Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102220
Diego Conti , Alejandro Gil-García
We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products ; we work at the level of the Lie algebra . In particular we consider the structures induced on by existing pseudo-Kähler structures on and ; we classify all semidirect products of this type with of dimension 4 and . 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 k-step nilpotent Lie algebras for every .
{"title":"Pseudo-Kähler and hypersymplectic structures on semidirect products","authors":"Diego Conti , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102203
Homare Tadano
We establish a Souplet–Zhang type local gradient estimate for positive solutions to the fast diffusion equation associated with the Witten Laplacian on an n-dimensional Riemannian manifold when the N-Bakry–Émery Ricci curvature with is bounded from below by a non-positive constant. When the N-Bakry–Émery Ricci curvature is reduced to the Ricci curvature, our result refines the Souplet–Zhang type local gradient estimate by X. Zhu (2011) [10]. 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 n-dimensional non-compact Riemannian manifold with non-negative N-Bakry–Émery Ricci curvature with .
{"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><</mo><mi>m</mi><mo><</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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102225
Tiancheng Xia
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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102216
Francisco G.S. Carvalho , Barnabé P. Lima , Paulo A. Sousa , Bruno V.M. Vieira
In a remarkable work [35], Wei established estimates for the eigenvalues of the Laplacian on closed submanifolds embedded in a unit sphere . In this study, we extend these results to the eigenvalues of the p-Laplacian. As a consequence, we provide new characterizations of the sphere . Additionally, we derive integral inequalities in terms of the norm of the second fundamental form of M and the first non-zero eigenvalue of the p-Laplacian, thereby generalizing the results previously established by Santos and Soares [11] for hypersurfaces.
{"title":"Lower estimates for the length of the second fundamental form of submanifolds","authors":"Francisco G.S. Carvalho , Barnabé P. Lima , Paulo A. Sousa , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102222
Xavier Ramos Olivé , Shoo Seto
We prove a global gradient estimate to positive solutions of the nonlinear parabolic equation 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.
{"title":"Gradient estimates of a nonlinear parabolic equation under integral Bakry-Émery Ricci condition","authors":"Xavier Ramos Olivé , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102224
E. Falbel , J.M. Veloso
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 .
{"title":"A global invariant for path structures and second order differential equations","authors":"E. Falbel , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102221
Naoya Suda
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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102217
Guangyue Huang, Qi Guo, Bingqing Ma
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 must be of Einstein.
{"title":"Rigidity of closed vacuum static spaces","authors":"Guangyue Huang, Qi Guo, 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}
Pub Date : 2025-02-01DOI: 10.1016/j.difgeo.2024.102223
Biqiang Zhao
Let be a complete pseudo-Hermitian (2m+1)-manifold. In this paper, we derive the subgradient estimates for the positive solutions of the equation on complete noncompact pseudo-Hermitian manifolds without the commutation condition.
{"title":"Subgradient estimates for the equation Δbu+aulogu+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}
Pub Date : 2025-01-31DOI: 10.1016/j.difgeo.2024.102219
Shin-young Kim , Kyeong-Dong Park
The double Cayley Grassmannian is a unique smooth equivariant completion with Picard number one of the 14-dimensional exceptional complex Lie group , 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.
{"title":"Deformation rigidity of the double Cayley Grassmannian","authors":"Shin-young Kim , 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}
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