{"title":"Calabi type functionals for coupled Kähler–Einstein metrics","authors":"Satoshi Nakamura","doi":"10.1007/s10455-023-09913-0","DOIUrl":"10.1007/s10455-023-09913-0","url":null,"abstract":"<div><p>We introduce the coupled Ricci–Calabi functional and the coupled H-functional which measure how far a Kähler metric is from a coupled Kähler–Einstein metric in the sense of Hultgren–Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09913-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42429595","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}
{"title":"Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map","authors":"Antonio Bueno, Rafael López","doi":"10.1007/s10455-023-09910-3","DOIUrl":"10.1007/s10455-023-09910-3","url":null,"abstract":"<div><p>Given a <span>(C^1)</span> function <span>(mathcal {H})</span> defined in the unit sphere <span>(mathbb {S}^2)</span>, an <span>(mathcal {H})</span>-surface <i>M</i> is a surface in the Euclidean space <span>(mathbb {R}^3)</span> whose mean curvature <span>(H_M)</span> satisfies <span>(H_M(p)=mathcal {H}(N_p))</span>, <span>(pin M)</span>, where <i>N</i> is the Gauss map of <i>M</i>. Given a closed simple curve <span>(Gamma subset mathbb {R}^3)</span> and a function <span>(mathcal {H})</span>, in this paper we investigate the geometry of compact <span>(mathcal {H})</span>-surfaces spanning <span>(Gamma )</span> in terms of <span>(Gamma )</span>. Under mild assumptions on <span>(mathcal {H})</span>, we prove non-existence of closed <span>(mathcal {H})</span>-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on <span>(mathcal {H})</span> that ensure that if <span>(Gamma )</span> is a circle, then <i>M</i> is a rotational surface. We also establish the existence of estimates of the area of <span>(mathcal {H})</span>-surfaces in terms of the height of the surface.\u0000</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43478109","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}
{"title":"Anti-quasi-Sasakian manifolds","authors":"D. Di Pinto, G. Dileo","doi":"10.1007/s10455-023-09907-y","DOIUrl":"10.1007/s10455-023-09907-y","url":null,"abstract":"<div><p>We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely Kähler almost contact metric manifolds <span>((M,varphi , xi ,eta ,g))</span>, quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the <span>(varphi )</span>-invariance and the <span>(varphi )</span>-anti-invariance of the 2-form <span>(textrm{d}eta )</span>. A Boothby–Wang type theorem allows to obtain aqS structures on principal circle bundles over Kähler manifolds endowed with a closed (2, 0)-form. We characterize aqS manifolds with constant <span>(xi )</span>-sectional curvature equal to 1: they admit an <span>(Sp(n)times 1)</span>-reduction of the frame bundle such that the manifold is transversely hyperkähler, carrying a second aqS structure and a null Sasakian <span>(eta )</span>-Einstein structure. We show that aqS manifolds with constant sectional curvature are necessarily flat and cokähler. Finally, by using a metric connection with torsion, we provide a sufficient condition for an aqS manifold to be locally decomposable as the Riemannian product of a Kähler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (<i>M</i>, <i>g</i>) cannot be locally symmetric.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09907-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45705860","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}
{"title":"Levi-flat CR structures on compact Lie groups","authors":"Howard Jacobowitz, Max Reinhold Jahnke","doi":"10.1007/s10455-023-09909-w","DOIUrl":"10.1007/s10455-023-09909-w","url":null,"abstract":"<div><p>Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09909-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42624146","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}
{"title":"Integral decompositions of varifolds","authors":"Hsin-Chuang Chou","doi":"10.1007/s10455-023-09908-x","DOIUrl":"10.1007/s10455-023-09908-x","url":null,"abstract":"<div><p>This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. However, the decompositions may fail to be unique. Furthermore, this result can be generalized by replacing the class of integral varifolds with some classes of rectifiable varifolds whose density is uniformly bounded from below; for these classes, we also prove a general version of the compactness theorem for integral varifolds.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42138858","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}
{"title":"Extra-twisted connected sum (G_2)-manifolds","authors":"Johannes Nordström","doi":"10.1007/s10455-023-09893-1","DOIUrl":"10.1007/s10455-023-09893-1","url":null,"abstract":"<div><p>We present a construction of closed 7-manifolds of holonomy <span>(G_2)</span>, which generalises Kovalev’s twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author use this to exhibit examples of closed 7-manifolds with disconnected moduli space of holonomy <span>(G_2)</span> metrics.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09893-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50475153","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}
{"title":"Existence results for a super Toda system","authors":"Aleks Jevnikar, Ruijun Wu","doi":"10.1007/s10455-023-09899-9","DOIUrl":"10.1007/s10455-023-09899-9","url":null,"abstract":"<div><p>We solve a super Toda system on a closed Riemann surface of genus <span>(gamma >1)</span> and with some particular spin structures. This generalizes the min–max methods and results for super Liouville equations and gives new existence results for super Toda systems.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09899-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44027650","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}
{"title":"Homogeneous Einstein metrics and butterflies","authors":"Christoph Böhm, Megan M. Kerr","doi":"10.1007/s10455-023-09905-0","DOIUrl":"10.1007/s10455-023-09905-0","url":null,"abstract":"<div><p>In 2012, M. M. Graev associated to a compact homogeneous space <i>G</i>/<i>H</i> a nerve <span>({text {X}}_{G/H})</span>, whose non-contractibility implies the existence of a <i>G</i>-invariant Einstein metric on <i>G</i>/<i>H</i>. The nerve <span>({text {X}}_{G/H})</span> is a compact, semi-algebraic set, defined purely Lie theoretically by intermediate subgroups. In this paper we present a detailed description of the work of Graev and the curvature estimates given by Böhm in 2004.\u0000</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"63 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42812695","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}
{"title":"Sobolev inequalities and convergence for Riemannian metrics and distance functions","authors":"B. Allen, E. Bryden","doi":"10.1007/s10455-023-09906-z","DOIUrl":"10.1007/s10455-023-09906-z","url":null,"abstract":"<div><p>If one thinks of a Riemannian metric, <span>(g_1)</span>, analogously as the gradient of the corresponding distance function, <span>(d_1)</span>, with respect to a background Riemannian metric, <span>(g_0)</span>, then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper, we study the sub-critical case <span>(p < frac{m}{2})</span> where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an <span>(L^{frac{p}{2}})</span> bound on a Riemannian metric implies an <span>(L^q)</span> bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov’s conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"63 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43867965","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}