Differential Geometry and its Applications最新文献

{"title":"Higher-power harmonic maps, instantons and Yang-Mills theory","authors":"Elias Knack,&nbsp;Henrik Naujoks","doi":"10.1016/j.difgeo.2025.102240","DOIUrl":"10.1016/j.difgeo.2025.102240","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span> be two pseudo-Riemannian manifolds. We study field theoretic properties of higher-power harmonic maps (also called <em>r</em>-harmonic maps) <span><math><mi>φ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></math></span>, which are a natural generalization of standard harmonic maps first introduced by C. Wood. In particular, we discuss the coupled system of higher-power harmonic maps and the Einstein-Hilbert action and prove a sufficient condition for a map to be <em>r</em>-harmonic, which is highly motivated by classical field equations like the harmonic map equation or the Yang-Mills equation. Furthermore, we derive an instanton theory for <em>r</em>-harmonic maps on 2<em>r</em>-dimensional base manifolds and investigate conformal properties of general higher-power harmonic maps. Finally, since the theory of higher-power harmonic maps bears striking similarities with Yang-Mills theory, we provide a comprehensive comparison between the two theories which explains in more detail surprisingly many analogies.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102240"},"PeriodicalIF":0.6,"publicationDate":"2025-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143580361","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}

{"title":"Connection blocking in quotients of Sol","authors":"Reza Bidar","doi":"10.1016/j.difgeo.2025.102241","DOIUrl":"10.1016/j.difgeo.2025.102241","url":null,"abstract":"<div><div>Let <em>G</em> be a connected Lie group and <span><math><mi>Γ</mi><mo>⊂</mo><mi>G</mi></math></span> a lattice. Connection curves of the homogeneous space <span><math><mi>M</mi><mo>=</mo><mi>G</mi><mo>/</mo><mi>Γ</mi></math></span> are the orbits of one parameter subgroups of <em>G</em>. To <em>block</em> a pair of points <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>M</mi></math></span> is to find a <em>finite</em> set <span><math><mi>B</mi><mo>⊂</mo><mi>M</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></math></span> such that every connecting curve joining <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> intersects <em>B</em>. The homogeneous space <em>M</em> is <em>blockable</em> if every pair of points in <em>M</em> can be blocked, otherwise we call it <em>non-blockable</em>.</div><div><em>Sol</em> is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, and together with the left invariant metric <span><math><mi>d</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>z</mi></mrow></msup><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>z</mi></mrow></msup><mi>d</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all lattice quotients of <em>Sol</em> are non-blockable. In particular, we show that for any lattice <span><math><mi>Γ</mi><mo>⊂</mo><mi>S</mi><mi>o</mi><mi>l</mi></math></span>, the set of non-blockable pairs is a dense subset of <span><math><mi>S</mi><mi>o</mi><mi>l</mi><mo>/</mo><mi>Γ</mi><mo>×</mo><mi>S</mi><mi>o</mi><mi>l</mi><mo>/</mo><mi>Γ</mi></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102241"},"PeriodicalIF":0.6,"publicationDate":"2025-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143580362","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}

{"title":"Chern-Simons-Higgs type equations on canonically compactifiable graphs","authors":"Longsong Jia ,&nbsp;Chang Li ,&nbsp;Yanlin Li ,&nbsp;Bin Wang","doi":"10.1016/j.difgeo.2025.102237","DOIUrl":"10.1016/j.difgeo.2025.102237","url":null,"abstract":"<div><div>In this paper, we prove existence results of solutions to three kinds of Chern-Simons-Higgs type equations, including mean field equations and Chern-Simons-Higgs equations as well as the generalized Chern-Simons-Higgs equations on canonically compactifiable graphs, which is a special infinite graphs giving inclusive relationship between Banach spaces on graphs. The paper mainly employs variational principles in Banach spaces as well as upper and lower solutions method, with the main challenge being the lack of finite bound of number of vertices and other certain properties, leading to difficulties of estimates of bound for functionals. We choose suitable restrict spaces in Lagrange multiplier theorem and use Moser-Trudinger inequalities to overcome these difficulties.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102237"},"PeriodicalIF":0.6,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143528711","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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}