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":"8 1","pages":"0"},"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.n1.a1
Chuangxun Cheng
{"title":"On $pi$-divisible $mathcal{O}$-modules over fields of characteristic $p$","authors":"Chuangxun Cheng","doi":"10.4310/ajm.2023.v27.n1.a1","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n1.a1","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392032","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 : 2022-07-06DOI: 10.4310/ajm.2021.v25.n5.a6
V. V. Bavula, V. Bekkert, V. Futorny
For the algebra $mathbb{I}_n = K {langle x_1, dotsc, x_n, partial_1, dotsc, partial_n, int_1, dotsc, int_n rangle}$ of polynomial integrodifferential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $mathbb{I}_n$‑modules is given. It is proven that the category of weight $mathbb{I}_n$‑modules is semisimple. An explicit description of generalized weight $mathbb{I}_n$‑modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $mathbb{I}_n$‑modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $mathbb{I}_n$‑modules is given. In the wild case ‘natural‘ tame subcategories are considered with explicit description of indecomposable modules. For an arbitrary ring $R$, we introduce the concept of absolutely prime $R$‑module (a nonzero $R$‑module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is proven that every generalized weight $mathbb{I}_n$‑module is a unique sum of absolutely prime modules. It is also shown that every indecomposable generalized weight $mathbb{I}_n$‑module is equidimensional. A criterion is given for a generalized weight $mathbb{I}_n$‑module to be finitely generated.
{"title":"Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $mathbb{I}_n$","authors":"V. V. Bavula, V. Bekkert, V. Futorny","doi":"10.4310/ajm.2021.v25.n5.a6","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n5.a6","url":null,"abstract":"For the algebra $mathbb{I}_n = K {langle x_1, dotsc, x_n, partial_1, dotsc, partial_n, int_1, dotsc, int_n rangle}$ of polynomial integrodifferential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $mathbb{I}_n$‑modules is given. It is proven that the category of weight $mathbb{I}_n$‑modules is semisimple. An explicit description of generalized weight $mathbb{I}_n$‑modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $mathbb{I}_n$‑modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $mathbb{I}_n$‑modules is given. In the wild case ‘natural‘ tame subcategories are considered with explicit description of indecomposable modules. For an arbitrary ring $R$, we introduce the concept of <i>absolutely prime</i> $R$‑module (a nonzero $R$‑module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is proven that every generalized weight $mathbb{I}_n$‑module is a unique sum of absolutely prime modules. It is also shown that every indecomposable generalized weight $mathbb{I}_n$‑module is equidimensional. A criterion is given for a generalized weight $mathbb{I}_n$‑module to be finitely generated.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538226","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 : 2022-07-06DOI: 10.4310/ajm.2021.v25.n5.a2
Mingyi Zhang
We give an explicit formula for the Hodge filtration on the $mathscr{D}_X$-module $mathcal{O}_X (*Z) f^{1-alpha}$ associated to the effective $mathbb{Q}$-divisor $D = alpha cdot Z$, where $0 lt alpha leq 1$ and $Z = (f = 0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic function that is convenient and has non-degenerate Newton boundary.
{"title":"Hodge filtration and Hodge ideals for $mathbb{Q}$-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities","authors":"Mingyi Zhang","doi":"10.4310/ajm.2021.v25.n5.a2","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n5.a2","url":null,"abstract":"We give an explicit formula for the Hodge filtration on the $mathscr{D}_X$-module $mathcal{O}_X (*Z) f^{1-alpha}$ associated to the effective $mathbb{Q}$-divisor $D = alpha cdot Z$, where $0 lt alpha leq 1$ and $Z = (f = 0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic function that is convenient and has non-degenerate Newton boundary.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"198 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538191","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 : 2022-07-06DOI: 10.4310/ajm.2021.v25.n5.a5
Dmitri Burago, Dong Chen
We show that given any Liouville direction and flat Finsler torus, one can make a $C^infty$‑small perturbation on an arbitrarily small disc to get a nondense geodesic in the given direction.
{"title":"Finsler perturbation with nondense geodesics with irrational directions","authors":"Dmitri Burago, Dong Chen","doi":"10.4310/ajm.2021.v25.n5.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n5.a5","url":null,"abstract":"We show that given any Liouville direction and flat Finsler torus, one can make a $C^infty$‑small perturbation on an arbitrarily small disc to get a nondense geodesic in the given direction.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"198 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538202","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 : 2022-04-18DOI: 10.4310/ajm.2021.v25.n6.a5
Xu Xu, Chao Zheng
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo [18], who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo’s conjecture was solved by Bobenko-Pinkall-Springborn [3], who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn’s proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in 3-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. In this paper, we give an elementary and short variational proof of the global rigidity of vertex scaling without involving 3-dimensional hyperbolic geometry. The method is based on continuity of eigenvalues of matrices and the extension of convex functions.
{"title":"A new proof for global rigidity of vertex scaling on polyhedral surfaces","authors":"Xu Xu, Chao Zheng","doi":"10.4310/ajm.2021.v25.n6.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n6.a5","url":null,"abstract":"The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo [18], who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo’s conjecture was solved by Bobenko-Pinkall-Springborn [3], who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn’s proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in 3-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. In this paper, we give an elementary and short variational proof of the global rigidity of vertex scaling without involving 3-dimensional hyperbolic geometry. The method is based on continuity of eigenvalues of matrices and the extension of convex functions.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47416504","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 : 2022-01-01DOI: 10.4310/ajm.2022.v26.n4.a1
M. Alim, Florian Beck, Laura Fredrickson
{"title":"Parabolic Higgs bundles, $tt^ast$ connections and opers","authors":"M. Alim, Florian Beck, Laura Fredrickson","doi":"10.4310/ajm.2022.v26.n4.a1","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n4.a1","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391965","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 : 2022-01-01DOI: 10.4310/ajm.2022.v26.n3.a2
Sören Kleine
{"title":"On the Iwasawa invariants of non-cotorsion Selmer groups","authors":"Sören Kleine","doi":"10.4310/ajm.2022.v26.n3.a2","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n3.a2","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391952","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 : 2022-01-01DOI: 10.4310/ajm.2022.v26.n5.a1
P. Giblin, Graham M. Reeve
{"title":"Contact of circles with surfaces: Answers to a question of Montaldi","authors":"P. Giblin, Graham M. Reeve","doi":"10.4310/ajm.2022.v26.n5.a1","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n5.a1","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392021","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 : 2022-01-01DOI: 10.4310/ajm.2022.v26.n2.a1
M. Jotz
{"title":"Obstructions to representations up to homotopy and ideals","authors":"M. Jotz","doi":"10.4310/ajm.2022.v26.n2.a1","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n2.a1","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392338","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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