Differential Geometry and its Applications最新文献

{"title":"A note on the lower bounds of the first nonzero Steklov eigenvalue on compact manifolds","authors":"Yiwei Liu,&nbsp;Yi-Hu Yang","doi":"10.1016/j.difgeo.2026.102429","DOIUrl":"10.1016/j.difgeo.2026.102429","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> be an <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional smooth compact connected Riemannian manifold with smooth boundary Σ, satisfying that <span><math><msub><mrow><mtext>Ric</mtext></mrow><mrow><mi>Ω</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span> and Σ is strictly convex, more precisely, its second fundamental form <span><math><mi>h</mi><mo>≥</mo><mi>c</mi><msub><mrow><mi>g</mi></mrow><mrow><mi>Σ</mi></mrow></msub></math></span> for some positive constant <em>c</em>. Escobar (1997) <span><span>[3]</span></span> considered the first nonzero Steklov eigenvalue <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of <span><math><mo>(</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> and proved that <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mi>c</mi></math></span> when <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> when <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>. He then conjectured (Escobar, 1999 <span><span>[4]</span></span>) that the first nonzero Steklov eigenvalue <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mi>c</mi></math></span>. Very recently, Xia and Xiong (2024) <span><span>[21]</span></span> confirmed Escobar's conjecture in the case that Ω has nonnegative sectional curvature, by constructing a weight function and using appropriate integral identities. In this paper, we construct a new weight function under certain sectional curvature assumptions and provide some new lower bounds for the first nonzero Steklov eigenvalue, which can be considered as generalizations of the results of Escobar and Xia-Xiong. As an application of the weight function, we also consider lower bound estimate of the first nonzero Steklov eigenvalue under conformal transformations.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"104 ","pages":"Article 102429"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148854510","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":"An improved upper bound for the second eigenvalue on tori","authors":"Fan Kang","doi":"10.1016/j.difgeo.2026.102428","DOIUrl":"10.1016/j.difgeo.2026.102428","url":null,"abstract":"<div><div>In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> on a torus <em>T</em>, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>,</mo><mi>g</mi><mo>)</mo></math></span> on any flat torus <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Our bound improves the general estimate <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>,</mo><mi>g</mi><mo>)</mo><mo>≤</mo><mn>4</mn><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>,</mo><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></math></span> obtained in <span><span>[4]</span></span>, <span><span>[13]</span></span> in the case of the torus. As applications, we derive a uniform upper bound <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>,</mo><mi>g</mi><mo>)</mo><mo>&lt;</mo><mfrac><mrow><mn>16</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></mfrac></math></span> for any torus <em>T</em> and any metric <em>g</em>, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>,</mo><mi>g</mi><mo>)</mo></math></span> on the specific family of flat tori <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <span><math><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>≤</mo><mi>b</mi><mo>≤</mo><mn>1.76</mn></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"104 ","pages":"Article 102428"},"PeriodicalIF":0.7,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148854509","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 flow method to isoperimetric inequality for mean convex star-shaped capillary hypersurfaces in a cone","authors":"Guanghan Li,&nbsp;Yifan Yang","doi":"10.1016/j.difgeo.2025.102329","DOIUrl":"10.1016/j.difgeo.2025.102329","url":null,"abstract":"<div><div>In this paper, the Minkowski formula and the Heintze-Karcher inequality are obtained for hypersurfaces with capillary boundary in a cone. Then we study a type of inverse mean curvature flow in a cone, as well as its long-time existence and convergence. As a result we derive the capillary isoperimetric inequality for mean convex star-shaped hypersurfaces with capillary boundary in a cone.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102329"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145957878","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":"Locally Levi-flat statistical submanifolds","authors":"Mirjana Milijević ,&nbsp;Sara Miri","doi":"10.1016/j.difgeo.2026.102345","DOIUrl":"10.1016/j.difgeo.2026.102345","url":null,"abstract":"<div><div>We define and study locally Levi-flat CR statistical submanifolds of maximal CR dimension within holomorphic statistical manifolds of constant holomorphic sectional curvature. Our definition generalizes the corresponding notion in Kähler geometry. Moreover, we establish a relationship between Levi-flatness and holomorphic sectional curvature. In particular, we prove that if a locally Levi-flat CR statistical submanifold admits a certain geometric configuration, namely, when a vector field derived from the complex structure is an eigenvector of both shape operators, then the ambient curvature must be strictly negative.