Pub Date : 2026-01-09DOI: 10.1016/j.difgeo.2026.102331
Xian-Dong Sun
Recently, Brendle proved an isoperimetric inequality for a compact n-dimensional submanifold M of with boundary ∂M[1]. In this short note, we give a better constant when . In particular, our inequality shows that Brendle's inequality is never optimal when .
{"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":"2026-01-09","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}
Pub Date : 2025-12-22DOI: 10.1016/j.difgeo.2025.102324
Rankin Shane
We explore the natural analogues of the Brylinski condition, strong Lefschetz condition, and dδ-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 dδ-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.
{"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":"2025-12-22","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}
Pub Date : 2025-12-22DOI: 10.1016/j.difgeo.2025.102323
Jihyeon Lee , Sanghun Lee
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.
{"title":"Rigidity of three-manifolds with boundary via modified Hawking mass","authors":"Jihyeon Lee , 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":"2025-12-22","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}
Pub Date : 2025-12-22DOI: 10.1016/j.difgeo.2025.102325
Santiago R. Simanca
<div><div>If <em>M</em> is a closed manifold, and <em>K</em> is a smooth triangulation of <em>M</em>, Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> assigning the value 1 mod 2 to each dual cell. We provide the pair <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span> with an arbitrary Riemannian metric <em>g</em>, and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span>. The representative of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is determined by <span><math><mi>det</mi><mo></mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>, the <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>s computed in a frame that is locally defined at each dual 1-cell; the representatives of the even classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> are determined by the Chern-Gauss-Bonnet density 2<em>k</em>-form of locally defined totally geodesic oriented 2<em>k</em> manifolds with boundary associated to each dual 2<em>k</em>-cell; and the representatives of the odd classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented <span><math><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> manifold with boundary associated to each dual <span><math><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-cell. If <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>J</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> is Hermitian, we prove that the metric representative of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> so obtained is the <span><math><mi>Z</mi><mo>/</mo><mn>2</mn></math></span> reduction of the <em>k</em>th Chern class <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>J</mi><mo>)</mo></math></span> induced by the coefficient homomorphism, and that the metric representative of any odd degree class <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></m
如果M是一个封闭流形，K是M的一个光滑三角剖分，Whitney证明了所有的Stiefel-Whitney类都被指定为对偶单元复合体（K '）上的协链，对每个对偶单元赋值1 mod 2。我们为（M，K）对提供一个任意的黎曼度量g，并使用惠特尼准则来证明存在所有Stiefel-Whitney类w1(M)，…，wn(M)的关联代表。w1(M)的代表由det （gij）决定，gij是在每个双1单元局部定义的帧中计算的；偶类w2k(M)的代表由局部定义的完全测地取向2k流形的chen - gauss - bonnet密度2k形式确定，其边界与每个对偶2k单元相关联；奇数类w2k+1(M)的代表由一个局部定义的完全测地取向（2k+1）流形的边界球的超表面积形式决定，其边界与每个对偶(2k+1)-单元相关联。如果（M，J,g）是厄密的，我们证明了由此得到的w2k(M)的度规代表是由系数同态导出的第k个陈氏类ck（M，J）的Z/2约简，并且由此得到的任何奇次类w2k+1(M)的度规代表在上同调上是平凡的。
{"title":"Riemannian metric representatives of the Stiefel-Whitney classes","authors":"Santiago R. Simanca","doi":"10.1016/j.difgeo.2025.102325","DOIUrl":"10.1016/j.difgeo.2025.102325","url":null,"abstract":"<div><div>If <em>M</em> is a closed manifold, and <em>K</em> is a smooth triangulation of <em>M</em>, Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> assigning the value 1 mod 2 to each dual cell. We provide the pair <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span> with an arbitrary Riemannian metric <em>g</em>, and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span>. The representative of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is determined by <span><math><mi>det</mi><mo></mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>, the <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>s computed in a frame that is locally defined at each dual 1-cell; the representatives of the even classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> are determined by the Chern-Gauss-Bonnet density 2<em>k</em>-form of locally defined totally geodesic oriented 2<em>k</em> manifolds with boundary associated to each dual 2<em>k</em>-cell; and the representatives of the odd classes <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented <span><math><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> manifold with boundary associated to each dual <span><math><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-cell. If <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>J</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> is Hermitian, we prove that the metric representative of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> so obtained is the <span><math><mi>Z</mi><mo>/</mo><mn>2</mn></math></span> reduction of the <em>k</em>th Chern class <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>J</mi><mo>)</mo></math></span> induced by the coefficient homomorphism, and that the metric representative of any odd degree class <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></m","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102325"},"PeriodicalIF":0.7,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840301","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}
Pub Date : 2025-12-09DOI: 10.1016/j.difgeo.2025.102321
Danuzia Nascimento Figueirêdo, Mathieu Molitor
We characterize a class of 1-dimensional dually flat manifolds that are toric in the sense of [21]. 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.
