We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term.
{"title":"Christoffel-Minkowski flows","authors":"Paul Bryan, Mohammad N. Ivaki, J. Scheuer","doi":"10.1090/tran/8683","DOIUrl":"https://doi.org/10.1090/tran/8683","url":null,"abstract":"We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"285 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79568208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civita connection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given.
{"title":"On 3-dimensional almost Einstein manifolds with circulant structures","authors":"Iva Dokuzova","doi":"10.3906/mat-1904-97","DOIUrl":"https://doi.org/10.3906/mat-1904-97","url":null,"abstract":"A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civita connection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79841501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-05DOI: 10.15330/cmp.13.1.110-118
Soumendu Roy, S. Dey, A. Bhattacharyya
The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $eta$-Einstein soliton. We have studied some certain properties of Kenmotsu manifold admitting conformal $eta$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $eta$-Einstein soliton.
{"title":"A Kenmotsu metric as a conformal $eta$-Einstein soliton","authors":"Soumendu Roy, S. Dey, A. Bhattacharyya","doi":"10.15330/cmp.13.1.110-118","DOIUrl":"https://doi.org/10.15330/cmp.13.1.110-118","url":null,"abstract":"The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $eta$-Einstein soliton. We have studied some certain properties of Kenmotsu manifold admitting conformal $eta$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $eta$-Einstein soliton.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78360715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gromov asked if the bi-invariant metric on an $n$ dimensional compact Lie group is extremal compared to any other metrics. In this note, we prove that the bi-invariant metric on an $n$ dimensional compact connected semi-simple Lie group $G$ is extremal in the sense of Gromov when compared to the left invariant metrics. In fact the same result holds for a compact connected homogeneous Riemannian manifold $G/H$ with the Lie algebra of $G$ having trivial center.
{"title":"Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces","authors":"Yukai Sun, X. Dai","doi":"10.3842/sigma.2020.068","DOIUrl":"https://doi.org/10.3842/sigma.2020.068","url":null,"abstract":"Gromov asked if the bi-invariant metric on an $n$ dimensional compact Lie group is extremal compared to any other metrics. \u0000In this note, we prove that the bi-invariant metric on an $n$ dimensional compact connected semi-simple Lie group $G$ is extremal in the sense of Gromov when compared to the left invariant metrics. In fact the same result holds for a compact connected homogeneous Riemannian manifold $G/H$ with the Lie algebra of $G$ having trivial center.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89570628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kahler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kahler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kahler metric whose set of minimal Lagrangian orbits is $S$?
{"title":"Minimal Lagrangian tori and action-angle coordinates","authors":"Gonccalo Oliveira, R. Sena-Dias","doi":"10.1090/tran/8403","DOIUrl":"https://doi.org/10.1090/tran/8403","url":null,"abstract":"We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kahler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kahler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kahler metric whose set of minimal Lagrangian orbits is $S$?","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90045494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a closed countably $1$-rectifiable set in $mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.
{"title":"Existence and regularity theorems of one-dimensional Brakke flows","authors":"Lami Kim, Y. Tonegawa","doi":"10.4171/ifb/448","DOIUrl":"https://doi.org/10.4171/ifb/448","url":null,"abstract":"Given a closed countably $1$-rectifiable set in $mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73926036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-16DOI: 10.1142/s1793525321500199
Zhengjiang Lin, Ao Sun
We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the generic perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the generic perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.
{"title":"Bifurcation of perturbations of non-generic closed self-shrinkers","authors":"Zhengjiang Lin, Ao Sun","doi":"10.1142/s1793525321500199","DOIUrl":"https://doi.org/10.1142/s1793525321500199","url":null,"abstract":"We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the generic perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the generic perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85701786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.
{"title":"Polar tangential angles and free elasticae","authors":"Tatsuya Miura","doi":"10.3934/MINE.2021034","DOIUrl":"https://doi.org/10.3934/MINE.2021034","url":null,"abstract":"In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86796384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-14DOI: 10.2140/PJM.2021.310.487
Taotao Zheng
We study the continuity equation of the Gauduchon metrics and establish its interval of maximal existence, which extends the continuity equation of the Kahler metrics introduced by La Nave & Tian for and of the Hermitian metrics introduced by Sherman & Weinkove. Our method is based on the solution to the Gauduchon conjecture by Szekelyhidi, Tosatti & Weinkove.
{"title":"The continuity equation of the Gauduchon metrics","authors":"Taotao Zheng","doi":"10.2140/PJM.2021.310.487","DOIUrl":"https://doi.org/10.2140/PJM.2021.310.487","url":null,"abstract":"We study the continuity equation of the Gauduchon metrics and establish its interval of maximal existence, which extends the continuity equation of the Kahler metrics introduced by La Nave & Tian for and of the Hermitian metrics introduced by Sherman & Weinkove. Our method is based on the solution to the Gauduchon conjecture by Szekelyhidi, Tosatti & Weinkove.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87361806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-08DOI: 10.1142/s0219199720500893
D. Alekseevsky, J. Gutt, G. Manno, G. Moreno
Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $mathcal{E} subset J^k$. We describe a general method for constructing such invariant PDEs for $kgeq 2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H^{(k-1)}$ of the $(k-1)$-prolonged action of $G$. We apply this approach to describe invariant PDEs for hypersurfaces in the Euclidean space $mathbb{E}^{n+1 }$ and in the conformal space $mathbb{S}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $J^{k-1}$, and A2) the stabilizer $H^{(k-1)}subset G$ of the fibre $J^kto J^{k-1}$ must factorize via the group of translations of the fibre itself.
{"title":"A general method to construct invariant PDEs on homogeneous manifolds","authors":"D. Alekseevsky, J. Gutt, G. Manno, G. Moreno","doi":"10.1142/s0219199720500893","DOIUrl":"https://doi.org/10.1142/s0219199720500893","url":null,"abstract":"Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $mathcal{E} subset J^k$. We describe a general method for constructing such invariant PDEs for $kgeq 2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H^{(k-1)}$ of the $(k-1)$-prolonged action of $G$. We apply this approach to describe invariant PDEs for hypersurfaces in the Euclidean space $mathbb{E}^{n+1 }$ and in the conformal space $mathbb{S}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: \u0000A1) the group $G$ must have an open orbit in $J^{k-1}$, and \u0000A2) the stabilizer $H^{(k-1)}subset G$ of the fibre $J^kto J^{k-1}$ must factorize via the group of translations of the fibre itself.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78537850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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