International Journal of Mathematics最新文献

{"title":"Volume functionals on pseudoconvex hypersurfaces","authors":"Simon Donaldson, Fabian Lehmann","doi":"10.1142/s0129167x24410052","DOIUrl":"https://doi.org/10.1142/s0129167x24410052","url":null,"abstract":"<p>The focus of this paper is on a volume form defined on a pseudoconvex hypersurface <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> in a complex Calabi–Yau manifold (that is, a complex <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-manifold with a nowhere-vanishing holomorphic <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mstyle><mtext mathvariant=\"normal\">R</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>. We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> bounding compact domains <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Ω</mi><mo>⊂</mo><mi>Z</mi></math></span><span></span>. That is, we study critical points of the volume functional <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi><mo stretchy=\"false\">(</mo><mi>M</mi><mo stretchy=\"false\">)</mo></math></span><span></span> where the ordinary volume <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi mathvariant=\"normal\">Ω</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msup><mo>⊂</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">C</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in “most” directions but non-negative in directions corresponding to deformations of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msup></math></span><span></span> by holomorphic diffeomorphisms. We are led to conjecture a “minimax” characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case <span><m","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140601730","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":"Quantization of locally compact groups associated with essentially bijective 1-cocycles","authors":"Pierre Bieliavsky, Victor Gayral, Sergey Neshveyev, Lars Tuset","doi":"10.1142/s0129167x24500277","DOIUrl":"https://doi.org/10.1142/s0129167x24500277","url":null,"abstract":"<p>Given an extension <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>0</mn><mo>→</mo><mi>V</mi><mo>→</mo><mi>G</mi><mo>→</mo><mi>Q</mi><mo>→</mo><mn>1</mn></math></span><span></span> of locally compact groups, with <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi></math></span><span></span> abelian, and a compatible essentially bijective <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span>-cocycle <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>η</mi><mo>:</mo><mi>Q</mi><mo>→</mo><mover accent=\"true\"><mrow><mi>V</mi></mrow><mo>̂</mo></mover></math></span><span></span>, we define a dual unitary <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>-cocycle on <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> and show that the associated deformation of <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>Ĝ</mi></math></span><span></span> is a cocycle bicrossed product defined by a matched pair of subgroups of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>Q</mi><mo stretchy=\"false\">⋉</mo><mover accent=\"true\"><mrow><mi>V</mi></mrow><mo>̂</mo></mover></math></span><span></span>. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> on <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>Q</mi><mo stretchy=\"false\">)</mo></math></span><span></span> and a unitary quantization map <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>→</mo><mstyle><mtext mathvariant=\"normal\">HS</mtext></mstyle><mo stretchy=\"false\">(</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>Q</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> of Kohn–Nirenberg type.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"75 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140601722","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":"On codimension one holomorphic distributions on compact toric orbifolds","authors":"Arnulfo Miguel Rodríguez Peña","doi":"10.1142/s0129167x24500241","DOIUrl":"https://doi.org/10.1142/s0129167x24500241","url":null,"abstract":"<p>The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140601731","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":"Endpoint estimates of variation and oscillation operators associated with Zλ functions","authors":"Yongming Wen, Yanyan Han, Xianming Hou","doi":"10.1142/s0129167x24500253","DOIUrl":"https://doi.org/10.1142/s0129167x24500253","url":null,"abstract":"<p>This paper obtains weak-type estimates, limiting weak-type behaviors for variation operators associated with <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>Z</mi></mrow><mrow><mi>λ</mi></mrow></msup></math></span><span></span> functions. Besides, we give a new characterization of Hardy space via the boundedness of variation operators associated with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>Z</mi></mrow><mrow><mi>λ</mi></mrow></msup></math></span><span></span> functions.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314817","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":"Twistor space for local systems on an open curve","authors":"Carlos T. Simpson","doi":"10.1142/s0129167x24410131","DOIUrl":"https://doi.org/10.1142/s0129167x24410131","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi><mo>=</mo><mover accent=\"false\"><mrow><mi>X</mi></mrow><mo accent=\"true\">¯</mo></mover><mo stretchy=\"false\">−</mo><mi>D</mi></math></span><span></span> be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>. We extend this construction to the case of any rank <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>r</mi></math></span><span></span>, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"61 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314910","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":"Implosion, contraction and Moore–Tachikawa","authors":"Andrew Dancer, Frances Kirwan, Johan Martens","doi":"10.1142/s0129167x24410040","DOIUrl":"https://doi.org/10.1142/s0129167x24410040","url":null,"abstract":"<p>We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> to a general reductive group, and interpret it in the context of the Moore–Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalized to the hyperkähler or complex symplectic situation.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140200625","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":"Lebesgue points of functions in the complex Sobolev space","authors":"Gabriel Vigny, Duc-Viet Vu","doi":"10.1142/s0129167x24500149","DOIUrl":"https://doi.org/10.1142/s0129167x24500149","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>φ</mi></math></span><span></span> be a function in the complex Sobolev space <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>W</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>U</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>U</mi></math></span><span></span> is an open subset in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span><span></span>. We show that the complement of the set of Lebesgue points of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>φ</mi></math></span><span></span> is pluripolar. The key ingredient in our approach is to show that <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>φ</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>α</mi></mrow></msup></math></span><span></span> for <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo>∈</mo><mo stretchy=\"false\">[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math></span><span></span> is locally bounded from above by a plurisubharmonic function.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154580","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 characterization of quasipositive two-bridge knots","authors":"Burak Ozbagci, Stepan Orevkov","doi":"10.1142/s0129167x24500150","DOIUrl":"https://doi.org/10.1142/s0129167x24500150","url":null,"abstract":"<p>We prove a simple necessary and sufficient condition for a two-bridge knot <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi><mo stretchy=\"false\">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy=\"false\">)</mo></math></span><span></span> to be quasipositive, based on the continued fraction expansion of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo stretchy=\"false\">/</mo><mi>q</mi></math></span><span></span>. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"131 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154832","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":"Finsler metrizabilities and geodesic invariance","authors":"Ioan Bucataru, Oana Constantinescu","doi":"10.1142/s0129167x24500162","DOIUrl":"https://doi.org/10.1142/s0129167x24500162","url":null,"abstract":"<p>We demonstrate that various metrizability problems for Finsler sprays can be reformulated in terms of the geodesic invariance of two tensors, namely the metric and angular tensors. We show that a spray <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi></math></span><span></span> is the geodesic spray of some Finsler metric if and only if its metric tensor is geodesically invariant. Moreover, we establish that gyroscopic sprays constitute the largest class of sprays characterized by a geodesic-invariant angular metric. Scalar functions associated with these geodesically invariant tensors will also be invariant, thereby providing first integrals for the given spray.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154655","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 spectral flow and equivariant η-invariants on manifolds with boundary","authors":"Johnny Lim, Hang Wang","doi":"10.1142/s0129167x2450006x","DOIUrl":"https://doi.org/10.1142/s0129167x2450006x","url":null,"abstract":"<p>In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>η</mi></math></span><span></span>-invariants which play a fundamental role in the equivariant analog of Getzler’s spectral flow formula. As a consequence, we establish a relation between equivariant <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>η</mi></math></span><span></span>-invariants and equivariant Maslov triple indices in the splitting of manifolds.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154653","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}