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The Lax equation and weak regularity of asymptotic estimate Lie groups 渐近估计李群的Lax方程和弱正则性
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-04-05 DOI: 10.1007/s10455-023-09888-y
Maximilian Hanusch
{"title":"The Lax equation and weak regularity of asymptotic estimate Lie groups","authors":"Maximilian Hanusch","doi":"10.1007/s10455-023-09888-y","DOIUrl":"10.1007/s10455-023-09888-y","url":null,"abstract":"<div><p>We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor’s sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker–Campbell–Dynkin–Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09888-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44535978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geodesics on a K3 surface near the orbifold limit 接近轨道极限的K3表面上的测地线
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-04-03 DOI: 10.1007/s10455-023-09898-w
Jørgen Olsen Lye
{"title":"Geodesics on a K3 surface near the orbifold limit","authors":"Jørgen Olsen Lye","doi":"10.1007/s10455-023-09898-w","DOIUrl":"10.1007/s10455-023-09898-w","url":null,"abstract":"<div><p>This article studies Kummer K3 surfaces close to the orbifold limit. We improve upon estimates for the Calabi–Yau metrics due to Kobayashi. As an application, we study stable closed geodesics. We use the metric estimates to show how there are generally restrictions on the existence of such geodesics. We also show how there can exist stable, closed geodesics in some highly symmetric circumstances due to hyperkähler identities.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09898-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47912407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Conformal Bach flow 保形巴赫流
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-03-30 DOI: 10.1007/s10455-023-09897-x
Jiaqi Chen, Peng Lu, Jie Qing
{"title":"Conformal Bach flow","authors":"Jiaqi Chen,&nbsp;Peng Lu,&nbsp;Jie Qing","doi":"10.1007/s10455-023-09897-x","DOIUrl":"10.1007/s10455-023-09897-x","url":null,"abstract":"<div><p>In this article we introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi’s type <span>(L^2)</span>-estimate of derivatives of curvatures is derived. Furthermore, using the <span>(L^2)</span>-estimate and based on an idea from (Streets in Calc Var PDE 46:39–54, 2013) we show Shi’s pointwise estimate of derivatives of curvatures without assuming Sobolev constant bound.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44854729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields 偏对称2张量场协变导数散度下界的反例
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-03-21 DOI: 10.1007/s10455-023-09896-y
Stefano Borghini, Lorenzo Mazzieri
{"title":"Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields","authors":"Stefano Borghini,&nbsp;Lorenzo Mazzieri","doi":"10.1007/s10455-023-09896-y","DOIUrl":"10.1007/s10455-023-09896-y","url":null,"abstract":"<div><p>In Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022), an estimate for skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: Most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature (Xu and Ye in Invent Math 33(2):64, 2022) and in connection with the Besse conjecture (Yun and Hwang in Critical point equation on three-dimensional manifolds and the Besse conjecture). In the present note, we point out an issue in the argument proposed in Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022) and we provide a counterexample to the estimate.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09896-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49236501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Dispersive equations on asymptotically conical manifolds: time decay in the low-frequency regime 渐近圆锥流形上的色散方程:低频区的时间衰减
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-03-06 DOI: 10.1007/s10455-023-09887-z
Viviana Grasselli
{"title":"Dispersive equations on asymptotically conical manifolds: time decay in the low-frequency regime","authors":"Viviana Grasselli","doi":"10.1007/s10455-023-09887-z","DOIUrl":"10.1007/s10455-023-09887-z","url":null,"abstract":"<div><p>On an asymptotically conical manifold, we prove time decay estimates for the flow of the Schrödinger wave and Klein–Gordon equations via some differentiability properties of the spectral measure. To keep the paper at a reasonable length, we limit ourselves to the low-energy part of the spectrum, which is the one that dictates the decay rates. With this paper, we extend sharp estimates that are known in the asymptotically flat case (see Bouclet and Burq in Duke Math J 170(11):2575–2629, 2021, https://doi.org/10.1215/00127094-2020-0080) to this more general geometric framework and therefore recover the same decay properties as in the Euclidean case. The first step is to prove some resolvent estimates via a limiting absorption principle. It is at this stage that the proof of the previously mentioned authors fails, in particular when we try to recover a low-frequency positive commutator estimate. Once the resolvent estimates are established, we derive regularity for the spectral measure that in turn is applied to obtain the decay of the flows.