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Solitons of the mean curvature flow in S2×R 平均曲率的孤子流在S2×R
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-06-01 Epub Date: 2025-03-25 DOI: 10.1016/j.difgeo.2025.102243
Rafael López , Marian Ioan Munteanu
{"title":"Solitons of the mean curvature flow in S2×R","authors":"Rafael López ,&nbsp;Marian Ioan Munteanu","doi":"10.1016/j.difgeo.2025.102243","DOIUrl":"10.1016/j.difgeo.2025.102243","url":null,"abstract":"<div><div>A soliton of the mean curvature flow in the product space <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span> is a surface whose mean curvature <em>H</em> satisfies the equation <span><math><mi>H</mi><mo>=</mo><mo>〈</mo><mi>N</mi><mo>,</mo><mi>X</mi><mo>〉</mo></math></span>, where <em>N</em> is the unit normal of the surface and <em>X</em> is a Killing vector field of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mi>R</mi></math></span>. In this paper we consider the cases that <em>X</em> is the vector field tangent to the second factor and the vector field associated to rotations about an axis of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or rotations of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102243"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696897","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
Bour's theorem for helicoidal surfaces with singularities 带奇点的螺旋曲面的小时定理
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-06-01 Epub Date: 2025-04-03 DOI: 10.1016/j.difgeo.2025.102248
Yuki Hattori , Atsufumi Honda , Tatsuya Morimoto
{"title":"Bour's theorem for helicoidal surfaces with singularities","authors":"Yuki Hattori ,&nbsp;Atsufumi Honda ,&nbsp;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":"2025-06-01","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}
引用次数: 0
The geometric Toda equations for noncompact symmetric spaces 非紧致对称空间的几何Toda方程
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-06-01 Epub Date: 2025-04-16 DOI: 10.1016/j.difgeo.2025.102249
Ian McIntosh
{"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 &amp; 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":"2025-06-01","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}
引用次数: 0
Matrix Li-Yau-Hamilton estimates for nonlinear heat equations 非线性热方程的矩阵li - yu - hamilton估计
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-06-01 Epub Date: 2025-02-12 DOI: 10.1016/j.difgeo.2025.102236
Hao-Yue Liu , Sha Yao , Xin-An Ren
{"title":"Matrix Li-Yau-Hamilton estimates for nonlinear heat equations","authors":"Hao-Yue Liu ,&nbsp;Sha Yao ,&nbsp;Xin-An Ren","doi":"10.1016/j.difgeo.2025.102236","DOIUrl":"10.1016/j.difgeo.2025.102236","url":null,"abstract":"<div><div>In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such an estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estimates can be generalized to constrained cases.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102236"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386637","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
Spacelike foliations on Lorentz manifolds 洛伦兹流形上的空间类叶
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-06-01 Epub Date: 2025-01-31 DOI: 10.1016/j.difgeo.2025.102235
Aldir Brasil , Sharief Deshmukh , Euripedes da Silva , Paulo Sousa
{"title":"Spacelike foliations on Lorentz manifolds","authors":"Aldir Brasil ,&nbsp;Sharief Deshmukh ,&nbsp;Euripedes da Silva ,&nbsp;Paulo Sousa","doi":"10.1016/j.difgeo.2025.102235","DOIUrl":"10.1016/j.difgeo.2025.102235","url":null,"abstract":"<div><div>In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We investigate conditions for the leaves being stable, totally geodesic or totally umbilical. We consider that <span><math><msup><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> is equipped with a timelike closed conformal vector field <em>ξ</em>. If the foliation has constant mean curvature, we show that the leaves are stable. When the leaves are compact spacelike hypersurfaces we show that, under certain conditions, its are totally umbilic hypersurfaces. In the case of foliations by complete noncompact hypersurfaces, we using a Maximum Principle at infinity to conclude that the foliation is totally geodesic.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102235"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101468","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
Pseudo-Kähler and hypersymplectic structures on semidirect products Pseudo-Kähler和半直接产物上的超辛结构
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 Epub Date: 2024-12-11 DOI: 10.1016/j.difgeo.2024.102220
Diego Conti , Alejandro Gil-García
