Pub Date : 2020-05-24DOI: 10.4310/ajm.2021.v25.n3.a4
Teng Huang
In this article, we introduce and study the notion of a complete special holonomy manifold $(X,omega)$ which is given by a global perturbation potential function, i.e., there are a function $f$ and a smooth differential $omega'$ on $X$ such that $omega=mathcal{L}_{nabla f}omega+omega'$. We establish some vanishing theorems on the $L^{2}$ harmonic forms under some conditions on the global perturbation potential function.
{"title":"Global perturbation potential function on complete special holonomy manifolds","authors":"Teng Huang","doi":"10.4310/ajm.2021.v25.n3.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n3.a4","url":null,"abstract":"In this article, we introduce and study the notion of a complete special holonomy manifold $(X,omega)$ which is given by a global perturbation potential function, i.e., there are a function $f$ and a smooth differential $omega'$ on $X$ such that $omega=mathcal{L}_{nabla f}omega+omega'$. We establish some vanishing theorems on the $L^{2}$ harmonic forms under some conditions on the global perturbation potential function.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47126948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-20DOI: 10.4310/ajm.2021.v25.n2.a1
Jianing Li
In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $mathbb{Q}(sqrt[4]{-q}) $, where $qequiv 3bmod 4$ is a prime. When $qequiv 15bmod 16$, this result confirms a speculation of Coates-Li and has consequences for certain Iwasawa modules arising in their work.
{"title":"On the $2$-adic logarithm of units of certain totally imaginary quartic fields","authors":"Jianing Li","doi":"10.4310/ajm.2021.v25.n2.a1","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a1","url":null,"abstract":"In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $mathbb{Q}(sqrt[4]{-q}) $, where $qequiv 3bmod 4$ is a prime. When $qequiv 15bmod 16$, this result confirms a speculation of Coates-Li and has consequences for certain Iwasawa modules arising in their work.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44050870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-01DOI: 10.4310/ajm.2021.v25.n1.a4
R. Laterveer
We prove Bloch's conjecture for numerical Campedelli surfaces with fundamental group of order $9$.
我们证明了具有$9阶基群的数值Campedelli曲面的Bloch猜想。
{"title":"Bloch’s conjecture for some numerical Campedelli surfaces","authors":"R. Laterveer","doi":"10.4310/ajm.2021.v25.n1.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n1.a4","url":null,"abstract":"We prove Bloch's conjecture for numerical Campedelli surfaces with fundamental group of order $9$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47140031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-01DOI: 10.4310/ajm.2022.v26.n2.a5
Xiaoman Chen, Hongzhi Liu, Han Wang, Guoliang Yu
In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.
{"title":"Higher rho invariant and delocalized eta invariant at infinity","authors":"Xiaoman Chen, Hongzhi Liu, Han Wang, Guoliang Yu","doi":"10.4310/ajm.2022.v26.n2.a5","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n2.a5","url":null,"abstract":"In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49341426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-05DOI: 10.4310/ajm.2022.v26.n5.a5
B. Stratmann
Let $M$ be a manifold with an exact symplectic form $omega$. Then there is a nowhere vanishing primitive $beta$ for $omega$, i.e. $omega=mathrm{d}beta$.
{"title":"Nowhere vanishing primitive of a symplectic form","authors":"B. Stratmann","doi":"10.4310/ajm.2022.v26.n5.a5","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n5.a5","url":null,"abstract":"Let $M$ be a manifold with an exact symplectic form $omega$. Then there is a nowhere vanishing primitive $beta$ for $omega$, i.e. $omega=mathrm{d}beta$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42905430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-09DOI: 10.4310/ajm.2021.v25.n3.a1
Karl K. Brustad
We present an analytic approach on how to solve the problem $|nabla u|=f(u)$, $Delta u = g(u)$, in connected domains $Omegasubseteqmathbb{R}^n$.
我们提出了一种解析方法来解决连接域$Omegasubseteqmathbb{R}^n$中的问题$|nabla u|=f(u)$, $Delta u = g(u)$。
{"title":"Segre’s theorem. An analytic proof of a result in differential geometry","authors":"Karl K. Brustad","doi":"10.4310/ajm.2021.v25.n3.a1","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n3.a1","url":null,"abstract":"We present an analytic approach on how to solve the problem $|nabla u|=f(u)$, $Delta u = g(u)$, in connected domains $Omegasubseteqmathbb{R}^n$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44027893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/ajm.2020.v24.n2.a5
Vlad-Cristian Crisan, Katharina Müller
{"title":"The vanishing of the $mu$-invariant for split prime $mathbb{Z}_p$-extensions over imaginary quadratic fields","authors":"Vlad-Cristian Crisan, Katharina Müller","doi":"10.4310/ajm.2020.v24.n2.a5","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n2.a5","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/ajm.2020.v24.n3.a5
Hsiao-Fan Liu
{"title":"The star mean curvature flow on 3-sphere and hyperbolic 3-space","authors":"Hsiao-Fan Liu","doi":"10.4310/ajm.2020.v24.n3.a5","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n3.a5","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-29DOI: 10.4310/AJM.2021.v25.n6.a6
F. Podestà, Alberto Raffero
. We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G 2 -structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.
{"title":"Closed $mathrm{G}_2$-structures with a transitive reductive group of automorphisms","authors":"F. Podestà, Alberto Raffero","doi":"10.4310/AJM.2021.v25.n6.a6","DOIUrl":"https://doi.org/10.4310/AJM.2021.v25.n6.a6","url":null,"abstract":". We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G 2 -structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42635544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-09DOI: 10.4310/ajm.2021.v25.n1.a5
L. Borisov, Z. Han, Chengxi Wang
We study a pair of conjectures on better behaved GKZ hypergeometric systems of PDEs inspired by Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities. We prove the conjectures in the case of dimension two.
{"title":"On duality of certain GKZ hypergeometric systems","authors":"L. Borisov, Z. Han, Chengxi Wang","doi":"10.4310/ajm.2021.v25.n1.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n1.a5","url":null,"abstract":"We study a pair of conjectures on better behaved GKZ hypergeometric systems of PDEs inspired by Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities. We prove the conjectures in the case of dimension two.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42092575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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