behavior in Eulerian coordinates. To be precise we show that the time &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; solution map &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; mapping the initial value of the solution to its time &lt;em&gt;T&lt;/em&gt; value is nowhere locally uniformly continuous. In particular the solution map is nowhere locally Li","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102320"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Basic sections of LA-groupoids","authors":"Antonio Maglio ,&nbsp;Fabricio Valencia","doi":"10.1016/j.difgeo.2025.102313","DOIUrl":"10.1016/j.difgeo.2025.102313","url":null,"abstract":"<div><div>We define the notion of basic section of an <span><math><mi>LA</mi></math></span>-groupoid whose core-anchor map is injective. Such a notion turns out to be Morita invariant, so that it provides a simpler model for the sections of the stacky Lie algebroids presented by such <span><math><mi>LA</mi></math></span>-groupoids, yet equivalent to the well-known model provided by their multiplicative sections.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102313"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Toric dually flat manifolds and complex space forms","authors":"Danuzia Nascimento Figueirêdo,&nbsp;Mathieu Molitor","doi":"10.1016/j.difgeo.2025.102321","DOIUrl":"10.1016/j.difgeo.2025.102321","url":null,"abstract":"<div><div>We characterize a class of 1-dimensional dually flat manifolds that are toric in the sense of <span><span>[21]</span></span>. This class consists of manifolds whose torifications possess a high degree of symmetry and satisfy specific geometric and analytic properties common in information geometry. We show that these torifications correspond precisely to the complex space forms. The case where the manifold is an exponential family defined over a finite set is considered.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102321"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Examples of non-target-representable symplectic capacities","authors":"Yann Guggisberg ,&nbsp;Fabian Ziltener","doi":"10.1016/j.difgeo.2025.102309","DOIUrl":"10.1016/j.difgeo.2025.102309","url":null,"abstract":"<div><div>We give the first concrete example of a symplectic capacity that is not target-representable. The capacity is defined on the class of all compact and exact symplectic manifolds. This provides some concrete answer to a question by Cieliebak, Hofer, Latschev, and Schlenk.</div><div>We also show that a capacity on the class of all closed symplectic manifolds is not target-representable, if it is finite at certain symplectic manifolds. In particular, the Hofer-Zehnder capacity on the class of all closed symplectic manifolds is not target-representable.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102309"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145594846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on isoperimetric inequality for a minimal submanifold in Euclidean space","authors":"Xian-Dong Sun","doi":"10.1016/j.difgeo.2026.102331","DOIUrl":"10.1016/j.difgeo.2026.102331","url":null,"abstract":"<div><div>Recently, Brendle proved an isoperimetric inequality for a compact <em>n</em>-dimensional submanifold <em>M</em> of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msup></math></span> with boundary ∂<em>M</em> <span><span>[1]</span></span>. In this short note, we give a better constant when <span><math><mi>m</mi><mo>⩾</mo><mn>3</mn></math></span>. In particular, our inequality shows that Brendle's inequality is never optimal when <span><math><mi>m</mi><mo>⩾</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102331"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145938683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant meridians and parallels on the hyperboloid of one sheet","authors":"Jaume Llibre ,&nbsp;Adrian C. Murza","doi":"10.1016/j.difgeo.2025.102326","DOIUrl":"10.1016/j.difgeo.2025.102326","url":null,"abstract":"<div><div>We analyze the polynomial vector fields in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> that have the hyperboloid of one sheet as an invariant surface. We characterize the maximum number of invariant meridians and parallels of this hyperboloid that such polynomial vector fields can have, as a function of the degree of the polynomial vector field.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102326"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rigidity of three-manifolds with boundary via modified Hawking mass","authors":"Jihyeon Lee ,&nbsp;Sanghun Lee","doi":"10.1016/j.difgeo.2025.102323","DOIUrl":"10.1016/j.difgeo.2025.102323","url":null,"abstract":"<div><div>In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary strictly stable minimal surface, which locally maximizes the modified Hawking mass, is embedded in a three-dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable minimal surfaces. Then, we show a rigidity theorem for a three-dimensional Riemannian manifold with boundary. In particular, if a free boundary surface is minimal two-disk, then the three-dimensional Riemannian manifold is locally isometric to the half anti-de Sitter-Schwarzschild manifold.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102323"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence proofs for rotationally symmetric translating solutions to mean curvature flow","authors":"Hakar Raji,&nbsp;Oliver C. Schnürer","doi":"10.1016/j.difgeo.2025.102327","DOIUrl":"10.1016/j.difgeo.2025.102327","url":null,"abstract":"<div><div>There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102327"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}