Pub Date : 2024-04-10DOI: 10.1142/s0219199724500111
Antonio Alarcón, Ildefonso Castro-Infantes, Jorge Hidalgo
We prove that every open Riemann surface is the complex structure of a complete surface of constant mean curvature () in the three-dimensional hyperbolic space . We go further and establish a jet interpolation theorem for complete conformal immersions . As a consequence, we show the existence of complete densely immersed surfaces in with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in which is also established in this paper.
{"title":"Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure","authors":"Antonio Alarcón, Ildefonso Castro-Infantes, Jorge Hidalgo","doi":"10.1142/s0219199724500111","DOIUrl":"https://doi.org/10.1142/s0219199724500111","url":null,"abstract":"<p>We prove that every open Riemann surface <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> is the complex structure of a complete surface of constant mean curvature <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> (<span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">CMC-1</mtext></mstyle></math></span><span></span>) in the three-dimensional hyperbolic space <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span>. We go further and establish a jet interpolation theorem for complete conformal <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">CMC-1</mtext></mstyle></math></span><span></span> immersions <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi><mo>→</mo><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span>. As a consequence, we show the existence of complete densely immersed <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">CMC-1</mtext></mstyle></math></span><span></span> surfaces in <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">×</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span> which is also established in this paper.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"52 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-27DOI: 10.1142/s0219199724500123
Lucas Backes, Davor Dragičević, Mihály Pituk
It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.
{"title":"Shadowing, Hyers–Ulam stability and hyperbolicity for nonautonomous linear delay differential equations","authors":"Lucas Backes, Davor Dragičević, Mihály Pituk","doi":"10.1142/s0219199724500123","DOIUrl":"https://doi.org/10.1142/s0219199724500123","url":null,"abstract":"<p>It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-09DOI: 10.1142/s0219199724500044
Adam Czapliński, Andreas Krug, Manfred Lehn, Sönke Rollenske
We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalize Kodaira’s classification of singular elliptic fibers and thus call them extended ADE curves. On such a curve , we describe a compactified Jacobian and show that its components reflect the intersection graph of . This extends known results when is reduced, but new difficulties arise when is non-reduced. As an application, we get an explicit description of general singular fibers of certain Lagrangian fibrations of Beauville–Mukai type.
我们观察到,K3 曲面上充分正线性系统中的一般可还原曲线的形式概括了小平的奇异椭圆纤维分类,因此称其为扩展 ADE 曲线。在这样的曲线 C 上,我们描述了一个紧凑化的雅各比,并证明其分量反映了 C 的交点图。这扩展了 C 被还原时的已知结果,但在 C 未被还原时又出现了新的困难。作为应用,我们得到了对某些博维尔-穆凯类型拉格朗日纤维的一般奇异纤维的明确描述。
{"title":"Compactified Jacobians of extended ADE curves and Lagrangian fibrations","authors":"Adam Czapliński, Andreas Krug, Manfred Lehn, Sönke Rollenske","doi":"10.1142/s0219199724500044","DOIUrl":"https://doi.org/10.1142/s0219199724500044","url":null,"abstract":"<p>We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalize Kodaira’s classification of singular elliptic fibers and thus call them extended ADE curves. On such a curve <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi></math></span><span></span>, we describe a compactified Jacobian and show that its components reflect the intersection graph of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi></math></span><span></span>. This extends known results when <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi></math></span><span></span> is reduced, but new difficulties arise when <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi></math></span><span></span> is non-reduced. As an application, we get an explicit description of general singular fibers of certain Lagrangian fibrations of Beauville–Mukai type.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"7 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-28DOI: 10.1142/s0219199723500645
Wu-Hsiung Huang
