Differential Geometry and its Applications最新文献

{"title":"The strong Diederich-Fornæss index on C2 domains in Hermitian manifolds","authors":"Phillip S. Harrington","doi":"10.1016/j.difgeo.2025.102251","DOIUrl":"10.1016/j.difgeo.2025.102251","url":null,"abstract":"<div><div>For a relatively compact Stein domain Ω with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> boundary in a Hermitian manifold <em>M</em>, we consider the strong Diederich-Fornæss index, denoted <span><math><mi>D</mi><mi>F</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>: the supremum of all exponents <span><math><mn>0</mn><mo>&lt;</mo><mi>η</mi><mo>&lt;</mo><mn>1</mn></math></span> such that eigenvalues of the complex Hessian of <span><math><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>ρ</mi><mo>)</mo></mrow><mrow><mi>η</mi></mrow></msup></math></span> are bounded below by some positive multiple of <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>ρ</mi><mo>)</mo></mrow><mrow><mi>η</mi></mrow></msup></math></span> on Ω for some <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> defining function <em>ρ</em>. We will show that <span><math><mi>D</mi><mi>F</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is completely characterized by the existence of a Hermitian metric with curvature terms satisfying a certain inequality when restricted to the null-space of the Levi-form.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102251"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869102","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":"Curvature pinching for three-dimensional submanifolds in a Riemannian manifold","authors":"Juanru Gu ,&nbsp;Yao Lu ,&nbsp;Hongwei Xu ,&nbsp;Entao Zhao","doi":"10.1016/j.difgeo.2025.102234","DOIUrl":"10.1016/j.difgeo.2025.102234","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> be an oriented submanifold with parallel mean curvature vector in a complete simply connected Riemannian manifold <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span>. When the mean curvature <span><math><mi>H</mi><mo>=</mo><mn>0</mn></math></span>, i.e., <em>M</em> is minimal, we prove that there exists a constant <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, such that if <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mo>∈</mo><mo>[</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, and if <em>M</em> has a lower bound for Ricci curvature and an upper bound for scalar curvature, then <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span> is isometric to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span>. Moreover, <em>M</em> is the totally geodesic sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This is a generalization of Shen and Li's results <span><span>[10]</span></span>, <span><span>[14]</span></span>. When the ambient manifold is a space form, we improve the geometric rigidity theorem due to Xu-Gu <span><span>[19]</span></span> for the codimension is not more than 2 and <span><math><mi>H</mi><mo>≠</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102234"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101469","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":"Connection blocking in quotients of Sol","authors":"Reza Bidar","doi":"10.1016/j.difgeo.2025.102241","DOIUrl":"10.1016/j.difgeo.2025.102241","url":null,"abstract":"<div><div>Let <em>G</em> be a connected Lie group and <span><math><mi>Γ</mi><mo>⊂</mo><mi>G</mi></math></span> a lattice. Connection curves of the homogeneous space <span><math><mi>M</mi><mo>=</mo><mi>G</mi><mo>/</mo><mi>Γ</mi></math></span> are the orbits of one parameter subgroups of <em>G</em>. To <em>block</em> a pair of points <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>M</mi></math></span> is to find a <em>finite</em> set <span><math><mi>B</mi><mo>⊂</mo><mi>M</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></math></span> such that every connecting curve joining <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> intersects <em>B</em>. The homogeneous space <em>M</em> is <em>blockable</em> if every pair of points in <em>M</em> can be blocked, otherwise we call it <em>non-blockable</em>.</div><div><em>Sol</em> is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, and together with the left invariant metric <span><math><mi>d</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>z</mi></mrow></msup><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>z</mi></mrow></msup><mi>d</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all lattice quotients of <em>Sol</em> are non-blockable. In particular, we show that for any lattice <span><math><mi>Γ</mi><mo>⊂</mo><mi>S</mi><mi>o</mi><mi>l</mi></math></span>, the set of non-blockable pairs is a dense subset of <span><math><mi>S</mi><mi>o</mi><mi>l</mi><mo>/</mo><mi>Γ</mi><mo>×</mo><mi>S</mi><mi>o</mi><mi>l</mi><mo>/</mo><mi>Γ</mi></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102241"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143580362","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 hard Lefschetz duality for locally conformally almost Kähler manifolds","authors":"Shuho Kanda","doi":"10.1016/j.difgeo.2025.102239","DOIUrl":"10.1016/j.difgeo.2025.102239","url":null,"abstract":"<div><div>We prove the hard Lefschetz duality for locally conformally almost Kähler manifolds. This is a generalization of that for almost Kähler manifolds studied by Cirici and Wilson. We generalize the Kähler identities to prove the duality. Based on the result, we introduce the hard Lefschetz condition for locally conformally symplectic manifolds. As examples, we give solvmanifolds which do not satisfy the hard Lefschetz condition.