Differential Geometry and its Applications最新文献

{"title":"Existence and density results of conformal metrics with prescribed higher order Q-curvature on Sn","authors":"Zhongwei Tang ,&nbsp;Heming Wang ,&nbsp;Ning Zhou","doi":"10.1016/j.difgeo.2024.102172","DOIUrl":"10.1016/j.difgeo.2024.102172","url":null,"abstract":"<div><p>We prove some results on the density and multiplicity of positive solutions to the conformal <em>Q</em>-curvature equations <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>K</mi><msup><mrow><mi>v</mi></mrow><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></mrow></msup></math></span> on the <em>n</em>-dimensional standard unit sphere <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span> and <em>m</em> is an integer, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is the intertwining operator of order 2<em>m</em> and <em>K</em> is the prescribed <em>Q</em>-curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, Pohozaev identity, as well as the blow up argument for nonlinear integral equations, we construct arbitrarily many multi-bump solutions. In particular, we show the smooth positive <em>Q</em>-curvature functions of metrics conformal to <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are dense in the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> topology. Existence results of infinitely many positive solutions to the poly-harmonic equations <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></mrow></msup></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> being asymptotically periodic are also obtained.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102172"},"PeriodicalIF":0.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961513","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":"Diffeological submanifolds and their friends","authors":"Yael Karshon ,&nbsp;David Miyamoto ,&nbsp;Jordan Watts","doi":"10.1016/j.difgeo.2024.102170","DOIUrl":"10.1016/j.difgeo.2024.102170","url":null,"abstract":"<div><p>A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a <em>uniquely immersed submanifold</em>. Diffeology provides yet another intrinsic notion of submanifold: a <em>diffeological submanifold</em>.</p><p>We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the <em>initial morphisms</em> are exactly the (diffeological) <em>inductions</em>, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of <em>pseudo-immersions</em>.</p><p>We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective.</p><p>In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102170"},"PeriodicalIF":0.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951372","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":"Topology of toric gravitational instantons","authors":"Gustav Nilsson","doi":"10.1016/j.difgeo.2024.102171","DOIUrl":"10.1016/j.difgeo.2024.102171","url":null,"abstract":"<div><p>For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with toric symmetry, we express the signature of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102171"},"PeriodicalIF":0.6,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000640/pdfft?md5=1af94bc08a68f11151c59c10b99043ce&pid=1-s2.0-S0926224524000640-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961512","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":"Schwarz lemma for conformal parametrization of minimal graphs in M×R","authors":"David Kalaj","doi":"10.1016/j.difgeo.2024.102169","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102169","url":null,"abstract":"<div><p>We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold <span><math><mi>M</mi><mo>×</mo><mi>R</mi></math></span>, where <em>M</em> is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the <em>ϱ</em>-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102169"},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481516","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":"Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces","authors":"Isami Koga ,&nbsp;Yasuyuki Nagatomo","doi":"10.1016/j.difgeo.2024.102167","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102167","url":null,"abstract":"<div><p>We classify equivariant harmonic maps of the complex projective spaces <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective <em>line</em>, we have one parameter family of those maps. (This result is already shown in <span>[2]</span> and <span>[4]</span> in other ways). However, when <span><math><mi>m</mi><mo>≧</mo><mn>2</mn></math></span>, we will obtain the rigidity results.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102167"},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000603/pdfft?md5=50c3b21df49c5a546924763a29df2d65&pid=1-s2.0-S0926224524000603-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481519","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":"The curvature tensors associated with the gluing formula of the zeta-determinants for the Robin boundary condition","authors":"Klaus Kirsten ,&nbsp;Yoonweon Lee","doi":"10.1016/j.difgeo.2024.102165","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102165","url":null,"abstract":"<div><p>The gluing formula for the zeta-determinants of Laplacians with respect to the Robin boundary condition was proved in <span>[15]</span>. This formula contains a constant which is expressed by some curvature tensors on the cutting hypersurface including the scalar and principal curvatures. In this paper we compute this constant explicitly when the cutting hypersurface is a 2-dimensional closed submanifold in a closed Riemannian manifold, and discuss some related topics. We next use the conformal rescaling of the Riemannian metric to compute the value of the zeta function at zero associated to the generalized Dirichlet-to-Neumann operator defined by the Robin boundary condition on this cutting hypersurface.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102165"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481517","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":"Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains","authors":"Fouzia El Wassouli,&nbsp;Daoud Oukacha","doi":"10.1016/j.difgeo.2024.102168","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102168","url":null,"abstract":"<div><p>Let <span><math><mi>Ω</mi><mo>=</mo><mi>G</mi><mo>/</mo><mi>K</mi></math></span> be a bounded symmetric domain of non-compact type. In this paper the image of the Poisson transform on the degenerate principal series representations of <em>G</em> attached to the Shilov boundary of Ω is considered. We characterize the images in terms of the third-order Hua operators <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span>. When Ω is the exceptional domain of type <em>V</em>, we give the explicit formulas for the operators <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102168"},"PeriodicalIF":0.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141438667","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":"Sasakian geometry on sphere bundles II: Constant scalar curvature","authors":"Charles P. Boyer ,&nbsp;Christina W. Tønnesen-Friedman","doi":"10.1016/j.difgeo.2024.102166","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102166","url":null,"abstract":"<div><p>In a previous paper <span>[18]</span> the authors employed the fiber join construction of Yamazaki <span>[38]</span> together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman <span>[2]</span> to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem <span>[14]</span> that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102166"},"PeriodicalIF":0.6,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434496","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":"Rotationally invariant translators of the mean curvature flow in Einstein's static universe","authors":"Miguel Ortega ,&nbsp;Handan Yıldırım","doi":"10.1016/j.difgeo.2024.102153","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102153","url":null,"abstract":"<div><p>In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"95 ","pages":"Article 102153"},"PeriodicalIF":0.5,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000469/pdfft?md5=6bc58615dc3e4e9f74770ce03c1820e6&pid=1-s2.0-S0926224524000469-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244155","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":"Isoparametric hypersurfaces in product spaces of space forms","authors":"Dong Gao ,&nbsp;Hui Ma ,&nbsp;Zeke Yao","doi":"10.1016/j.difgeo.2024.102155","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102155","url":null,"abstract":"<div><p>We give a complete classification of isoparametric hypersurfaces in a product space <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup><mo>×</mo><msubsup><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> of 2-dimensional space forms for <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> with <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. In fact we prove that any isoparametric hypersurface in such a space has constant product angle function, which enables us to remove the condition of constant principal curvatures from the classification obtained recently by J.B.M. dos Santos and J.P. dos Santos.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"95 ","pages":"Article 102155"},"PeriodicalIF":0.5,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244156","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}