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Width estimate and doubly warped product 宽度估计和双重翘曲产品
arXiv: Differential Geometry Pub Date : 2020-03-03 DOI: 10.1090/tran/8263
Jintian Zhu
{"title":"Width estimate and doubly warped product","authors":"Jintian Zhu","doi":"10.1090/tran/8263","DOIUrl":"https://doi.org/10.1090/tran/8263","url":null,"abstract":"In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $mathbf R^2times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in $3$-spheres with positive sectional curvatures.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91119120","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}
引用次数: 26
Collapsing Calabi-Yau manifolds 坍缩的Calabi-Yau流形
arXiv: Differential Geometry Pub Date : 2020-03-02 DOI: 10.4310/SDG.2018.v23.n1.a8
Valentino Tosatti
{"title":"Collapsing Calabi-Yau manifolds","authors":"Valentino Tosatti","doi":"10.4310/SDG.2018.v23.n1.a8","DOIUrl":"https://doi.org/10.4310/SDG.2018.v23.n1.a8","url":null,"abstract":"We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83604439","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}
引用次数: 11
Parabolic Frequency on Manifolds 流形上的抛物频率
arXiv: Differential Geometry Pub Date : 2020-02-25 DOI: 10.1093/IMRN/RNAB052
T. Colding, W. Minicozzi
{"title":"Parabolic Frequency on Manifolds","authors":"T. Colding, W. Minicozzi","doi":"10.1093/IMRN/RNAB052","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB052","url":null,"abstract":"We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on all manifolds and no curvature assumption is needed. When the manifold is Euclidean space and the drift operator is the Ornstein-Uhlenbeck operator this can been seen to imply Poon's frequency monotonicity for the ordinary heat equation. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three circles theorem about log convexity of holomorphic functions on $CC$. From the monotonicity, we get parabolic unique continuation and backward uniqueness.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82040852","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}
引用次数: 10
The derivative of global surface-holonomy for a non-abelian gerbe 非阿贝尔格布的整体曲面完整的导数
arXiv: Differential Geometry Pub Date : 2020-02-22 DOI: 10.1016/J.DIFGEO.2021.101737
Cheyne Glass
{"title":"The derivative of global surface-holonomy for a non-abelian gerbe","authors":"Cheyne Glass","doi":"10.1016/J.DIFGEO.2021.101737","DOIUrl":"https://doi.org/10.1016/J.DIFGEO.2021.101737","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77710793","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}
引用次数: 1
The Schwarzian derivative and conformal transformation on Finsler manifolds Finsler流形上的Schwarzian导数与保角变换
arXiv: Differential Geometry Pub Date : 2020-02-19 DOI: 10.4134/JKMS.J190436
B. Bidabad, Faranak Sedighi
{"title":"The Schwarzian derivative and conformal transformation on Finsler manifolds","authors":"B. Bidabad, Faranak Sedighi","doi":"10.4134/JKMS.J190436","DOIUrl":"https://doi.org/10.4134/JKMS.J190436","url":null,"abstract":"Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"256 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76179198","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}
引用次数: 1
ON PSEUDO-HERMITIAN MAGNETIC CURVES IN SASAKIAN MANIFOLDS 关于sasaki流形中的伪厄米特磁曲线
arXiv: Differential Geometry Pub Date : 2020-02-12 DOI: 10.22190/FUMI2005291G
S. Guvenc, Cihan Ozgur
{"title":"ON PSEUDO-HERMITIAN MAGNETIC CURVES IN SASAKIAN MANIFOLDS","authors":"S. Guvenc, Cihan Ozgur","doi":"10.22190/FUMI2005291G","DOIUrl":"https://doi.org/10.22190/FUMI2005291G","url":null,"abstract":"We define pseudo-Hermitian magnetic curves in Sasakian manifolds endowed with the Tanaka-Webster connection. After we give a complete classification theorem, we construct parametrizations of pseudo-Hermitian magnetic curves in $mathbb{R}^{2n+1}(-3)$.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79429934","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}
引用次数: 0
Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow 拉格朗日相算子的坦凹性及其在切线拉格朗日相流中的应用
arXiv: Differential Geometry Pub Date : 2020-02-12 DOI: 10.1142/s0129167x20501165
Ryosuke Takahashi
{"title":"Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow","authors":"Ryosuke Takahashi","doi":"10.1142/s0129167x20501165","DOIUrl":"https://doi.org/10.1142/s0129167x20501165","url":null,"abstract":"We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated $(1,1)$-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a $C$-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79445696","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}
引用次数: 15
Conformal infinitesimal variations of submanifolds
arXiv: Differential Geometry Pub Date : 2020-02-06 DOI: 10.1016/J.DIFGEO.2021.101721
M. Dajczer, M. I. Jimenez
{"title":"Conformal infinitesimal variations of submanifolds","authors":"M. Dajczer, M. I. Jimenez","doi":"10.1016/J.DIFGEO.2021.101721","DOIUrl":"https://doi.org/10.1016/J.DIFGEO.2021.101721","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72938599","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}
引用次数: 2
Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry 保形几何中完全非线性退化椭圆边值问题的非唯一性
arXiv: Differential Geometry Pub Date : 2020-02-01 DOI: 10.1016/j.difgeo.2020.101688
Zhengyang Shan
{"title":"Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry","authors":"Zhengyang Shan","doi":"10.1016/j.difgeo.2020.101688","DOIUrl":"https://doi.org/10.1016/j.difgeo.2020.101688","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87144319","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}
引用次数: 1
A survey on spectral embeddings and their application in data analysis 光谱嵌入及其在数据分析中的应用综述
arXiv: Differential Geometry Pub Date : 2020-01-31 DOI: 10.5802/tsg.369
David Tewodrose
{"title":"A survey on spectral embeddings and their application in data analysis","authors":"David Tewodrose","doi":"10.5802/tsg.369","DOIUrl":"https://doi.org/10.5802/tsg.369","url":null,"abstract":"The aim of this survey is to present some aspects of the Berard-Besson-Gallot spectral embeddings of a closed Riemannian manifold from their origins in Riemannian geometry to more recent applications in data analysis.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"193 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83493100","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}
引用次数: 2
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