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102239"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constraint vector bundles and reduction of Lie (bi-)algebroids","authors":"Marvin Dippell ,&nbsp;David Kern","doi":"10.1016/j.difgeo.2025.102242","DOIUrl":"10.1016/j.difgeo.2025.102242","url":null,"abstract":"<div><div>We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102242"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New classes of Finsler metrics: The birth of new projective invariant","authors":"Nasrin Sadeghzadeh","doi":"10.1016/j.difgeo.2025.102250","DOIUrl":"10.1016/j.difgeo.2025.102250","url":null,"abstract":"<div><div>This paper presents a pioneering projective invariant in Finsler geometry, introducing a new class of Finsler metrics that are preserved under projective transformations. The newly formulated weakly generalized Douglas-Weyl <span><math><mo>(</mo><mi>W</mi><mo>−</mo><mi>G</mi><mi>D</mi><mi>W</mi><mo>)</mo></math></span> equation facilitates the generalization of generalized Douglas-Weyl <span><math><mo>(</mo><mi>G</mi><mi>D</mi><mi>W</mi><mo>)</mo></math></span>-metrics into the broader category of <span><math><mi>W</mi><mo>−</mo><mi>G</mi><mi>D</mi><mi>W</mi></math></span>-metrics, which encompasses all <em>GDW</em>-metrics. Within this class, there are also two additional subclasses: generalized weakly-Weyl metrics, characterized by a milder form of Weyl curvature, and generalized <span><math><mover><mrow><mi>D</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>-metrics, defined by a less strict version of Douglas curvature. The paper provides a comprehensive overview of these generalized class of Finsler metrics and elucidates their properties, supported by detailed examples.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102250"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825585","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":"Chern-Simons-Higgs type equations on canonically compactifiable graphs","authors":"Longsong Jia ,&nbsp;Chang Li ,&nbsp;Yanlin Li ,&nbsp;Bin Wang","doi":"10.1016/j.difgeo.2025.102237","DOIUrl":"10.1016/j.difgeo.2025.102237","url":null,"abstract":"<div><div>In this paper, we prove existence results of solutions to three kinds of Chern-Simons-Higgs type equations, including mean field equations and Chern-Simons-Higgs equations as well as the generalized Chern-Simons-Higgs equations on canonically compactifiable graphs, which is a special infinite graphs giving inclusive relationship between Banach spaces on graphs. The paper mainly employs variational principles in Banach spaces as well as upper and lower solutions method, with the main challenge being the lack of finite bound of number of vertices and other certain properties, leading to difficulties of estimates of bound for functionals. We choose suitable restrict spaces in Lagrange multiplier theorem and use Moser-Trudinger inequalities to overcome these difficulties.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102237"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143528711","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":"Mean curvature flow with pinched curvature integral","authors":"Yongheng Han","doi":"10.1016/j.difgeo.2025.102244","DOIUrl":"10.1016/j.difgeo.2025.102244","url":null,"abstract":"<div><div>If Σ is an <em>n</em>-dimensional noncompact self-shrinker and the second fundamental form of Σ is <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> integrable for <span><math><mi>p</mi><mo>≥</mo><mi>n</mi></math></span>, we show that Σ is asymptotic to a regular cone. We also prove long-time existence of the mean curvature flow starting from complete manifolds with bounded curvature and small total curvature.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102244"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679853","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 manifold of polygons degenerated to segments","authors":"Manuel A. Espinosa-García ,&nbsp;Ahtziri González ,&nbsp;Yesenia Villicaña-Molina","doi":"10.1016/j.difgeo.2025.102247","DOIUrl":"10.1016/j.difgeo.2025.102247","url":null,"abstract":"<div><div>In this paper we study the space <span><math><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of <em>n</em>-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and describe its topology in terms of the manifold <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of <em>n</em>-gons degenerated to segments and with the first vertex at 0. We show that <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of <span><math><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by the diagonal action of the affine complex group and the re-enumeration of the vertices is described.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102247"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deformation rigidity of the double Cayley Grassmannian","authors":"Shin-young Kim ,&nbsp;Kyeong-Dong Park","doi":"10.1016/j.difgeo.2024.102219","DOIUrl":"10.1016/j.difgeo.2024.102219","url":null,"abstract":"<div><div>The double Cayley Grassmannian is a unique smooth equivariant completion with Picard number one of the 14-dimensional exceptional complex Lie group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and it parametrizes eight-dimensional isotropic subalgebras of the complexified bi-octonions. We show the rigidity of the double Cayley Grassmannian under Kähler deformations. This means that for any smooth projective family of complex manifolds over a connected base of which one fiber is biholomorphic to the double Cayley Grassmannian, all other fibers are biholomorphic to the double Cayley Grassmannian.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102219"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101467","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}