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Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature bound
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-29 DOI: 10.1016/j.difgeo.2024.102218
Maria Gordina , Gunhee Cho
{"title":"Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature bound","authors":"Maria Gordina ,&nbsp;Gunhee Cho","doi":"10.1016/j.difgeo.2024.102218","DOIUrl":"10.1016/j.difgeo.2024.102218","url":null,"abstract":"<div><div>We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102218"},"PeriodicalIF":0.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746975","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
Integral Ricci curvature bounds for possibly collapsed spaces with Ricci curvature bounded from below 里奇曲率自下而上有界的可能坍缩空间的里奇曲率积分界值
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-21 DOI: 10.1016/j.difgeo.2024.102214
Michael Smith
{"title":"Integral Ricci curvature bounds for possibly collapsed spaces with Ricci curvature bounded from below","authors":"Michael Smith","doi":"10.1016/j.difgeo.2024.102214","DOIUrl":"10.1016/j.difgeo.2024.102214","url":null,"abstract":"<div><div>Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for <span><math><mi>q</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span> we show the existence of bounds on the local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102214"},"PeriodicalIF":0.6,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723027","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
Conformal surface splines 共形表面花键
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-21 DOI: 10.1016/j.difgeo.2024.102200
Yousuf Soliman , Ulrich Pinkall , Peter Schröder
{"title":"Conformal surface splines","authors":"Yousuf Soliman ,&nbsp;Ulrich Pinkall ,&nbsp;Peter Schröder","doi":"10.1016/j.difgeo.2024.102200","DOIUrl":"10.1016/j.difgeo.2024.102200","url":null,"abstract":"<div><div>We introduce a family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. Our free boundary conditions fix the metric on the boundary, up to a global scale, and admit a discretization compatible with discrete conformal equivalence. We also introduce constraints on the conformal scale factor, enforcing rigidity of the geometry in regions of interest, and describe how in the presence of point constraints the conformal class encodes knot points of the spline that can be directly manipulated. To control the tangent planes, we introduce flux constraints balancing the internal material stresses. The collection of these point constraints provide intuitive controls for exploring a subspace of conformal immersions interpolating a fixed set of points in space. We demonstrate the applicability of our framework to geometric modeling, mathematical visualization, and form finding.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102200"},"PeriodicalIF":0.6,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706899","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
Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle 具有微小对数切线束的复流形上的对数卡坦几何
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-19 DOI: 10.1016/j.difgeo.2024.102213
Indranil Biswas , Sorin Dumitrescu , Archana S. Morye
{"title":"Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle","authors":"Indranil Biswas ,&nbsp;Sorin Dumitrescu ,&nbsp;Archana S. Morye","doi":"10.1016/j.difgeo.2024.102213","DOIUrl":"10.1016/j.difgeo.2024.102213","url":null,"abstract":"<div><div>Let <em>M</em> be a compact complex manifold, and <span><math><mi>D</mi><mspace></mspace><mo>⊂</mo><mspace></mspace><mi>M</mi></math></span> a reduced normal crossing divisor on it, such that the logarithmic tangent bundle <span><math><mi>T</mi><mi>M</mi><mo>(</mo><mo>−</mo><mi>log</mi><mo>⁡</mo><mi>D</mi><mo>)</mo></math></span> is holomorphically trivial. Let <span><math><mi>A</mi></math></span> denote the maximal connected subgroup of the group of all holomorphic automorphisms of <em>M</em> that preserve the divisor <em>D</em>. Take a holomorphic Cartan geometry <span><math><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>Θ</mi><mo>)</mo></math></span> of type <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mspace></mspace><mi>H</mi><mo>)</mo></math></span> on <em>M</em>, where <span><math><mi>H</mi><mspace></mspace><mo>⊂</mo><mspace></mspace><mi>G</mi></math></span> are complex Lie groups. We prove that <span><math><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>Θ</mi><mo>)</mo></math></span> is isomorphic to <span><math><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><msub><mrow><mi>E</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>ρ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>Θ</mi><mo>)</mo></math></span> for every <span><math><mi>ρ</mi><mspace></mspace><mo>∈</mo><mspace></mspace><mi>A</mi></math></span> if and only if the principal <em>H</em>–bundle <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> admits a logarithmic connection Δ singular on <em>D</em> such that Θ is preserved by the connection Δ.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102213"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706900","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
A characterization of parallel surfaces in Minkowski space via minimal and maximal surfaces 通过最小和最大曲面表征闵科夫斯基空间中的平行曲面
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-15 DOI: 10.1016/j.difgeo.2024.102204
José Eduardo Núñez Ortiz, Gabriel Ruiz-Hernández
{"title":"A characterization of parallel surfaces in Minkowski space via minimal and maximal surfaces","authors":"José Eduardo Núñez Ortiz,&nbsp;Gabriel Ruiz-Hernández","doi":"10.1016/j.difgeo.2024.102204","DOIUrl":"10.1016/j.difgeo.2024.102204","url":null,"abstract":"<div><div>We give a characterization of parallel surfaces in the three dimensional Minkowski space. We consider the following construction on a non degenerate surface <em>M</em>. Given a non degenerate curve in the surface we have the ruled surface orthogonal to <em>M</em> along the curve. We prove that if this orthogonal surface is either maximal or minimal then the curve is a geodesic of <em>M</em>. Moreover such geodesic is either a planar line of curvature of <em>M</em> or it has both constant curvature and constant no zero torsion. A first result says that if <em>M</em> is a surface such that through every point pass two non degenerate geodesics, both with constant curvature and torsion, then the surface is parallel. Our main result says that if <em>M</em> is a surface then through every point pass three non degenerate curves whose associated ruled orthogonal surfaces are either maximal or minimal if and only if <em>M</em> is a parallel surface.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102204"},"PeriodicalIF":0.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660250","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