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102345"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079471","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":"Z2-torus actions on positively curved manifolds","authors":"Farida Ghazawneh","doi":"10.1016/j.difgeo.2026.102332","DOIUrl":"10.1016/j.difgeo.2026.102332","url":null,"abstract":"<div><div>Kennard, Khalili Samani, and Searle showed that for a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-torus, <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, acting on a closed, positively curved Riemannian <em>n</em>-manifold, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, with a non-empty fixed point set for <em>n</em> large enough and <em>r</em> approximately half the dimension of <em>M</em>, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is homotopy equivalent to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>R</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>, or a lens space. In this paper, we lower <em>r</em> to approximately <span><math><mn>2</mn><mi>n</mi><mo>/</mo><mn>5</mn></math></span> and show that we still obtain the same result.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102332"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981244","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":"Cyclic Riemannian Lie groups: description and curvatures","authors":"Fatima-Ezzahrae Abid ,&nbsp;Saïd Benayadi ,&nbsp;Mohamed Boucetta ,&nbsp;Hamza El Ouali ,&nbsp;Hicham Lebzioui","doi":"10.1016/j.difgeo.2026.102392","DOIUrl":"10.1016/j.difgeo.2026.102392","url":null,"abstract":"<div><div>A cyclic Riemannian Lie group is a Lie group <em>G</em> equipped with a left-invariant Riemannian metric <em>h</em> that satisfies <span><math><msub><mrow><mo>∮</mo></mrow><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mi>h</mi><mo>(</mo><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo><mo>,</mo><mi>Z</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> for any left-invariant vector fields <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></math></span>. The initial concept and exploration of these Lie groups were presented in Gadea et al. (2015) <span><span>[4]</span></span>. This paper builds upon the results from the aforementioned study by providing a complete description of cyclic Riemannian Lie groups and an in-depth analysis of their various curvatures.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102392"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148183858","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":"Density-valued symplectic forms from a multisymplectic viewpoint","authors":"Laura Leski ,&nbsp;Leonid Ryvkin","doi":"10.1016/j.difgeo.2026.102334","DOIUrl":"10.1016/j.difgeo.2026.102334","url":null,"abstract":"<div><div>We give an intrinsic characterization of multisymplectic manifolds that have the linear type of density-valued symplectic forms in each tangent space, prove Darboux-type theorems for these forms, and investigate their symmetries.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102334"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981243","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":"On the local classification of four-dimensional Lorentzian real reductive pairs","authors":"Marco Castrillón López ,&nbsp;Pedro M. Gadea ,&nbsp;Boris P. Komrakov ,&nbsp;M. Eugenia Rosado María","doi":"10.1016/j.difgeo.2026.102333","DOIUrl":"10.1016/j.difgeo.2026.102333","url":null,"abstract":"<div><div>One important piece of work in the classifications started by the seminal works of S. Lie <span><span>[16]</span></span>, <span><span>[17]</span></span> is the classification of four-dimensional Lorentzian real reductive pairs. This classification appeared, except for one paper, as preprints of the University of Oslo, where moreover many proofs and implications are (necessarily, due to their length) greatly abridged.</div><div>Given the relevance of these classifications, we think that an article on the origin, context, methods and relevance of that classification is in order. This is precisely the aim of the present paper. We intend to fill the gaps in the exposition of the ideas that structure these proofs.</div><div>On the other hand, motivated by the physical applications, we studied in <span><span>[3]</span></span> which of Lorentzian symmetric pairs furnish connected simply-connected Einstein-Yang-Mills spaces, obtaining 10 spaces. Since the calculations are rather long (some one hundred fifty pages, only for these cases), we confine ourselves in the present paper to carefully check the arguments for those 10 cases.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102333"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981241","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 transverse density bundle and modular classes of Lie groupoids","authors":"Marius Crainic ,&nbsp;João Nuno Mestre","doi":"10.1016/j.difgeo.2026.102335","DOIUrl":"10.1016/j.difgeo.2026.102335","url":null,"abstract":"<div><div>In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>A</mi></mrow><mrow><mtext>tr</mtext></mrow></msubsup></math></span> instead of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>, which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is not really an object associated with the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).</div><div>We provide a simple construction of the representation of a Lie groupoid on its transverse volume, orientation, and density bundles in terms of (good) functors on vector spaces. We also extend the modular class by a Stiefel-Whitney class that controls the transverse orientability of a Lie groupoid</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102335"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981242","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 first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds","authors":"Fagui Li ,&nbsp;Junrong Yan","doi":"10.1016/j.difgeo.2026.102330","DOIUrl":"10.1016/j.difgeo.2026.102330","url":null,"abstract":"<div><div>Let Σ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold <em>N</em>. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of <em>N</em>, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on Σ. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of Σ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. <strong>18</strong> (1983), 559–562.] to the non-minimal case.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102330"},"PeriodicalIF":0.7,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976689","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}