Differential Geometry and its Applications最新文献

{"title":"Modal fracture of higher groups","authors":"David Jaz Myers","doi":"10.1016/j.difgeo.2024.102176","DOIUrl":"10.1016/j.difgeo.2024.102176","url":null,"abstract":"<div><p>In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle <em>k</em>-gerbes with connection and describe their modal fracture hexagon.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102176"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142011821","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 volume of conformally flat manifolds as hypersurfaces in the light-cone","authors":"Riku Kishida","doi":"10.1016/j.difgeo.2024.102173","DOIUrl":"10.1016/j.difgeo.2024.102173","url":null,"abstract":"<div><p>In this paper, we focus on a conformally flat Riemannian manifold <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> of dimension <em>n</em> isometrically immersed into the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional light-cone <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is not only immersed in <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> but also isometrically realized as a hypersurface of a certain null hypersurface <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> in the Minkowski spacetime, which is different from <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Moreover, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> has a volume-maximizing property in <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102173"},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141984634","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":"A non-Vaisman LCK solvmanifold associated to a one-dimensional extension of a 2-step nilmanifold","authors":"Hiroshi Sawai","doi":"10.1016/j.difgeo.2024.102174","DOIUrl":"10.1016/j.difgeo.2024.102174","url":null,"abstract":"<div><p>The purpose of this paper is to determine a locally conformal Kähler solvmanifold such that its associated solvable Lie group is a one-dimensional extension of a 2-step nilpotent Lie group.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102174"},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141964440","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":"Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space","authors":"O. Oğulcan Tuncer","doi":"10.1016/j.difgeo.2024.102175","DOIUrl":"10.1016/j.difgeo.2024.102175","url":null,"abstract":"<div><p>We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are well-defined even at singular points of the curve. These moving frames enable us to define evolutes and focal surfaces of pseudo-spherical framed immersions. We investigate the singularity properties of these evolutes and focal surfaces. We then reveal that the evolute of a pseudo-spherical framed immersion is the set of singular points of its focal surface. We also interpret evolutes and focal surfaces as the discriminant and the secondary discriminant sets of certain height functions, which allows us to explain evolutes and focal surfaces as wavefronts from the viewpoint of Legendrian singularity theory. Examples are provided to flesh out our results, and we use the hyperbolic Hopf map to visualize these examples.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102175"},"PeriodicalIF":0.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950695","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":"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}