Pub Date : 2023-01-01DOI: 10.4310/ajm.2023.v27.n1.a5
Kyle Broder
{"title":"The Schwarz lemma in Kähler and non-Kähler geometry","authors":"Kyle Broder","doi":"10.4310/ajm.2023.v27.n1.a5","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n1.a5","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392109","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n3.a4
Ashay Burungale, Laurent Clozel
The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if this ring in Noetherian (a natural assumption considered by Hida) it is free over the ring of Witt vectors of $k$. This however imposes natural conditions on certain $mu$-invariants.
{"title":"Ordinary deformations are unobstructed in the cyclotomic limit","authors":"Ashay Burungale, Laurent Clozel","doi":"10.4310/ajm.2023.v27.n3.a4","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n3.a4","url":null,"abstract":"The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if this ring in Noetherian (a natural assumption considered by Hida) it is free over the ring of Witt vectors of $k$. This however imposes natural conditions on certain $mu$-invariants.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783348","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n3.a2
Kota Hattori
We study the geometric quantization on $K3$ surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the $K3$ surfaces and a family of hyper-Kahler structures tending to large complex structure limit, and show a spectral convergence of the $bar{partial}$-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.
{"title":"Spectral convergence in geometric quantization on $K3$ surfaces","authors":"Kota Hattori","doi":"10.4310/ajm.2023.v27.n3.a2","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n3.a2","url":null,"abstract":"We study the geometric quantization on $K3$ surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the $K3$ surfaces and a family of hyper-Kahler structures tending to large complex structure limit, and show a spectral convergence of the $bar{partial}$-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135508160","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n2.a4
Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo
{"title":"A criteria for classification of weighted dual graphs of singularities and its application","authors":"Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo","doi":"10.4310/ajm.2023.v27.n2.a4","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n2.a4","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302919","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}
{"title":"Terracini locus for three points on a Segre variety","authors":"Edoardo Ballico, Alessandra Bernardi, Pierpaola Santarsiero","doi":"10.4310/ajm.2023.v27.n3.a3","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n3.a3","url":null,"abstract":"We introduce the notion of r-th Terracini locus of a variety and we compute it for at most three points on a Segre variety.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784449","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n2.a3
King Leung Lee
In this short note, we study some coupled metric equations in terms of moment map. As a consequence, we will give an alternate moment map picture of coupled cscK equation and a special case of coupled Kahler Yang-Mill's equation.
{"title":"Moment map for coupled equations of Kähler forms and curvature","authors":"King Leung Lee","doi":"10.4310/ajm.2023.v27.n2.a3","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n2.a3","url":null,"abstract":"In this short note, we study some coupled metric equations in terms of moment map. As a consequence, we will give an alternate moment map picture of coupled cscK equation and a special case of coupled Kahler Yang-Mill's equation.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302922","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n1.a4
Xiuqiong Chen, S. Yau
{"title":"On the stability of linear feedback particle filter","authors":"Xiuqiong Chen, S. Yau","doi":"10.4310/ajm.2023.v27.n1.a4","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n1.a4","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392100","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n2.a1
Gregory Arone, Vyacheslav Krushkal
Given an $m$-dimensional CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a Euclidean space ${mathbb R}^d$. For $2$-complexes in ${mathbb R}^4$ a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. The focus in this paper is on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case we show the two approaches give the same result, and also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.
{"title":"Embedding obstructions in $mathbb{R}^d$ from the Goodwillie–Weiss calculus and Whitney disks","authors":"Gregory Arone, Vyacheslav Krushkal","doi":"10.4310/ajm.2023.v27.n2.a1","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n2.a1","url":null,"abstract":"Given an $m$-dimensional CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a Euclidean space ${mathbb R}^d$. For $2$-complexes in ${mathbb R}^4$ a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. The focus in this paper is on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case we show the two approaches give the same result, and also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181171","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n2.a2
Sven Hirsch, Martin Man-Chun Li
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.
{"title":"Contracting convex surfaces by mean curvature flow with free boundary on convex barriers","authors":"Sven Hirsch, Martin Man-Chun Li","doi":"10.4310/ajm.2023.v27.n2.a2","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n2.a2","url":null,"abstract":"We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302921","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 : 2023-01-01DOI: 10.4310/ajm.2023.v27.n3.a1
Tang-Kai Lee
The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.
{"title":"Compactness and rigidity of self-shrinking surfaces","authors":"Tang-Kai Lee","doi":"10.4310/ajm.2023.v27.n3.a1","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n3.a1","url":null,"abstract":"The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135508445","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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