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Ramification and unicity theorems for Gauss maps of complete space-like stationary surfaces in four-dimensional Lorentz-Minkowski space
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-27 DOI: 10.1016/j.difgeo.2025.102238
Li Ou
{"title":"Ramification and unicity theorems for Gauss maps of complete space-like stationary surfaces in four-dimensional Lorentz-Minkowski space","authors":"Li Ou","doi":"10.1016/j.difgeo.2025.102238","DOIUrl":"10.1016/j.difgeo.2025.102238","url":null,"abstract":"<div><div>In this paper, we investigate value distribution properties for Gauss maps of space-like stationary surfaces in four-dimensional Lorentz-Minkowski space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>, focusing on aspects such as the total weight of totally ramified values and unicity properties. We obtain not only general conclusions analogous to those in four-dimensional Euclidean space, but also results for space-like stationary surfaces with rational graphical Gauss image, which is an extension of degenerate space-like stationary surfaces.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102238"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510165","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 hard Lefschetz duality for locally conformally almost Kähler manifolds
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-27 DOI: 10.1016/j.difgeo.2025.102239
Shuho Kanda
{"title":"The hard Lefschetz duality for locally conformally almost Kähler manifolds","authors":"Shuho Kanda","doi":"10.1016/j.difgeo.2025.102239","DOIUrl":"10.1016/j.difgeo.2025.102239","url":null,"abstract":"<div><div>We prove the hard Lefschetz duality for locally conformally almost Kähler manifolds. This is a generalization of that for almost Kähler manifolds studied by Cirici and Wilson. We generalize the Kähler identities to prove the duality. Based on the result, we introduce the hard Lefschetz condition for locally conformally symplectic manifolds. As examples, we give solvmanifolds which do not satisfy the hard Lefschetz condition.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102239"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510166","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
Matrix Li-Yau-Hamilton estimates for nonlinear heat equations
IF 0.6 4区 数学
Differential Geometry and its Applications Pub 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":"2025-02-12","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
Curvature pinching for three-dimensional submanifolds in a Riemannian manifold
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2025-02-05 DOI: 10.1016/j.difgeo.2025.102234
Juanru Gu , Yao Lu , Hongwei Xu , Entao Zhao
{"title":"Curvature pinching for three-dimensional submanifolds in a Riemannian manifold","authors":"Juanru Gu ,&nbsp;Yao Lu ,&nbsp;Hongwei Xu ,&nbsp;Entao Zhao","doi":"10.1016/j.difgeo.2025.102234","DOIUrl":"10.1016/j.difgeo.2025.102234","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> be an oriented submanifold with parallel mean curvature vector in a complete simply connected Riemannian manifold <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span>. When the mean curvature <span><math><mi>H</mi><mo>=</mo><mn>0</mn></math></span>, i.e., <em>M</em> is minimal, we prove that there exists a constant <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, such that if <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>N</mi></mrow></msub><mo>∈</mo><mo>[</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, and if <em>M</em> has a lower bound for Ricci curvature and an upper bound for scalar curvature, then <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span> is isometric to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></msup></math></span>. Moreover, <em>M</em> is the totally geodesic sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This is a generalization of Shen and Li's results <span><span>[10]</span></span>, <span><span>[14]</span></span>. When the ambient manifold is a space form, we improve the geometric rigidity theorem due to Xu-Gu <span><span>[19]</span></span> for the codimension is not more than 2 and <span><math><mi>H</mi><mo>≠</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102234"},"PeriodicalIF":0.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143101469","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
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
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
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
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
Gradient estimates of a nonlinear parabolic equation under integral Bakry-Émery Ricci condition
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
A global invariant for path structures and second order differential equations
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
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