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}
Pub Date : 2024-11-11DOI: 10.1016/j.difgeo.2024.102209
Esfandiar Nava-Yazdani
In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.
{"title":"On geodesics in the spaces of constrained curves","authors":"Esfandiar Nava-Yazdani","doi":"10.1016/j.difgeo.2024.102209","DOIUrl":"10.1016/j.difgeo.2024.102209","url":null,"abstract":"<div><div>In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102209"},"PeriodicalIF":0.6,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660373","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 : 2024-10-23DOI: 10.1016/j.difgeo.2024.102193
Michael Albanese , Giuseppe Barbaro , Mehdi Lejmi
We generalize Kähler–Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map [25], and show that their existence is an obstruction to the existence of first-Chern–Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the 4-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on 2n-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler–Ricci solitons. In particular, we partially extend Matsushima's theorem [41] to compact first-Chern–Einstein almost-Kähler manifolds.
{"title":"Generalized almost-Kähler–Ricci solitons","authors":"Michael Albanese , Giuseppe Barbaro , Mehdi Lejmi","doi":"10.1016/j.difgeo.2024.102193","DOIUrl":"10.1016/j.difgeo.2024.102193","url":null,"abstract":"<div><div>We generalize Kähler–Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map <span><span>[25]</span></span>, and show that their existence is an obstruction to the existence of first-Chern–Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the 4-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on 2<em>n</em>-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler–Ricci solitons. In particular, we partially extend Matsushima's theorem <span><span>[41]</span></span> to compact first-Chern–Einstein almost-Kähler manifolds.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102193"},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533161","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 : 2024-10-22DOI: 10.1016/j.difgeo.2024.102192
Laiyuan Gao , Horst Martini , Deyan Zhang
A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant k-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is k-symmetric.
引入了一种非局部曲率流来演化平面中的局部凸曲线。研究证明，任何初始局部凸曲线的非局部曲率流都有一个全局解，它保持了演化曲线的局部凸性和弹性能量，而且随着时间的无穷大，曲线会收敛到一条具有恒定 k 阶宽度的光滑局部凸曲线。特别是，当且仅当初始局部凸曲线是 k 对称曲线时，极限曲线是一个多重圆。
{"title":"Deforming locally convex curves into curves of constant k-order width","authors":"Laiyuan Gao , Horst Martini , Deyan Zhang","doi":"10.1016/j.difgeo.2024.102192","DOIUrl":"10.1016/j.difgeo.2024.102192","url":null,"abstract":"<div><div>A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant <em>k</em>-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is <em>k</em>-symmetric.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102192"},"PeriodicalIF":0.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533160","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}
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