{"title":"Toric dually flat manifolds and complex space forms","authors":"Danuzia Nascimento Figueirêdo, 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":"2025-12-09","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}
Pub Date : 2025-12-08DOI: 10.1016/j.difgeo.2025.102322
Hongmei Zhu , Lumin Song
In this paper, we study two different kinds of scalar curvature in Finsler geometry, which are defined by two different kinds of Ricci curvature tensor. For Kropina metrics, we show that they are equivalent among weakly isotropic scalar curvature, isotropic scalar curvature, weakly isotropic Ricci curvature tensor, weakly isotropic Ricci curvature, isotropic Ricci curvature tensor and isotropic Ricci curvature.
{"title":"On the scalar curvature of Kropina metrics II","authors":"Hongmei Zhu , Lumin Song","doi":"10.1016/j.difgeo.2025.102322","DOIUrl":"10.1016/j.difgeo.2025.102322","url":null,"abstract":"<div><div>In this paper, we study two different kinds of scalar curvature in Finsler geometry, which are defined by two different kinds of Ricci curvature tensor. For Kropina metrics, we show that they are equivalent among weakly isotropic scalar curvature, isotropic scalar curvature, weakly isotropic Ricci curvature tensor, weakly isotropic Ricci curvature, isotropic Ricci curvature tensor and isotropic Ricci curvature.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102322"},"PeriodicalIF":0.7,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748054","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}
Pub Date : 2025-12-04DOI: 10.1016/j.difgeo.2025.102320
Hasan İnci
<div><div>In this paper we consider the “symplectic” version of the Euler equations studied by Ebin <span><span>[7]</span></span>. We show that these equations are globally well-posed on the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>></mo><mn>2</mn><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></math></span>. The mechanism underlying global well-posedness has similarities to the case of the 2D Euler equations. Moreover we consider the group of symplectomorphisms <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> of Sobolev type <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> preserving the symplectic form <span><math><mi>ω</mi><mo>=</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∧</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∧</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>. We show that <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> is a closed analytic submanifold of the full group <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> of diffeomorphisms of Sobolev type <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> preserving the orientation. We prove that the symplectic version of the Euler equations has a Lagrangian formulation on <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> as an analytic second order ODE in the manner of the Euler-Arnold formalism <span><span>[1]</span></span>. 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 <span><math><mi>T</mi><mo>></mo><mn>0</mn></math></span> solution map <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>↦</mo><mi>u</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> mapping the initial value of the solution to its time <em>T</em> value is nowhere locally uniformly continuous. In particular the solution map is nowhere locally Li
{"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":"<div><div>In this paper we consider the “symplectic” version of the Euler equations studied by Ebin <span><span>[7]</span></span>. We show that these equations are globally well-posed on the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>></mo><mn>2</mn><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></math></span>. The mechanism underlying global well-posedness has similarities to the case of the 2D Euler equations. Moreover we consider the group of symplectomorphisms <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> of Sobolev type <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> preserving the symplectic form <span><math><mi>ω</mi><mo>=</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∧</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∧</mo><mi>d</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>. We show that <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> is a closed analytic submanifold of the full group <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> of diffeomorphisms of Sobolev type <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> preserving the orientation. We prove that the symplectic version of the Euler equations has a Lagrangian formulation on <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>)</mo></math></span> as an analytic second order ODE in the manner of the Euler-Arnold formalism <span><span>[1]</span></span>. 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 <span><math><mi>T</mi><mo>></mo><mn>0</mn></math></span> solution map <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>↦</mo><mi>u</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> mapping the initial value of the solution to its time <em>T</em> 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":"2025-12-04","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":"Classification of conformal minimal immersions from S2 to G(2,N;C) with parallel second fundamental form","authors":"Xiaoxiang Jiao , Mingyan Li","doi":"10.1016/j.difgeo.2025.102312","DOIUrl":"10.1016/j.difgeo.2025.102312","url":null,"abstract":"<div><div>In this paper, we determine all conformal minimal immersions of 2-spheres in complex Grassmann manifold <span><math><mi>G</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>N</mi><mo>;</mo><mi>C</mi><mo>)</mo></math></span> with parallel second fundamental form.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102312"},"PeriodicalIF":0.7,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623137","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}
Pub Date : 2025-11-28DOI: 10.1016/j.difgeo.2025.102313
Antonio Maglio , Fabricio Valencia
We define the notion of basic section of an -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 -groupoids, yet equivalent to the well-known model provided by their multiplicative sections.
{"title":"Basic sections of LA-groupoids","authors":"Antonio Maglio , 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":"2025-11-28","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}
Pub Date : 2025-11-26DOI: 10.1016/j.difgeo.2025.102309
Yann Guggisberg , Fabian Ziltener
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.
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.
{"title":"Examples of non-target-representable symplectic capacities","authors":"Yann Guggisberg , 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":"2025-11-26","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}
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