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Abundance and SYZ conjecture in families of hyperkahler manifolds 超卡勒流形族中的丰度和 SYZ 猜想
arXiv - MATH - Differential Geometry Pub Date : 2024-09-13 DOI: arxiv-2409.09142
Andrey Soldatenkov, Misha Verbitsky
{"title":"Abundance and SYZ conjecture in families of hyperkahler manifolds","authors":"Andrey Soldatenkov, Misha Verbitsky","doi":"arxiv-2409.09142","DOIUrl":"https://doi.org/arxiv-2409.09142","url":null,"abstract":"Let $L$ be a holomorphic line bundle on a hyperkahler manifold $M$, with\u0000$c_1(L)$ nef and not big. SYZ conjecture predicts that $L$ is semiample. We\u0000prove that this is true, assuming that $(M,L)$ has a deformation $(M',L')$ with\u0000$L'$ semiample. We introduce a version of the Teichmuller space that\u0000parametrizes pairs $(M,L)$ up to isotopy. We prove a version of the global\u0000Torelli theorem for such Teichmuller spaces and use it to deduce the\u0000deformation invariance of semiampleness.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254100","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
Operator $Δ-aS$ on warped product manifolds 翘积流形上的算子 $Δ-aS$
arXiv - MATH - Differential Geometry Pub Date : 2024-09-13 DOI: arxiv-2409.08818
Ezequiel Barbosa, Mateus Souza, Celso Viana
{"title":"Operator $Δ-aS$ on warped product manifolds","authors":"Ezequiel Barbosa, Mateus Souza, Celso Viana","doi":"arxiv-2409.08818","DOIUrl":"https://doi.org/arxiv-2409.08818","url":null,"abstract":"In this work we studied the stability of the family of operators\u0000$L_a=Delta-aS$, $ainmathbb R$, in a warped product of an infinite interval\u0000or real line by one compact manifold, where $Delta$ is the Laplacian and $S$\u0000is the scalar curvature of the resulting manifold.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254099","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
The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields 流形上正对称张量场的迈克尔-西蒙-索博列夫不等式
arXiv - MATH - Differential Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.08011
Yuting Wu, Chengyang Yi, Yu Zheng
{"title":"The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields","authors":"Yuting Wu, Chengyang Yi, Yu Zheng","doi":"arxiv-2409.08011","DOIUrl":"https://doi.org/arxiv-2409.08011","url":null,"abstract":"We prove the Michael-Simon-Sobolev inequality for smooth symmetric uniformly\u0000positive definite (0, 2)-tensor fields on compact submanifolds with or without\u0000boundary in Riemannian manifolds with nonnegative sectional curvature by the\u0000Alexandrov-Bakelman-Pucci (ABP) method. It should be a generalization of S.\u0000Brendle in [2].","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198968","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
Nonexistence of closed timelike geodesics in Kerr spacetimes 克尔空间中不存在封闭的时间似大地线
arXiv - MATH - Differential Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.09094
Giulio Sanzeni
{"title":"Nonexistence of closed timelike geodesics in Kerr spacetimes","authors":"Giulio Sanzeni","doi":"arxiv-2409.09094","DOIUrl":"https://doi.org/arxiv-2409.09094","url":null,"abstract":"The Kerr-star spacetime is the extension over the horizons and in the\u0000negative radial region of the Kerr spacetime. Despite the presence of closed\u0000timelike curves below the inner horizon, we prove that the timelike geodesics\u0000cannot be closed in the Kerr-star spacetime. Since the existence of closed null\u0000geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)],\u0000this result shows the absence of closed causal geodesics in the Kerr-star\u0000spacetime.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254098","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
A functional for Spin(7) forms Spin(7) 形式的函数
arXiv - MATH - Differential Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.08274
Calin Iuliu Lazaroiu, C. S. Shahbazi
{"title":"A functional for Spin(7) forms","authors":"Calin Iuliu Lazaroiu, C. S. Shahbazi","doi":"arxiv-2409.08274","DOIUrl":"https://doi.org/arxiv-2409.08274","url":null,"abstract":"We characterize the set of all conformal Spin(7) forms on an oriented and\u0000spin Riemannian eight-manifold $(M,g)$ as solutions to a homogeneous algebraic\u0000equation of degree two for the self-dual four-forms of $(M,g)$. When $M$ is\u0000compact, we use this result to construct a functional whose self-dual critical\u0000set is precisely the set of all Spin(7) structures on $M$. Furthermore, the\u0000natural coupling of this potential to the Einstein-Hilbert action gives a\u0000functional whose self-dual critical points are conformally Ricci-flat Spin(7)\u0000structures. Our proof relies on the computation of the square of an irreducible\u0000and chiral real spinor as a section of a bundle of real algebraic varieties\u0000sitting inside the K\"ahler-Atiyah bundle of $(M,g)$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198967","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
A review of compact geodesic orbit manifolds and the g.o. condition for $SU(5)/s(U(2)times U(2))$ 紧凑大地轨道流形和$SU(5)/s(U(2)times U(2))$的g.o.条件回顾
arXiv - MATH - Differential Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.08247
Andreas Arvanitoyeorgos, Nikolaos Panagiotis Souris, Marina Statha