<p>In this paper, we establish a “global” Morse index theorem. Given a hypersurface <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> of constant mean curvature, immersed in <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></msup></math></span><span></span>. Consider a continuous deformation of “generalized” Lipschitz domain <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span><span></span> enlarging in <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>. The topological type of <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span><span></span> is permitted to change along <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>t</mi></math></span><span></span>, so that <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span><span></span> has an arbitrary shape which can “reach afar” in <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>, i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>t</mi></math></span><span></span> of the Sobolev space <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span></span> of variation functions on <span><math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span><span></span>, as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of <span><math altimg="eq-00012.gif" display="inline" overflow="scroll"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span><span></span> in <span><math altimg="eq-00013.gif" display="inline" overflow="scroll"><mi>t</mi></math></span><span></span> to yield the required continuities of <span><math altimg="eq-00014.gif" display="inline" overflow="scroll"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span></span> and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in <span><math altimg="eq-00015.gif" display="
{"title":"A global Morse index theorem and applications to Jacobi fields on CMC surfaces","authors":"Wu-Hsiung Huang","doi":"10.1142/s0219199723500645","DOIUrl":"https://doi.org/10.1142/s0219199723500645","url":null,"abstract":"<p>In this paper, we establish a “global” Morse index theorem. Given a hypersurface <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> of constant mean curvature, immersed in <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msup></math></span><span></span>. Consider a continuous deformation of “generalized” Lipschitz domain <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> enlarging in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>. The topological type of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is permitted to change along <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>, so that <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> has an arbitrary shape which can “reach afar” in <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>, i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span> of the Sobolev space <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span></span> of variation functions on <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> in <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span> to yield the required continuities of <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span></span> and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in <span><math altimg=\"eq-00015.gif\" display=\"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"25 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-23DOI: 10.1142/s0219199724500032
Fulin Chen, Haisheng Li, Shaobin Tan
In this paper, we study nullity- toroidal extended affine Lie algebras in the context of vertex algebras and their -coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and -coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of a class of irreducible modules for the variant Lie algebras.
{"title":"Toroidal extended affine Lie algebras and vertex algebras","authors":"Fulin Chen, Haisheng Li, Shaobin Tan","doi":"10.1142/s0219199724500032","DOIUrl":"https://doi.org/10.1142/s0219199724500032","url":null,"abstract":"<p>In this paper, we study nullity-<span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span> toroidal extended affine Lie algebras in the context of vertex algebras and their <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi></math></span><span></span>-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ϕ</mi></math></span><span></span>-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of a class of irreducible modules for the variant Lie algebras.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"5 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-23DOI: 10.1142/s0219199723500633
Dražen Adamović, Kazuya Kawasetsu, David Ridout
The Bershadsky–Polyakov algebras are the subregular quantum Hamiltonian reductions of the affine vertex operator algebras associated with . In (D. Adamović, K. Kawasetsu and D. Ridout, A realisation of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys.111 (2021) 38, arXiv:2007.00396 [math.QA]), we realized these algebras in terms of the regular reduction, Zamolodchikov’s W3-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky–Polyakov modules has the property that the result is generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of (Z. Fehily, K. Kawasetsu and D. Ridout, Classifying relaxed highest-weight modules for admissible-level Bershadsky–Polyakov algebras, Comm. Math. Phys.385 (2021) 859–904, arXiv:2007.03917 [math.RT]) for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level , which is new.
