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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 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
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 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
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 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
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 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
A global invariant for path structures and second order differential equations 路径结构和二阶微分方程的全局不变量
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 DOI: 10.1016/j.difgeo.2024.102224
E. Falbel , J.M. Veloso
{"title":"A global invariant for path structures and second order differential equations","authors":"E. Falbel ,&nbsp;J.M. Veloso","doi":"10.1016/j.difgeo.2024.102224","DOIUrl":"10.1016/j.difgeo.2024.102224","url":null,"abstract":"<div><div>We study a global invariant for path structures which is obtained as a secondary invariant from a Cartan connection on a canonical bundle associated to a path structure. This invariant is computed in examples which are defined in terms of reductions of the path structure. In particular we give a formula for this global invariant for second order differential equations defined on a torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102224"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176225","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
Gradient estimates of a nonlinear parabolic equation under integral Bakry-Émery Ricci condition 积分Bakry-Émery Ricci条件下非线性抛物方程的梯度估计
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 DOI: 10.1016/j.difgeo.2024.102222
Xavier Ramos Olivé , Shoo Seto
{"title":"Gradient estimates of a nonlinear parabolic equation under integral Bakry-Émery Ricci condition","authors":"Xavier Ramos Olivé ,&nbsp;Shoo Seto","doi":"10.1016/j.difgeo.2024.102222","DOIUrl":"10.1016/j.difgeo.2024.102222","url":null,"abstract":"<div><div>We prove a global gradient estimate to positive solutions of the nonlinear parabolic equation <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>a</mi><mi>u</mi><mi>ln</mi><mo>⁡</mo><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>b</mi><mi>u</mi></math></span> under an integral Bakry-Émery Ricci condition on compact weighted manifolds. The elliptic version of the equation arises in the study of gradient Ricci solitons and in this paper we consider the parabolic version.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102222"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176203","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
Ricci flow of discrete surfaces of revolution, and relation to constant Gaussian curvature 离散旋转曲面的里奇流，以及与常数高斯曲率的关系
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 DOI: 10.1016/j.difgeo.2024.102221
Naoya Suda
{"title":"Ricci flow of discrete surfaces of revolution, and relation to constant Gaussian curvature","authors":"Naoya Suda","doi":"10.1016/j.difgeo.2024.102221","DOIUrl":"10.1016/j.difgeo.2024.102221","url":null,"abstract":"<div><div>Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution have a geometric realization for the Ricci flow that approaches the constant Gaussian curvature surfaces we have parametrized.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102221"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176202","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
Rigidity of closed vacuum static spaces 封闭真空静态空间的刚性
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 DOI: 10.1016/j.difgeo.2024.102217
Guangyue Huang, Qi Guo, Bingqing Ma
{"title":"Rigidity of closed vacuum static spaces","authors":"Guangyue Huang,&nbsp;Qi Guo,&nbsp;Bingqing Ma","doi":"10.1016/j.difgeo.2024.102217","DOIUrl":"10.1016/j.difgeo.2024.102217","url":null,"abstract":"<div><div>In this paper, we study the rigidity results of closed vacuum static spaces. By introducing a trace-free three tensor, we provide a necessary condition that such spaces with the dimensional scope <span><math><mn>3</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>5</mn></math></span> must be of Einstein.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102217"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176182","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
Subgradient estimates for the equation Δbu+aulog⁡u+bu=0 on complete noncompact pseudo-Hermitian manifolds 完全非紧伪厄米流形上方程Δbu+aulog²u+bu=0的次梯度估计
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-01 DOI: 10.1016/j.difgeo.2024.102223
Biqiang Zhao
{"title":"Subgradient estimates for the equation Δbu+aulog⁡u+bu=0 on complete noncompact pseudo-Hermitian manifolds","authors":"Biqiang Zhao","doi":"10.1016/j.difgeo.2024.102223","DOIUrl":"10.1016/j.difgeo.2024.102223","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>H</mi><mi>M</mi><mo>,</mo><mi>J</mi><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> be a complete pseudo-Hermitian (2m+1)-manifold. In this paper, we derive the subgradient estimates for the positive solutions of the equation <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>a</mi><mi>u</mi><mi>log</mi><mo>⁡</mo><mi>u</mi><mo>+</mo><mi>b</mi><mi>u</mi><mo>=</mo><mn>0</mn></math></span> on complete noncompact pseudo-Hermitian manifolds without the commutation condition.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"98 ","pages":"Article 102223"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176183","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
Deformation rigidity of the double Cayley Grassmannian 变形刚度双凯利格拉斯曼
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-01-31 DOI: 10.1016/j.difgeo.2024.102219
Shin-young Kim , Kyeong-Dong Park
{"title":"Deformation rigidity of the double Cayley Grassmannian","authors":"Shin-young Kim ,&nbsp;Kyeong-Dong Park","doi":"10.1016/j.difgeo.2024.102219","DOIUrl":"10.1016/j.difgeo.2024.102219","url":null,"abstract":"<div><div>The double Cayley Grassmannian is a unique smooth equivariant completion with Picard number one of the 14-dimensional exceptional complex Lie group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and it parametrizes eight-dimensional isotropic subalgebras of the complexified bi-octonions. We show the rigidity of the double Cayley Grassmannian under Kähler deformations. This means that for any smooth projective family of complex manifolds over a connected base of which one fiber is biholomorphic to the double Cayley Grassmannian, all other fibers are biholomorphic to the double Cayley Grassmannian.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102219"},"PeriodicalIF":0.6,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101467","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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