Pub Date : 2023-04-19DOI: 10.1007/s10455-023-09902-3
Bruno Caldeira, Luiz Hartmann, Boris Vertman
The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Suárez-Serrato and Tapie to a non-compact setting. In the appendix we specify our analysis to a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case we obtain stronger estimates for the short time solution using microlocal methods.
{"title":"Normalized Yamabe flow on manifolds with bounded geometry","authors":"Bruno Caldeira, Luiz Hartmann, Boris Vertman","doi":"10.1007/s10455-023-09902-3","DOIUrl":"10.1007/s10455-023-09902-3","url":null,"abstract":"<div><p>The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Suárez-Serrato and Tapie to a non-compact setting. In the appendix we specify our analysis to a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case we obtain stronger estimates for the short time solution using microlocal methods.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"63 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09902-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44015268","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}
Pub Date : 2023-04-13DOI: 10.1007/s10455-023-09900-5
Chanyoung Sung
A smooth closed manifold M is called almost Ricci-flat if
$$begin{aligned} inf _g||text {Ric}_g||_infty cdot text {diam}_g(M)^2=0 end{aligned}$$
where (text {Ric}_g) and (text {diam}_g), respectively, denote the Ricci tensor and the diameter of g and g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold M which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.
{"title":"Kummer-type constructions of almost Ricci-flat 5-manifolds","authors":"Chanyoung Sung","doi":"10.1007/s10455-023-09900-5","DOIUrl":"10.1007/s10455-023-09900-5","url":null,"abstract":"<div><p>A smooth closed manifold <i>M</i> is called almost Ricci-flat if </p><div><div><span>$$begin{aligned} inf _g||text {Ric}_g||_infty cdot text {diam}_g(M)^2=0 end{aligned}$$</span></div></div><p>where <span>(text {Ric}_g)</span> and <span>(text {diam}_g)</span>, respectively, denote the Ricci tensor and the diameter of <i>g</i> and <i>g</i> runs over all Riemannian metrics on <i>M</i>. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold <i>M</i> which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"63 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09900-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45084153","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}
Pub Date : 2023-04-05DOI: 10.1007/s10455-023-09888-y
Maximilian Hanusch
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.
{"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":"63 3","pages":""},"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}
Pub Date : 2023-04-03DOI: 10.1007/s10455-023-09898-w
Jørgen Olsen Lye
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.
{"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":"63 3","pages":""},"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}
Pub Date : 2023-03-30DOI: 10.1007/s10455-023-09897-x
Jiaqi Chen, Peng Lu, Jie Qing
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 (L^2)-estimate of derivatives of curvatures is derived. Furthermore, using the (L^2)-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.
本文引入保角巴赫流,并建立了它在闭流形上的适定性。我们也获得了它向后的独特性。为了尝试研究保角巴赫流的长期行为,假设曲率和压力函数是有界的,导出了曲率导数的全局和局部施型(L^2)估计。此外,使用(L^2)-估计,并基于(Streets in Calc Var PDE 46:39–541013)中的一个想法,我们展示了施对曲率导数的逐点估计,而不假设Sobolev常数界。
{"title":"Conformal Bach flow","authors":"Jiaqi Chen, Peng Lu, 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":"63 2","pages":""},"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}
Pub Date : 2023-03-21DOI: 10.1007/s10455-023-09896-y
Stefano Borghini, Lorenzo Mazzieri
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.
{"title":"Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields","authors":"Stefano Borghini, 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":"63 2","pages":""},"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}
Pub Date : 2023-03-06DOI: 10.1007/s10455-023-09887-z
Viviana Grasselli
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.
在渐近锥流形上,我们通过谱测度的一些可微性质证明了薛定谔波和克莱因-戈登方程流的时间衰减估计。为了使论文保持合理的长度,我们将自己限制在光谱的低能量部分,这就是决定衰变率的部分。在本文中,我们扩展了渐近平坦情况下已知的尖锐估计(见Bouclet和Burq在Duke Math J 170(11):2575–26292021,https://doi.org/10.1215/00127094-2020-0080)从而恢复与欧几里得情况相同的衰变特性。第一步是通过极限吸收原理证明一些预解估计。正是在这个阶段,前面提到的作者的证明失败了,特别是当我们试图恢复低频正换向器估计时。一旦建立了预解估计,我们就导出了光谱测量的正则性,然后应用该正则性来获得流的衰减。
{"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":"63 2","pages":""},"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}
Pub Date : 2023-03-02DOI: 10.1007/s10455-023-09892-2
Seungsu Hwang, Gabjin Yun
{"title":"Correction to: Besse conjecture with positive isotropic curvature","authors":"Seungsu Hwang, 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":"63 2","pages":""},"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}
Pub Date : 2023-03-02DOI: 10.1007/s10455-023-09890-4
Vicente Cortés, Jeremias Ehlert, Alexander S. Haupt, David Lindemann
We classify all left-invariant pseudo-Riemannian Einstein metrics on (textrm{SL}(2,mathbb {R})times textrm{SL}(2,mathbb {R})) 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.
{"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, Jeremias Ehlert, Alexander S. Haupt, 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":"63 2","pages":""},"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}
Pub Date : 2023-02-27DOI: 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":"63 2","pages":""},"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}