Pub Date : 2020-04-06DOI: 10.2422/2036-2145.202005_021
Zakarias Sjostrom Dyrefelt
Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric in at least one Kahler class $[omega] in H^{1,1}(X,mathbb{R})$. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song (arXiv:1805.06863) and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture.
{"title":"Existence of cscK metrics on smooth minimal models","authors":"Zakarias Sjostrom Dyrefelt","doi":"10.2422/2036-2145.202005_021","DOIUrl":"https://doi.org/10.2422/2036-2145.202005_021","url":null,"abstract":"Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric in at least one Kahler class $[omega] in H^{1,1}(X,mathbb{R})$. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song (arXiv:1805.06863) and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86572523","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 fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.
{"title":"A steady length function for Ricci flows","authors":"J. Jordan","doi":"10.1090/proc/15202","DOIUrl":"https://doi.org/10.1090/proc/15202","url":null,"abstract":"A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"54 5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79466210","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 paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation of geometric curvature tensors under $*-$ Ricci flow. In the last two section of the paper, we have found the $Im$-functional and $omega-$ functional for $*-$ Ricci flow respectively.
{"title":"SOME RESULTS ON ∗−RICCI FLOW","authors":"Dipankar Debnath, Nirabhra Basu","doi":"10.22190/FUMI2005305D","DOIUrl":"https://doi.org/10.22190/FUMI2005305D","url":null,"abstract":"In this paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation of geometric curvature tensors under $*-$ Ricci flow. In the last two section of the paper, we have found the $Im$-functional and $omega-$ functional for $*-$ Ricci flow respectively.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76152057","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 : 2020-04-01DOI: 10.22034/KJM.2020.235131.1873
D. Dey, P. Majhi
In the present paper, we characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (HPCV) fields. We have shown that if an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field $V$ such that $phi V = 0$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. As a corollary of this we obtain few classifications of an almost Kenmotsu manifold to be a Kenmotsu manifold and also prove that the integral manifolds of D are totally umbilical submanifolds of $M^{2n+1}$. Further, we prove that if an almost Kenmotsu manifold with positive constant $xi$-sectional curvature admits a non-zero HPCV field $V$, then either $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval or isometric to a sphere. Moreover, a $(k,mu)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $phi V = 0$ is either locally isometric to $mathbb{H}^{n+1}(-4) times mathbb{R}^n$ or $V$ is an eigenvector of $h'$. Finally, an example is presented.
在本文中，我们刻画了具有全纯平面共形矢量场的几乎Kenmotsu流形。我们证明了如果一个几乎Kenmotsu流形$M^{2n+1}$允许一个非零HPCV场$V$使得$phi V = 0$，那么$M^{2n+1}$是一个几乎Kaehler流形与开区间的局部翘曲积。作为这一结论的推论，我们得到了几乎Kenmotsu流形为Kenmotsu流形的几个分类，并证明了D的积分流形是$M^{2n+1}$的完全脐带子流形。进一步证明了如果一个具有正常数$xi$ -截面曲率的几乎Kenmotsu流形存在一个非零HPCV场$V$，那么$M^{2n+1}$要么是一个几乎Kaehler流形与球面的开区间或等距的局部翘曲积。此外，承认HPCV场$V$的$(k,mu)'$ -几乎Kenmotsu流形使得$phi V = 0$与$mathbb{H}^{n+1}(-4) times mathbb{R}^n$局部等距或$V$是$h'$的特征向量。最后给出了一个实例。
{"title":"Almost Kenmotsu manifolds admitting certain vector fields","authors":"D. Dey, P. Majhi","doi":"10.22034/KJM.2020.235131.1873","DOIUrl":"https://doi.org/10.22034/KJM.2020.235131.1873","url":null,"abstract":"In the present paper, we characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (HPCV) fields. We have shown that if an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field $V$ such that $phi V = 0$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. As a corollary of this we obtain few classifications of an almost Kenmotsu manifold to be a Kenmotsu manifold and also prove that the integral manifolds of D are totally umbilical submanifolds of $M^{2n+1}$. Further, we prove that if an almost Kenmotsu manifold with positive constant $xi$-sectional curvature admits a non-zero HPCV field $V$, then either $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval or isometric to a sphere. Moreover, a $(k,mu)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $phi V = 0$ is either locally isometric to $mathbb{H}^{n+1}(-4) times mathbb{R}^n$ or $V$ is an eigenvector of $h'$. Finally, an example is presented.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80663449","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 : 2020-03-31DOI: 10.1016/j.na.2020.112174
Paul Bryan, Mohammad N. Ivaki, Julian Scheuer
{"title":"Parabolic approaches to curvature equations","authors":"Paul Bryan, Mohammad N. Ivaki, Julian Scheuer","doi":"10.1016/j.na.2020.112174","DOIUrl":"https://doi.org/10.1016/j.na.2020.112174","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80617867","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}
On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kaehler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for all eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
{"title":"Bounding the invariant spectrum when the\u0000 scalar curvature is non-negative","authors":"Stuart J. Hall, T. Murphy","doi":"10.1090/conm/756/15202","DOIUrl":"https://doi.org/10.1090/conm/756/15202","url":null,"abstract":"On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kaehler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for all eigenvalues of the invariant spectrum assuming non-negative scalar curvature.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74439521","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 paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm compares favorably with the widely applied Scale-Invariant Feature Transform (SIFT), Speeded Up Robust Features (SURF), and Affine-SIFT (ASIFT) methods.
{"title":"Feature Matching and Heat Flow in Centro-Affine Geometry","authors":"P. Olver, C. Qu, Yun Yang","doi":"10.3842/SIGMA.2020.093","DOIUrl":"https://doi.org/10.3842/SIGMA.2020.093","url":null,"abstract":"In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm compares favorably with the widely applied Scale-Invariant Feature Transform (SIFT), Speeded Up Robust Features (SURF), and Affine-SIFT (ASIFT) methods.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80302983","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 study the sub-Laplacian of the $15$-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space $mathbb{O}H^1$. We also obtain two integral representations for the corresponding subelliptic heat kernel.
{"title":"The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration","authors":"Fabrice Baudoin, Gunhee Cho","doi":"10.3842/SIGMA.2021.014","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.014","url":null,"abstract":"In this note, we study the sub-Laplacian of the $15$-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space $mathbb{O}H^1$. We also obtain two integral representations for the corresponding subelliptic heat kernel.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"2015 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87804789","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 paper, we define and classify minimal translation surfaces with respect to a kind of semi-symmetric metric connections and a kind of semi-symmetric non-metric connections in $mathbb{R}^3$ and $mathbb{R}^3_1$.
{"title":"Minimal translation surfaces with respect to semi-symmetric connections in $mathbb{R}^3$ and $mathbb{R}^3_1$","authors":"Yong Wang","doi":"10.4134/BKMS.B200732","DOIUrl":"https://doi.org/10.4134/BKMS.B200732","url":null,"abstract":"In this paper, we define and classify minimal translation surfaces with respect to a kind of semi-symmetric metric connections and a kind of semi-symmetric non-metric connections in $mathbb{R}^3$ and $mathbb{R}^3_1$.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79857341","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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