Pub Date : 2025-03-25DOI: 10.1016/j.difgeo.2025.102243
Rafael López , Marian Ioan Munteanu
A soliton of the mean curvature flow in the product space is a surface whose mean curvature H satisfies the equation , where N is the unit normal of the surface and X is a Killing vector field of . In this paper we consider the cases that X is the vector field tangent to the second factor and the vector field associated to rotations about an axis of , 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 .
{"title":"Solitons of the mean curvature flow in S2×R","authors":"Rafael López , 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":"2025-03-25","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}
Pub Date : 2025-03-21DOI: 10.1016/j.difgeo.2025.102244
Yongheng Han
If Σ is an n-dimensional noncompact self-shrinker and the second fundamental form of Σ is integrable for , we show that Σ is asymptotic to a regular cone. We also prove long-time existence of the mean curvature flow starting from complete manifolds with bounded curvature and small total curvature.
{"title":"Mean curvature flow with pinched curvature integral","authors":"Yongheng Han","doi":"10.1016/j.difgeo.2025.102244","DOIUrl":"10.1016/j.difgeo.2025.102244","url":null,"abstract":"<div><div>If Σ is an <em>n</em>-dimensional noncompact self-shrinker and the second fundamental form of Σ is <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> integrable for <span><math><mi>p</mi><mo>≥</mo><mi>n</mi></math></span>, we show that Σ is asymptotic to a regular cone. We also prove long-time existence of the mean curvature flow starting from complete manifolds with bounded curvature and small total curvature.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102244"},"PeriodicalIF":0.6,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679853","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 : 2025-03-12DOI: 10.1016/j.difgeo.2025.102242
Marvin Dippell , David Kern
We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.
{"title":"Constraint vector bundles and reduction of Lie (bi-)algebroids","authors":"Marvin Dippell , David Kern","doi":"10.1016/j.difgeo.2025.102242","DOIUrl":"10.1016/j.difgeo.2025.102242","url":null,"abstract":"<div><div>We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"99 ","pages":"Article 102242"},"PeriodicalIF":0.6,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601229","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 : 2025-03-09DOI: 10.1016/j.difgeo.2025.102240
Elias Knack, Henrik Naujoks
Let and be two pseudo-Riemannian manifolds. We study field theoretic properties of higher-power harmonic maps (also called r-harmonic maps) , 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 r-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 r-harmonic maps on 2r-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.
{"title":"Higher-power harmonic maps, instantons and Yang-Mills theory","authors":"Elias Knack, 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}
Pub Date : 2025-03-09DOI: 10.1016/j.difgeo.2025.102241
Reza Bidar
Let G be a connected Lie group and a lattice. Connection curves of the homogeneous space are the orbits of one parameter subgroups of G. To block a pair of points is to find a finite set such that every connecting curve joining and intersects B. The homogeneous space M is blockable if every pair of points in M can be blocked, otherwise we call it non-blockable.
Sol is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to , and together with the left invariant metric includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all lattice quotients of Sol are non-blockable. In particular, we show that for any lattice , the set of non-blockable pairs is a dense subset of .
{"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}
Pub Date : 2025-03-03DOI: 10.1016/j.difgeo.2025.102237
Longsong Jia , Chang Li , Yanlin Li , Bin Wang
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.
{"title":"Chern-Simons-Higgs type equations on canonically compactifiable graphs","authors":"Longsong Jia , Chang Li , Yanlin Li , 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}
Pub Date : 2025-02-27DOI: 10.1016/j.difgeo.2025.102238
Li Ou
In this paper, we investigate value distribution properties for Gauss maps of space-like stationary surfaces in four-dimensional Lorentz-Minkowski space , 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.
{"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}
Pub Date : 2025-02-27DOI: 10.1016/j.difgeo.2025.102239
Shuho Kanda
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.
{"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}
Pub Date : 2025-02-12DOI: 10.1016/j.difgeo.2025.102236
Hao-Yue Liu , Sha Yao , Xin-An Ren
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.
本文研究了非线性热方程的矩阵li - you - hamilton估计。首先，我们在一个具有固定Kähler度量的Kähler流形上得到了这样的估计。然后，我们考虑了在重新标度的Kähler-Ricci流下，对具有Kähler度量演化的Kähler流形的估计。这两种估计都可以推广到约束情况。
{"title":"Matrix Li-Yau-Hamilton estimates for nonlinear heat equations","authors":"Hao-Yue Liu , Sha Yao , 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}
Pub Date : 2025-02-05DOI: 10.1016/j.difgeo.2025.102234
Juanru Gu , Yao Lu , Hongwei Xu , Entao Zhao
Let be an oriented submanifold with parallel mean curvature vector in a complete simply connected Riemannian manifold . When the mean curvature , i.e., M is minimal, we prove that there exists a constant , such that if , and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then is isometric to . Moreover, M is the totally geodesic sphere . This is a generalization of Shen and Li's results [10], [14]. When the ambient manifold is a space form, we improve the geometric rigidity theorem due to Xu-Gu [19] for the codimension is not more than 2 and .
{"title":"Curvature pinching for three-dimensional submanifolds in a Riemannian manifold","authors":"Juanru Gu , Yao Lu , Hongwei Xu , 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}
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