A Frobenius integrability theorem for plane fields generated by quasiconformal deformations 准共形变形产生的平面场的弗罗贝尼斯可整性定理
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-14 DOI: 10.1016/j.difgeo.2024.102202
Slobodan N. Simić
{"title":"A Frobenius integrability theorem for plane fields generated by quasiconformal deformations","authors":"Slobodan N. Simić","doi":"10.1016/j.difgeo.2024.102202","DOIUrl":"10.1016/j.difgeo.2024.102202","url":null,"abstract":"<div><div>We generalize the classical Frobenius integrability theorem to plane fields of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span>, a regularity class introduced by Reimann <span><span>[9]</span></span> for vector fields in Euclidean spaces. Reimann showed that a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> plane field (defined in a suitable way) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is integrable, with integral manifolds of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>Q</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102202"},"PeriodicalIF":0.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660249","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 existence of real nine-dimensional manifolds which include classical one-parameter families of triply periodic minimal surfaces 存在包含三周期极小曲面的经典一参数族的实九维流形
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-14 DOI: 10.1016/j.difgeo.2024.102212
Norio Ejiri, Toshihiro Shoda
{"title":"The existence of real nine-dimensional manifolds which include classical one-parameter families of triply periodic minimal surfaces","authors":"Norio Ejiri,&nbsp;Toshihiro Shoda","doi":"10.1016/j.difgeo.2024.102212","DOIUrl":"10.1016/j.difgeo.2024.102212","url":null,"abstract":"<div><div>Triply periodic minimal surfaces have been studied in many fields of natural science, and in particular, many one-parameter families of triply periodic minimal surfaces of genus three have been considered. In 1990s, the moduli theory of triply periodic minimal surfaces established by C. Arezzo and G. P. Pirola <span><span>[1]</span></span>, <span><span>[14]</span></span>, and they studied a relationship between the nullity of a minimal surface and the differential of its real period map from the viewpoint of complex geometry. The present paper develops their theory in terms of a real differential geometric aspect, and, by applying the classical transversal property to the real period map, we obtain the numerical evidence for the existence of real nine-dimensional manifolds of triply periodic minimal surfaces which include such one-parameter families. For each case that the transversal property fails, we give values of parameters from which new one-parameter families of triply periodic minimal surfaces issue.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102212"},"PeriodicalIF":0.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660372","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 weakly Einstein submanifolds in space forms satisfying certain equalities 关于满足某些等式的空间形式中的弱爱因斯坦子曼形体
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-13 DOI: 10.1016/j.difgeo.2024.102208
Jihun Kim, JeongHyeong Park
{"title":"On weakly Einstein submanifolds in space forms satisfying certain equalities","authors":"Jihun Kim,&nbsp;JeongHyeong Park","doi":"10.1016/j.difgeo.2024.102208","DOIUrl":"10.1016/j.difgeo.2024.102208","url":null,"abstract":"<div><div>We classify weakly Einstein submanifolds in space forms that satisfy Chen's equality. We also give a classification of weakly Einstein hypersurfaces in space forms that satisfy the semisymmetric condition. In addition, we discuss some characterizations of weakly Einstein submanifolds in space forms whose normal connection is flat.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102208"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660374","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
Isometric and anti-isometric classes of timelike minimal surfaces in Lorentz–Minkowski space 洛伦兹-闵科夫斯基空间中时间拟极小曲面的等距类和反等距类
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-13 DOI: 10.1016/j.difgeo.2024.102210
Shintaro Akamine
{"title":"Isometric and anti-isometric classes of timelike minimal surfaces in Lorentz–Minkowski space","authors":"Shintaro Akamine","doi":"10.1016/j.difgeo.2024.102210","DOIUrl":"10.1016/j.difgeo.2024.102210","url":null,"abstract":"<div><div>Isometric class of minimal surfaces in the Euclidean 3-space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> has the rigidity: if two simply connected minimal surfaces are isometric, then one of them is congruent to a surface in the specific one-parameter family, called the associated family, of the other. On the other hand, the situation for surfaces with Lorentzian metrics is different. In this paper, we show that there exist two timelike minimal surfaces in the Lorentz-Minkowski 3-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> that are isometric each other but one of which does not belong to the congruent class of the associated family of the other. We also prove a rigidity theorem for isometric and anti-isometric classes of timelike minimal surfaces under the assumption that surfaces have no flat points.</div><div>Moreover, we show how symmetries of such surfaces propagate for various deformations including isometric and anti-isometric deformations. In particular, some conservation laws of symmetry for Goursat transformations are discussed.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102210"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660375","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
Globality of the DPW construction for Smyth potentials in the case of SU1,1 SU1,1情况下斯密斯电势的DPW构造的全局性
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-13 DOI: 10.1016/j.difgeo.2024.102211
Tadashi Udagawa
{"title":"Globality of the DPW construction for Smyth potentials in the case of SU1,1","authors":"Tadashi Udagawa","doi":"10.1016/j.difgeo.2024.102211","DOIUrl":"10.1016/j.difgeo.2024.102211","url":null,"abstract":"<div><div>We construct harmonic maps into <span><math><msub><mrow><mi>SU</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> starting from Smyth potentials <em>ξ</em>, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution <em>L</em> of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>L</mi><mo>=</mo><mi>ξ</mi></math></span>. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that <em>L</em> can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman <span><span>[5]</span></span>, while avoiding the general isomonodromy theory used by Guest-Its-Lin <span><span>[11]</span></span>, <span><span>[12]</span></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102211"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660368","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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