{"title":"Pseudo-Kähler and hypersymplectic structures on semidirect products","authors":"Diego Conti ,&nbsp;Alejandro Gil-García","doi":"10.1016/j.difgeo.2024.102220","DOIUrl":"10.1016/j.difgeo.2024.102220","url":null,"abstract":"<div><div>We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products <span><math><mi>G</mi><mo>⋊</mo><mi>H</mi></math></span>; we work at the level of the Lie algebra <span><math><mi>g</mi><mo>⋊</mo><mi>h</mi></math></span>. In particular we consider the structures induced on <span><math><mi>g</mi><mo>⋊</mo><mi>h</mi></math></span> by existing pseudo-Kähler structures on <span><math><mi>g</mi></math></span> and <span><math><mi>h</mi></math></span>; we classify all semidirect products of this type with <span><math><mi>g</mi></math></span> of dimension 4 and <span><math><mi>h</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. In the hypersymplectic setting, we consider a more general construction on semidirect products. We construct a large class of hypersymplectic Lie algebras whose underlying complex structure is not abelian as well as non-flat hypersymplectic metrics on <em>k</em>-step nilpotent Lie algebras for every <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102220"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176224","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
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 : 2025-02-01 Epub 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":"2025-02-01","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
The deformation of the balanced cone and its degeneration 平衡锥的变形及其退化
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 Epub Date: 2025-01-06 DOI: 10.1016/j.difgeo.2024.102225
Tiancheng Xia
{"title":"The deformation of the balanced cone and its degeneration","authors":"Tiancheng Xia","doi":"10.1016/j.difgeo.2024.102225","DOIUrl":"10.1016/j.difgeo.2024.102225","url":null,"abstract":"<div><div>In this paper, we briefly review the relationship between the degeneration of the balanced cone and the degeneration of the Gauduchon cone. After that, the lower semi-continuity of the balanced cone under deformation is proved.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102225"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176204","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 Souplet–Zhang type gradient estimate for the fast diffusion equation associated with the Witten Laplacian 与Witten Laplacian相关的快速扩散方程的Souplet-Zhang型梯度估计
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 Epub Date: 2024-12-06 DOI: 10.1016/j.difgeo.2024.102203
Homare Tadano
{"title":"A Souplet–Zhang type gradient estimate for the fast diffusion equation associated with the Witten Laplacian","authors":"Homare Tadano","doi":"10.1016/j.difgeo.2024.102203","DOIUrl":"10.1016/j.difgeo.2024.102203","url":null,"abstract":"<div><div>We establish a Souplet–Zhang type local gradient estimate for positive solutions <span><math><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> to the fast diffusion equation associated with the Witten Laplacian<span><span><span><math><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>V</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mspace></mspace><mn>1</mn><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mo>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span></span></span> on an <em>n</em>-dimensional Riemannian manifold <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> when the <em>N</em>-Bakry–Émery Ricci curvature with <span><math><mi>N</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span> is bounded from below by a non-positive constant. When the <em>N</em>-Bakry–Émery Ricci curvature is reduced to the Ricci curvature, our result refines the Souplet–Zhang type local gradient estimate by X. Zhu (2011) <span><span>[10]</span></span>. As an application, we prove a Liouville type theorem for positive ancient solutions to the fast diffusion equation associated with the Witten Laplacian on an <em>n</em>-dimensional non-compact Riemannian manifold <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with non-negative <em>N</em>-Bakry–Émery Ricci curvature with <span><math><mi>N</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102203"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176201","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
Lower estimates for the length of the second fundamental form of submanifolds 子流形第二种基本形式长度的较低估计
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 Epub Date: 2024-12-02 DOI: 10.1016/j.difgeo.2024.102216
Francisco G.S. Carvalho , Barnabé P. Lima , Paulo A. Sousa , Bruno V.M. Vieira
{"title":"Lower estimates for the length of the second fundamental form of submanifolds","authors":"Francisco G.S. Carvalho ,&nbsp;Barnabé P. Lima ,&nbsp;Paulo A. Sousa ,&nbsp;Bruno V.M. Vieira","doi":"10.1016/j.difgeo.2024.102216","DOIUrl":"10.1016/j.difgeo.2024.102216","url":null,"abstract":"<div><div>In a remarkable work <span><span>[35]</span></span>, Wei established estimates for the eigenvalues of the Laplacian on closed submanifolds <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> embedded in a unit sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msup></math></span>. In this study, we extend these results to the eigenvalues of the <em>p</em>-Laplacian. As a consequence, we provide new characterizations of the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Additionally, we derive integral inequalities in terms of the norm of the second fundamental form of <em>M</em> and the first non-zero eigenvalue of the <em>p</em>-Laplacian, thereby generalizing the results previously established by Santos and Soares <span><span>[11]</span></span> for hypersurfaces.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102216"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176223","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
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