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The strong Diederich-Fornæss index on C2 domains in Hermitian manifolds 厄米流形中C2域上的强Diederich-Fornæss指标
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-24 DOI: 10.1016/j.difgeo.2025.102251
Phillip S. Harrington
{"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":"2025-04-24","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}
引用次数: 0
The geometric Toda equations for noncompact symmetric spaces 非紧致对称空间的几何Toda方程
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-16 DOI: 10.1016/j.difgeo.2025.102249
Ian McIntosh
{"title":"The geometric Toda equations for noncompact symmetric spaces","authors":"Ian McIntosh","doi":"10.1016/j.difgeo.2025.102249","DOIUrl":"10.1016/j.difgeo.2025.102249","url":null,"abstract":"<div><div>This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of <em>τ</em>-primitive harmonic maps from a surface into a homogeneous space <span><math><mi>G</mi><mo>/</mo><mi>T</mi></math></span> for which <em>G</em> is a noncomplex noncompact simple real Lie group, <em>τ</em> is the Coxeter automorphism which Drinfel'd &amp; Sokolov assigned to each affine Dynkin diagram, and <em>T</em> is the compact torus fixed pointwise by <em>τ</em>. Here <em>τ</em> may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface Σ as equations for a metric on a holomorphic principal <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span>-bundle <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> over Σ whose Chern connection, when combined with a holomorphic field <em>φ</em>, produces a <em>G</em>-connection which is flat precisely when the Toda equations hold. The second purpose is to establish when stability criteria for the pair <span><math><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> can be used to prove the existence of solutions. We classify those real forms of the Toda equations for which this pair is a principal pair and we call these <em>totally noncompact</em> Toda pairs: stability theory then gives algebraic conditions for the existence of solutions. Every solution to the geometric Toda equations has a corresponding <em>G</em>-Higgs bundle. We explain how to construct this <em>G</em>-Higgs bundle directly from the Toda pair and show that Baraglia's cyclic Higgs bundles arise from a very special case of totally noncompact cyclic Toda pairs.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102249"},"PeriodicalIF":0.6,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835095","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}
引用次数: 0
New classes of Finsler metrics: The birth of new projective invariant 芬斯勒度量的新类别：新射影不变量的诞生
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-14 DOI: 10.1016/j.difgeo.2025.102250
Nasrin Sadeghzadeh
{"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":"2025-04-14","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}
引用次数: 0
On the integration of Manin pairs
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-04 DOI: 10.1016/j.difgeo.2025.102246
David Li-Bland, Eckhard Meinrenken
{"title":"On the integration of Manin pairs","authors":"David Li-Bland,&nbsp;Eckhard Meinrenken","doi":"10.1016/j.difgeo.2025.102246","DOIUrl":"10.1016/j.difgeo.2025.102246","url":null,"abstract":"<div><div>It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid <em>A</em>: The source-simply connected Lie groupoid <span><math><mi>G</mi><mo>⇉</mo><mi>M</mi></math></span> integrating <em>A</em> automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the ‘quasi’ settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs <span><math><mo>(</mo><mi>E</mi><mo>,</mo><mi>A</mi><mo>)</mo></math></span>, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid <em>G</em> integrating <em>A</em> is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102246"},"PeriodicalIF":0.6,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768267","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}
引用次数: 0
Bour's theorem for helicoidal surfaces with singularities
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-03 DOI: 10.1016/j.difgeo.2025.102248
Yuki Hattori , Atsufumi Honda , Tatsuya Morimoto
{"title":"Bour's theorem for helicoidal surfaces with singularities","authors":"Yuki Hattori ,&nbsp;Atsufumi Honda ,&nbsp;Tatsuya Morimoto","doi":"10.1016/j.difgeo.2025.102248","DOIUrl":"10.1016/j.difgeo.2025.102248","url":null,"abstract":"<div><div>In this paper, by generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge and, more generally, every generic <em>n</em>-type edge, which is invariant under a helicoidal motion in Euclidean 3-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102248"},"PeriodicalIF":0.6,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759123","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}
引用次数: 0
The manifold of polygons degenerated to segments
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-04-01 DOI: 10.1016/j.difgeo.2025.102247
Manuel A. Espinosa-García , Ahtziri González , Yesenia Villicaña-Molina
{"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":"2025-04-01","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}
引用次数: 0
Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-03-27 DOI: 10.1016/j.difgeo.2025.102245
Tomasz Goliński , Fernand Pelletier
{"title":"Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure","authors":"Tomasz Goliński ,&nbsp;Fernand Pelletier","doi":"10.1016/j.difgeo.2025.102245","DOIUrl":"10.1016/j.difgeo.2025.102245","url":null,"abstract":"<div><div>We consider regulated curves in a Banach bundle whose projection on the basis is continuous with regulated derivative. We build a Banach manifold structure on the set of such curves. This result was previously obtained for the case of strong Riemannian Banach manifold and absolutely continuous curves in <span><span>[16]</span></span>. The essential argument used was the existence of a “local addition” on such a manifold. Our proof is true for any Banach manifold. In the second part of the paper the problems of controllability will be discussed.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102245"},"PeriodicalIF":0.6,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143706392","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}
引用次数: 0
Solitons of the mean curvature flow in S2×R
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-03-25 DOI: 10.1016/j.difgeo.2025.102243
Rafael López , Marian Ioan Munteanu
{"title":"Solitons of the mean curvature flow in S2×R","authors":"Rafael López ,&nbsp;Marian Ioan Munteanu","doi":"10.1016/j.difgeo.2025.102243","DOIUrl":"10.1016/j.difgeo.2025.102243","url":null,"abstract":"<div><div>A soliton of the mean curvature flow in the product space <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span> is a surface whose mean curvature <em>H</em> satisfies the equation <span><math><mi>H</mi><mo>=</mo><mo>〈</mo><mi>N</mi><mo>,</mo><mi>X</mi><mo>〉</mo></math></span>, where <em>N</em> is the unit normal of the surface and <em>X</em> is a Killing vector field of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span>. In this paper we consider the cases that <em>X</em> is the vector field tangent to the second factor and the vector field associated to rotations about an axis of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or rotations of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102243"},"PeriodicalIF":0.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696897","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}
引用次数: 0
Mean curvature flow with pinched curvature integral
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-03-21 DOI: 10.1016/j.difgeo.2025.102244
Yongheng Han
{"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":"2025-03-21","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}
引用次数: 0
Constraint vector bundles and reduction of Lie (bi-)algebroids
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-03-12 DOI: 10.1016/j.difgeo.2025.102242
Marvin Dippell , David Kern
{"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":"2025-03-12","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}
引用次数: 0
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