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09887-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47503903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to: Besse conjecture with positive isotropic curvature 修正：具有正各向同性曲率的Besse猜想
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-03-02 DOI: 10.1007/s10455-023-09892-2
Seungsu Hwang, Gabjin Yun
{"title":"Correction to: Besse conjecture with positive isotropic curvature","authors":"Seungsu Hwang,&nbsp;Gabjin Yun","doi":"10.1007/s10455-023-09892-2","DOIUrl":"10.1007/s10455-023-09892-2","url":null,"abstract":"","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45678275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Classification of left-invariant Einstein metrics on (textrm{SL}(2,mathbb {R})times textrm{SL}(2,mathbb {R})) that are bi-invariant under a one-parameter subgroup 单参数子群下双不变的（textrm｛SL｝（2，mathbb｛R｝）timestextrm{SL｝
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-03-02 DOI: 10.1007/s10455-023-09890-4
Vicente Cortés, Jeremias Ehlert, Alexander S. Haupt, David Lindemann
{"title":"Classification of left-invariant Einstein metrics on (textrm{SL}(2,mathbb {R})times textrm{SL}(2,mathbb {R})) that are bi-invariant under a one-parameter subgroup","authors":"Vicente Cortés,&nbsp;Jeremias Ehlert,&nbsp;Alexander S. Haupt,&nbsp;David Lindemann","doi":"10.1007/s10455-023-09890-4","DOIUrl":"10.1007/s10455-023-09890-4","url":null,"abstract":"<div><p>We classify all left-invariant pseudo-Riemannian Einstein metrics on <span>(textrm{SL}(2,mathbb {R})times textrm{SL}(2,mathbb {R}))</span> that are bi-invariant under a one-parameter subgroup. We find that there are precisely two such metrics up to homothety, the Killing form and a nearly pseudo-Kähler metric.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09890-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50441997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to: Hypercohomologies of truncated twisted holomorphic de Rham complexes 对截断扭曲全纯de Rham复合物的超上同调的修正
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-02-27 DOI: 10.1007/s10455-023-09891-3
Lingxu Meng
{"title":"Correction to: Hypercohomologies of truncated twisted holomorphic de Rham complexes","authors":"Lingxu Meng","doi":"10.1007/s10455-023-09891-3","DOIUrl":"10.1007/s10455-023-09891-3","url":null,"abstract":"","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50517371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rigidity results for Riemannian twistor spaces under vanishing curvature conditions 曲率消失条件下黎曼扭转空间的刚度结果
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-02-23 DOI: 10.1007/s10455-023-09889-x
G. Catino, D. Dameno, P. Mastrolia
{"title":"Rigidity results for Riemannian twistor spaces under vanishing curvature conditions","authors":"G. Catino,&nbsp;D. Dameno,&nbsp;P. Mastrolia","doi":"10.1007/s10455-023-09889-x","DOIUrl":"10.1007/s10455-023-09889-x","url":null,"abstract":"<div><p>In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that <span>(mathbb {C}mathbb {P}^3)</span> is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09889-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44737636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gauss maps of harmonic and minimal great circle fibrations 调和和最小大圆振动的高斯图
IF 0.7 3区 数学
Annals of Global Analysis and Geometry Pub Date : 2023-02-13 DOI: 10.1007/s10455-023-09886-0
Ioannis Fourtzis, Michael Markellos, Andreas Savas-Halilaj
{"title":"Gauss maps of harmonic and minimal great circle fibrations","authors":"Ioannis Fourtzis,&nbsp;Michael Markellos,&nbsp;Andreas Savas-Halilaj","doi":"10.1007/s10455-023-09886-0","DOIUrl":"10.1007/s10455-023-09886-0","url":null,"abstract":"<div><p>We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere <span>(mathbb {S}^3)</span>. We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on <span>(mathbb {S}^3)</span>, whose integral curves are great circles, is a Hopf vector field.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09886-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47929403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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