{"title":"A remark on deformation of Gromov non-squeezing","authors":"Yasha Savelyev","doi":"10.1016/j.difgeo.2025.102262","DOIUrl":"10.1016/j.difgeo.2025.102262","url":null,"abstract":"<div><div>Let <span><math><mi>R</mi><mo>,</mo><mi>r</mi></math></span> be as in the classical Gromov non-squeezing theorem, and let <span><math><mi>ϵ</mi><mo>=</mo><mo>(</mo><mi>π</mi><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>π</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>/</mo><mi>π</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> (w.r.t. the standard metric) <em>ϵ</em>-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when <span><math><mi>R</mi><mo><</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mi>r</mi></math></span>. Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102262"},"PeriodicalIF":0.6,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190117","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":"Generalized Alexandrov theorems in spacetimes with integral conditions","authors":"Kwok-Kun Kwong , Xianfeng Wang","doi":"10.1016/j.difgeo.2025.102254","DOIUrl":"10.1016/j.difgeo.2025.102254","url":null,"abstract":"<div><div>We investigate integral conditions involving the mean curvature vector <span><math><mover><mrow><mi>H</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span> or mixed higher-order mean curvatures, to determine when a codimension-two submanifold Σ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in <span><span>[18]</span></span> by relaxing the curvature conditions on Σ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for Σ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102254"},"PeriodicalIF":0.6,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143941901","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":"Time in classical and quantum mechanics","authors":"J. Muñoz-Díaz, R.J. Alonso-Blanco","doi":"10.1016/j.difgeo.2025.102253","DOIUrl":"10.1016/j.difgeo.2025.102253","url":null,"abstract":"<div><div>In this article we study the nature of time in Mechanics. The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation, implies the existence of an <em>absolute time-duration</em> in the sense of Newton. There is a second notion of time for conservative systems which makes the Hamiltonian action evolves at a constant rate. In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory. Then, we propose two different ways to reach the time dependent Schrödinger equation. One way consists of considering a “time constraint” on a free system. The other way is based on the point of view of Hertz, by considering the system as a projection of a free system. In the later manner, the “time” appearing in the Schrödinger equation is a linear combination of the time-duration with the “time” quotient of the action by the energy on each solution of the Hamilton-Jacobi equation. Both of them are based on a rule of quantization that we explain in Section 4.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102253"},"PeriodicalIF":0.6,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143941897","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 closed minimal hypersurfaces in S5","authors":"Pengpeng Cheng, Tongzhu Li","doi":"10.1016/j.difgeo.2025.102252","DOIUrl":"10.1016/j.difgeo.2025.102252","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> be a closed immersed minimal hypersurface with constant squared length of the second fundamental form <em>S</em> in a 5-dimensional sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>. In this paper, we prove that if the 3-mean curvature <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and the number <em>g</em> of the distinct principal curvatures are constant, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> is an isoparametric hypersurface, and the value of <em>S</em> can only be <span><math><mn>0</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>12</mn></math></span>. This result supports Chern Conjecture.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102252"},"PeriodicalIF":0.6,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922663","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 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><</mo><mi>η</mi><mo><</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}
{"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 & 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}
{"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}
{"title":"On the integration of Manin pairs","authors":"David Li-Bland, 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}
{"title":"Bour's theorem for helicoidal surfaces with singularities","authors":"Yuki Hattori , Atsufumi Honda , 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}
Manuel A. Espinosa-García , Ahtziri González , Yesenia Villicaña-Molina
{"title":"The manifold of polygons degenerated to segments","authors":"Manuel A. Espinosa-García , Ahtziri González , 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}