{"title":"A review of compact geodesic orbit manifolds and the g.o. condition for $SU(5)/s(U(2)times U(2))$","authors":"Andreas Arvanitoyeorgos, Nikolaos Panagiotis Souris, Marina Statha","doi":"arxiv-2409.08247","DOIUrl":"https://doi.org/arxiv-2409.08247","url":null,"abstract":"Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds\u0000$(M,g)$ whose geodesics are integral curves of Killing vector fields.\u0000Equivalently, there exists a Lie group $G$ of isometries of $(M,g)$ such that\u0000any geodesic $gamma$ has the simple form $gamma(t)=e^{tX}cdot p$, where $e$\u0000denotes the exponential map on $G$. The classification of g.o. manifolds is a\u0000longstanding problem in Riemannian geometry. In this brief survey, we present\u0000some recent results and open questions on the subject focusing on the compact\u0000case. In addition we study the geodesic orbit condition for the space\u0000$SU(5)/s(U(2)times U(2))$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198972","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
Hypersurfaces of $mathbb{S}^3 times mathbb{R}$ and $mathbb{H}^3 times mathbb{R}$ with constant principal curvatures 主曲率恒定的 $mathbb{S}^3 times mathbb{R}$ 和 $mathbb{H}^3 times mathbb{R}$ 的超曲面
arXiv - MATH - Differential Geometry Pub Date : 2024-09-12 DOI: arxiv-2409.07978
Fernando Manfio, João Batista Marques dos Santos, João Paulo dos Santos, Joeri Van der Veken
{"title":"Hypersurfaces of $mathbb{S}^3 times mathbb{R}$ and $mathbb{H}^3 times mathbb{R}$ with constant principal curvatures","authors":"Fernando Manfio, João Batista Marques dos Santos, João Paulo dos Santos, Joeri Van der Veken","doi":"arxiv-2409.07978","DOIUrl":"https://doi.org/arxiv-2409.07978","url":null,"abstract":"We classify the hypersurfaces of $mathbb{Q}^3timesmathbb{R}$ with three\u0000distinct constant principal curvatures, where $varepsilon in {1,-1}$ and\u0000$mathbb{Q}^3$ denotes the unit sphere $mathbb{S}^3$ if $varepsilon = 1$,\u0000whereas it denotes the hyperbolic space $mathbb{H}^3$ if $varepsilon = -1$.\u0000We show that they are cylinders over isoparametric surfaces in $mathbb{Q}^3$,\u0000filling an intriguing gap in the existing literature. We also prove that the\u0000hypersurfaces with constant principal curvatures of\u0000$mathbb{Q}^3timesmathbb{R}$ are isoparametric. Furthermore, we provide the\u0000complete classification of the extrinsically homogeneous hypersurfaces in\u0000$mathbb{Q}^3timesmathbb{R}$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198970","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
Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds 非紧凑流形中规定平均曲率超曲面的最小-最大构造
arXiv - MATH - Differential Geometry Pub Date : 2024-09-11 DOI: arxiv-2409.07330
Douglas Stryker
{"title":"Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds","authors":"Douglas Stryker","doi":"arxiv-2409.07330","DOIUrl":"https://doi.org/arxiv-2409.07330","url":null,"abstract":"We develop a min-max theory for hypersurfaces of prescribed mean curvature in\u0000noncompact manifolds, applicable to prescription functions that do not change\u0000sign outside a compact set. We use this theory to prove new existence results\u0000for closed prescribed mean curvature hypersurfaces in Euclidean space and\u0000complete finite area constant mean curvature hypersurfaces in finite volume\u0000manifolds.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198971","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
On The Triviality Of $m$-Modified Conformal Vector Fields 论m$修正共形矢量场的琐碎性
arXiv - MATH - Differential Geometry Pub Date : 2024-09-11 DOI: arxiv-2409.07607
Rahul Poddar, Ramesh Sharma
{"title":"On The Triviality Of $m$-Modified Conformal Vector Fields","authors":"Rahul Poddar, Ramesh Sharma","doi":"arxiv-2409.07607","DOIUrl":"https://doi.org/arxiv-2409.07607","url":null,"abstract":"We prove that a compact Riemannian manifold $M$ does not admit any\u0000non-trivial $m$-modified homothetic vector fields. In the corresponding case of\u0000an $m$-modified conformal vector field $V$, we establish an inequality that\u0000implies the triviality of $V$. Further, we demonstrate that an affine Killing\u0000$m$-modified conformal vector field on a non-compact Riemannian manifold $M$\u0000must be trivial. Finally, we show that an $m$-modified gradient conformal\u0000vector field is trivial under the assumptions of polynomial volume growth and\u0000convergence to zero at infinity.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198969","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
Renormalized Yang-Mills Energy on Poincaré-Einstein Manifolds 波因卡内-爱因斯坦流形上的重正化杨-米尔斯能量
arXiv - MATH - Differential Geometry Pub Date : 2024-09-11 DOI: arxiv-2409.06995
A. R. Gover, E. Latini, A. Waldron, Y. Zhang
{"title":"Renormalized Yang-Mills Energy on Poincaré-Einstein Manifolds","authors":"A. R. Gover, E. Latini, A. Waldron, Y. Zhang","doi":"arxiv-2409.06995","DOIUrl":"https://doi.org/arxiv-2409.06995","url":null,"abstract":"We prove that the renormalized Yang-Mills energy on six dimensional\u0000Poincar'e-Einstein spaces can be expressed as the bulk integral of a local,\u0000pointwise conformally invariant integrand. We show that the latter agrees with\u0000the corresponding anomaly boundary integrand in the seven dimensional\u0000renormalized Yang-Mills energy. Our methods rely on a generalization of the\u0000Chang-Qing-Yang method for computing renormalized volumes of\u0000Poincar'e-Einstein manifolds, as well as known scattering theory results for\u0000Schr\"odinger operators with short range potentials.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199034","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
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