Pub Date : 2018-07-03DOI: 10.1007/S12220-020-00447-6
A. Fino, Alberto Raffero
{"title":"A Class of Eternal Solutions to the G$$_{mathbf 2}$$-Laplacian Flow","authors":"A. Fino, Alberto Raffero","doi":"10.1007/S12220-020-00447-6","DOIUrl":"https://doi.org/10.1007/S12220-020-00447-6","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"1 1","pages":"1-20"},"PeriodicalIF":1.1,"publicationDate":"2018-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-020-00447-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47392086","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 : 2018-05-24DOI: 10.1007/978-3-030-34953-0_24
A. Zinger
{"title":"Some Questions in the Theory of Pseudoholomorphic Curves","authors":"A. Zinger","doi":"10.1007/978-3-030-34953-0_24","DOIUrl":"https://doi.org/10.1007/978-3-030-34953-0_24","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"49 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83522151","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 : 2018-03-20DOI: 10.1007/978-3-030-34953-0_2
W. Ballmann, Henrik Matthiesen, Panagiotis Polymerakis
{"title":"Bottom of Spectra and Amenability of Coverings","authors":"W. Ballmann, Henrik Matthiesen, Panagiotis Polymerakis","doi":"10.1007/978-3-030-34953-0_2","DOIUrl":"https://doi.org/10.1007/978-3-030-34953-0_2","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"40 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91302777","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 : 2018-03-19DOI: 10.1007/978-3-030-34953-0_6
A. Futaki, Hajime Ono
{"title":"On the Existence Problem of Einstein–Maxwell Kähler Metrics","authors":"A. Futaki, Hajime Ono","doi":"10.1007/978-3-030-34953-0_6","DOIUrl":"https://doi.org/10.1007/978-3-030-34953-0_6","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"10 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79955074","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 : 2018-01-01Epub Date: 2017-10-16DOI: 10.1007/s12220-017-9942-9
Zakarias Sjöström Dyrefelt
We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.
证明了具有超越上同调类的常数标量曲率Kähler (cscK)流形是k -半稳定的，自然地推广了极化流形的情况。根据R. Berman, T. Darvas和C. Lu最近关于k能量的性质的结果，进一步得出具有离散自同构群的cscK流形是一致k稳定的。作为证明的主要步骤，我们在一般的Kähler设置下，建立了一个(广义的)Donaldson-Futaki不变量与k能量沿弱测地线射线渐近斜率的关系式。
{"title":"K-Semistability of cscK Manifolds with Transcendental Cohomology Class.","authors":"Zakarias Sjöström Dyrefelt","doi":"10.1007/s12220-017-9942-9","DOIUrl":"https://doi.org/10.1007/s12220-017-9942-9","url":null,"abstract":"<p><p>We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 4","pages":"2927-2960"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9942-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36822412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-01-01Epub Date: 2017-10-12DOI: 10.1007/s12220-017-9934-9
Maciej Dunajski, Thomas Mettler
Given a projective structure on a surface , we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle . The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on is the non-compact real form of the Fubini-Study metric on . We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.
{"title":"Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four.","authors":"Maciej Dunajski, Thomas Mettler","doi":"10.1007/s12220-017-9934-9","DOIUrl":"https://doi.org/10.1007/s12220-017-9934-9","url":null,"abstract":"<p><p>Given a projective structure on a surface <math><mi>N</mi></math> , we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space <i>M</i> of a certain rank 2 affine bundle <math><mrow><mi>M</mi> <mo>→</mo> <mi>N</mi></mrow> </math> . The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on <math> <msup><mrow><mi>RP</mi></mrow> <mn>2</mn></msup> </math> is the non-compact real form of the Fubini-Study metric on <math><mrow><mi>M</mi> <mo>=</mo> <mi>SL</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo> <mo>/</mo> <mi>GL</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo></mrow> </math> . We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 3","pages":"2780-2811"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9934-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37028597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-01-01Epub Date: 2017-07-05DOI: 10.1007/s12220-017-9888-y
Clara L Aldana, Julie Rowlett
We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.
{"title":"A Polyakov Formula for Sectors.","authors":"Clara L Aldana, Julie Rowlett","doi":"10.1007/s12220-017-9888-y","DOIUrl":"10.1007/s12220-017-9888-y","url":null,"abstract":"<p><p>We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"28 2","pages":"1773-1839"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9888-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37028900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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