伯沙德斯基-波利亚科夫代数是与𝔰𝔩3 相关的仿射顶点算子代数的亚规则量子哈密顿还原。在 (D. Adamović、K. Kawasetsu 和 D. Ridout, A realisation of the Bershadsky-Polyakov algebras and their relaxed modules, Lett.Math.Phys.111(2021)38,arXiv:2007.00396 [math.QA]),我们用正则还原、扎莫洛奇科夫的 W3-代数和等向晶格顶点算子代数实现了这些代数。我们还证明了松弛的最高权布尔夏德斯基-波利亚科夫模块的自然构造具有结果一般不可还原的性质。在这里,我们证明了当这种构造与谱流捻合相结合时,可以得到一组完整的不可还原权重模块,其权重空间是有限维的。这给出了 (Z. Fehily, K. Kawasetsu and D. Ridout, Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras, Comm. Math.Math.Phys.385(2021)859-904,arXiv:2007.03917 [math.RT]),并将此分类扩展到权重模块类别。我们还推导出了新的非可容许级 k=-73 的分类。
{"title":"Weight module classifications for Bershadsky–Polyakov algebras","authors":"Dražen Adamović, Kazuya Kawasetsu, David Ridout","doi":"10.1142/s0219199723500633","DOIUrl":"https://doi.org/10.1142/s0219199723500633","url":null,"abstract":"<p>The Bershadsky–Polyakov algebras are the subregular quantum Hamiltonian reductions of the affine vertex operator algebras associated with <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔰</mi><mi>𝔩</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span><span></span>. In (D. Adamović, K. Kawasetsu and D. Ridout, A realisation of the Bershadsky–Polyakov algebras and their relaxed modules, <i>Lett. Math. Phys.</i><b>111</b> (2021) 38, arXiv:2007.00396 [math.QA]), we realized these algebras in terms of the regular reduction, Zamolodchikov’s <i>W</i><sub>3</sub>-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky–Polyakov modules has the property that the result is generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of (Z. Fehily, K. Kawasetsu and D. Ridout, Classifying relaxed highest-weight modules for admissible-level Bershadsky–Polyakov algebras, <i>Comm. Math. Phys.</i><b>385</b> (2021) 859–904, arXiv:2007.03917 [math.RT]) for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>k</mi></mstyle><mo>=</mo><mo stretchy=\"false\">−</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span><span></span>, which is new.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"64 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-23DOI: 10.1142/s0219199724500019
Brian Grajales, Lino Grama, Rafaela F. Prado
In this paper, we examine the geodesics on adjoint orbits of that are equipped with -invariant metrics, where is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain -flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for .
在本文中,我们研究了SL(n,ℝ)邻接轨道上的大地线,这些轨道配备了SO(n)不变度量,其中SO(n)是最大紧凑子群。我们的主要技术包括将这一问题转化为某些 SO(n)-flag 流形切线束中的几何问题,并描述切线束上有关佐佐木度量的大地方程。此外,我们还利用 Lie Theory 的工具获得了对大地方程组的明确描述。我们详细分析了 SL(2,ℝ) 的情况。
{"title":"Geodesics on adjoint orbits of SL(n, ℝ)","authors":"Brian Grajales, Lino Grama, Rafaela F. Prado","doi":"10.1142/s0219199724500019","DOIUrl":"https://doi.org/10.1142/s0219199724500019","url":null,"abstract":"<p>In this paper, we examine the geodesics on adjoint orbits of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> that are equipped with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">SO</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-invariant metrics, where <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">SO</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">SO</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle><mo stretchy=\"false\">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"34 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-19DOI: 10.1142/s0219199723500657
Luis M. Briceño-Arias, Patrick L. Combettes, Francisco J. Silva
The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented.
{"title":"Perspective functions with nonlinear scaling","authors":"Luis M. Briceño-Arias, Patrick L. Combettes, Francisco J. Silva","doi":"10.1142/s0219199723500657","DOIUrl":"https://doi.org/10.1142/s0219199723500657","url":null,"abstract":"<p>The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140072619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-16DOI: 10.1142/s0219199724500056
Amin Esfahani, Achenef Tesfahun
In this paper, the sixth-order Boussinesq equation is studied. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the “bad” fourth term in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.
{"title":"Well-posedness and analyticity of solutions for the sixth-order Boussinesq equation","authors":"Amin Esfahani, Achenef Tesfahun","doi":"10.1142/s0219199724500056","DOIUrl":"https://doi.org/10.1142/s0219199724500056","url":null,"abstract":"<p>In this paper, the sixth-order Boussinesq equation is studied. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the “bad” fourth term <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Δ</mi><mi>u</mi></math></span><span></span> in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.</p>","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"88 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140076296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}