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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
Classification of conformally flat Moebius isoparametric submanifolds in the Euclidean space 欧几里得空间中保形平莫比乌斯等参数子平面的分类
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102201
M.S.R. Antas
{"title":"Classification of conformally flat Moebius isoparametric submanifolds in the Euclidean space","authors":"M.S.R. Antas","doi":"10.1016/j.difgeo.2024.102201","DOIUrl":"10.1016/j.difgeo.2024.102201","url":null,"abstract":"<div><div>The aim of this article is to classify umbilic-free isometric immersions <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, of a conformally flat manifold which are Moebius isoparametric.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102201"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660369","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
Left-invariant pseudo-Riemannian metrics on Lie groups: The null cone 李群上的左变伪黎曼度量：空锥
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102205
Sigbjørn Hervik
{"title":"Left-invariant pseudo-Riemannian metrics on Lie groups: The null cone","authors":"Sigbjørn Hervik","doi":"10.1016/j.difgeo.2024.102205","DOIUrl":"10.1016/j.difgeo.2024.102205","url":null,"abstract":"<div><div>We study left-invariant pseudo-Riemannian metrics on Lie groups using the moving bracket approach of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the <span><math><mi>G</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-action; i.e., Lie algebras <em>μ</em> where zero is in the closure of the orbits: <span><math><mn>0</mn><mo>∈</mo><mover><mrow><mi>G</mi><mo>⋅</mo><mi>μ</mi></mrow><mo>‾</mo></mover></math></span>. We provide examples of such Lie groups in various signatures and give some general results. For signatures <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some <span><math><mi>O</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102205"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660362","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
Singularities of discrete indefinite affine minimal surfaces 离散不定仿射极小曲面的奇点
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102206
Marcos Craizer
{"title":"Singularities of discrete indefinite affine minimal surfaces","authors":"Marcos Craizer","doi":"10.1016/j.difgeo.2024.102206","DOIUrl":"10.1016/j.difgeo.2024.102206","url":null,"abstract":"<div><div>A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and swallowtails. By discretizing the initial curves, one can obtain by the discrete Lelieuvre's formulas a discrete affine minimal surface with indefinite metric. The aim of this paper is to define the singular edges and vertices of the corresponding discrete asymptotic net in such a way that the most relevant properties of the singular set of the smooth version remain valid.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102206"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660371","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 flows of graphs sliding off to infinity in warped product manifolds 扭曲积流形中滑向无穷远的图的平均曲率流
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102207
Naotoshi Fujihara
{"title":"Mean curvature flows of graphs sliding off to infinity in warped product manifolds","authors":"Naotoshi Fujihara","doi":"10.1016/j.difgeo.2024.102207","DOIUrl":"10.1016/j.difgeo.2024.102207","url":null,"abstract":"<div><div>We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and <span><math><mi>R</mi></math></span>. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102207"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660370","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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