Pub Date : 2024-11-15DOI: 10.1016/j.difgeo.2024.102204
José Eduardo Núñez Ortiz, Gabriel Ruiz-Hernández
We give a characterization of parallel surfaces in the three dimensional Minkowski space. We consider the following construction on a non degenerate surface M. Given a non degenerate curve in the surface we have the ruled surface orthogonal to M along the curve. We prove that if this orthogonal surface is either maximal or minimal then the curve is a geodesic of M. Moreover such geodesic is either a planar line of curvature of M or it has both constant curvature and constant no zero torsion. A first result says that if M 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 M 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 M is a parallel surface.
我们给出了三维闵科夫斯基空间中平行曲面的特征。给定曲面中的一条非退化曲线，我们就有了沿该曲线与 M 正交的规则曲面。我们证明，如果这个正交曲面是最大或最小的，那么这条曲线就是 M 的一条大地线。此外，这条大地线要么是 M 的一条平面曲率线，要么具有恒定曲率和恒定无零扭。第一个结果表明，如果 M 是一个曲面，且每一点都经过两条非退化的大地线，且这两条大地线都具有恒定的曲率和扭转，那么这个曲面是平行的。我们的主要结果表明，如果 M 是一个曲面，那么通过每一点的三条非退化曲线，其相关的规则正交曲面要么是最大的，要么是最小的，当且仅当 M 是一个平行曲面。
{"title":"A characterization of parallel surfaces in Minkowski space via minimal and maximal surfaces","authors":"José Eduardo Núñez Ortiz, 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}
Pub Date : 2024-11-14DOI: 10.1016/j.difgeo.2024.102202
Slobodan N. Simić
We generalize the classical Frobenius integrability theorem to plane fields of class , a regularity class introduced by Reimann [9] for vector fields in Euclidean spaces. Reimann showed that a vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive plane field (defined in a suitable way) in is integrable, with integral manifolds of class .
{"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}
Pub Date : 2024-11-14DOI: 10.1016/j.difgeo.2024.102212
Norio Ejiri, Toshihiro Shoda
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 [1], [14], 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.
三周期极小曲面在自然科学的许多领域都得到了研究，特别是属三的三周期极小曲面的许多单参数族。20 世纪 90 年代，C. Arezzo 和 G. P. Pirola 建立了三周期极小曲面的模理论[1], [14]，他们从复几何的角度研究了极小曲面的无效性与其实周期映射微分之间的关系。本文从实微分几何的角度发展了他们的理论，并通过将经典的横向性质应用于实周期映射，得到了包括这种单参数族的三周期极小曲面的实九维流形存在的数值证据。在横向性质失效的每种情况下，我们都给出了参数值，由此产生了新的三重周期极小曲面的单参数族。
{"title":"The existence of real nine-dimensional manifolds which include classical one-parameter families of triply periodic minimal surfaces","authors":"Norio Ejiri, 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}
Pub Date : 2024-11-13DOI: 10.1016/j.difgeo.2024.102208
Jihun Kim, JeongHyeong Park
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.
{"title":"On weakly Einstein submanifolds in space forms satisfying certain equalities","authors":"Jihun Kim, 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}
Pub Date : 2024-11-13DOI: 10.1016/j.difgeo.2024.102210
Shintaro Akamine
Isometric class of minimal surfaces in the Euclidean 3-space 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 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.
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.
{"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}
Pub Date : 2024-11-13DOI: 10.1016/j.difgeo.2024.102211
Tadashi Udagawa
We construct harmonic maps into starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of . However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L 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 [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].
我们通过 DPW 方法，从斯迈势 ξ 开始，构建进入 SU1,1/U1 的谐波映射。在这种方法中，谐波映射是从 L-1dL=ξ 的解 L 的岩泽因子化得到的。然而，在非紧密群的情况下，岩泽因式分解并不总是全局的。我们证明 L 可以用贝塞尔函数来表示，并通过贝塞尔函数的渐近展开求解黎曼-希尔伯特问题，从而给出全局岩泽因式分解。与 Dorfmeister-Guest-Rossman [5] 的研究相比，我们通过这种方法更直接地证明了我们的求解的全局性，同时避免了 Guest-Its-Lin [11], [12] 所使用的一般等单调性理论。
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.difgeo.2024.102201
M.S.R. Antas
The aim of this article is to classify umbilic-free isometric immersions , , of a conformally flat manifold which are Moebius isoparametric.
本文旨在对莫比乌斯等参数的保角平坦流形的无脐等距沉浸 f:Mn→Rm, n≥4 进行分类。
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.difgeo.2024.102205
Sigbjørn Hervik
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 -action; i.e., Lie algebras μ where zero is in the closure of the orbits: . We provide examples of such Lie groups in various signatures and give some general results. For signatures and 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 action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given.
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.difgeo.2024.102206
Marcos Craizer
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.
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.difgeo.2024.102207
Naotoshi Fujihara
We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and . 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.
我们研究由封闭黎曼流形和 R 定义的翘曲积流形中的平均曲率流。在这样的翘曲积流形中，我们可以定义一个图的概念，称为大地图。我们证明，对于任何翘曲函数，曲线缩短流都会保留大地图，而对于某些单调凸翘曲函数，超曲面的平均曲率流也会保留大地图。特别是，我们考虑了一些在无穷远处归零的翘曲函数，这意味着曲线或超曲面沿着流动在无穷远处归于一点。在这种情况下，我们证明了流的长期存在性，以及曲率及其高阶导数沿流归零。
{"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}
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