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Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations 时空调和函数和爱因斯坦方程的三维渐近平坦初始数据的质量
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2020-02-04 DOI: 10.4310/jdg/1669998184
Sven Hirsch, Demetre Kazaras, M. Khuri
{"title":"Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations","authors":"Sven Hirsch, Demetre Kazaras, M. Khuri","doi":"10.4310/jdg/1669998184","DOIUrl":"https://doi.org/10.4310/jdg/1669998184","url":null,"abstract":"We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48166512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 29
The Weyl problem in warped product spaces 扭曲积空间中的Weyl问题
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2020-02-01 DOI: 10.4310/jdg/1580526016
Chunhe Li, Zhizhang Wang
{"title":"The Weyl problem in warped product spaces","authors":"Chunhe Li, Zhizhang Wang","doi":"10.4310/jdg/1580526016","DOIUrl":"https://doi.org/10.4310/jdg/1580526016","url":null,"abstract":"In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the standard sphere if we fixed its geometric center in the ambient space. Finally, we discuss a Shi-Tam type of inequality for the Schwarzschild manifold as an application of our findings.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"114 1","pages":"243-304"},"PeriodicalIF":2.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47159225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
A priori estimates of Toda systems, I: the Lie algebras of $mathbf{A}_n$, $mathbf{B}_n$, $mathbf{C}_n$ and $mathbf{G}_2$ Toda系统的先验估计I: $mathbf{A}_n$， $mathbf{B}_n$， $mathbf{C}_n$和$mathbf{G}_2$的李代数
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2020-02-01 DOI: 10.4310/jdg/1580526018
Changshou Lin, Wen Yang, X. Zhong
{"title":"A priori estimates of Toda systems, I: the Lie algebras of $mathbf{A}_n$, $mathbf{B}_n$, $mathbf{C}_n$ and $mathbf{G}_2$","authors":"Changshou Lin, Wen Yang, X. Zhong","doi":"10.4310/jdg/1580526018","DOIUrl":"https://doi.org/10.4310/jdg/1580526018","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"114 1","pages":"337-391"},"PeriodicalIF":2.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43827779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 12
Cohomology of contact loci 接触位点的同调
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-11-19 DOI: 10.4310/jdg/1649953456
Nero Budur, J. F. Bobadilla, Q. Lê, H. Nguyen
{"title":"Cohomology of contact loci","authors":"Nero Budur, J. F. Bobadilla, Q. Lê, H. Nguyen","doi":"10.4310/jdg/1649953456","DOIUrl":"https://doi.org/10.4310/jdg/1649953456","url":null,"abstract":"We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of the their tangent cones are homotopy equivalent.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43882131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds 无限能量等变谐波映射，支配和反德西特$3 -流形
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-11-16 DOI: 10.4310/jdg/1689262064
Nathaniel Sagman
{"title":"Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds","authors":"Nathaniel Sagman","doi":"10.4310/jdg/1689262064","DOIUrl":"https://doi.org/10.4310/jdg/1689262064","url":null,"abstract":"We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. \u0000We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichm{\"u}ller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter $3$-manifolds. We also present an application to the theory of minimal immersions into the Grassmanian of timelike planes in $mathbb{R}^{2,2}$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43155045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Codimension two holomorphic foliation 余维二全纯叶理
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-11-01 DOI: 10.4310/jdg/1573786970
D. Cerveau, A. L. Neto
{"title":"Codimension two holomorphic foliation","authors":"D. Cerveau, A. L. Neto","doi":"10.4310/jdg/1573786970","DOIUrl":"https://doi.org/10.4310/jdg/1573786970","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42929649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Extended Goldman symplectic structure in Fock–Goncharov coordinates Fock–Goncharov坐标系中的扩展Goldman辛结构
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-10-12 DOI: 10.4310/jdg/1689262061
M. Bertola, D. Korotkin
{"title":"Extended Goldman symplectic structure in Fock–Goncharov coordinates","authors":"M. Bertola, D. Korotkin","doi":"10.4310/jdg/1689262061","DOIUrl":"https://doi.org/10.4310/jdg/1689262061","url":null,"abstract":"Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant under natural transformations. We use this notion to identify the canonical non-degenerate extension of Goldman's symplectic form on the $SL(n)$ character variety with a the form associated to a suitable graph. Using the invariance of the form under natural moves, we utilize the parametrization of the character variety in terms of Fock--Goncharov coordinates and associate to it a canonical decorated triangulation. This allows us to show that these coordinates are log--canonical for the extended Goldman Poisson structure.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49014362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Convex $mathbb{RP}^2$ structures and cubic differentials under neck separation 凸$mathbb｛RP｝^2$结构和颈分离下的三次微分
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-10-01 DOI: 10.4310/jdg/1571882429
John C. Loftin
{"title":"Convex $mathbb{RP}^2$ structures and cubic differentials under neck separation","authors":"John C. Loftin","doi":"10.4310/jdg/1571882429","DOIUrl":"https://doi.org/10.4310/jdg/1571882429","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44757964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary 非紧边渐近平面流形的黎曼彭罗斯不等式
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-09-29 DOI: 10.4310/jdg/1686931603
T. Koerber
{"title":"The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary","authors":"T. Koerber","doi":"10.4310/jdg/1686931603","DOIUrl":"https://doi.org/10.4310/jdg/1686931603","url":null,"abstract":"In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $mathbb{R}^3$ and thus sheds some light on a conjecture made by Huisken.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Maximizing Steklov eigenvalues on surfaces 曲面上Steklov特征值的最大化
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2019-09-01 DOI: 10.4310/jdg/1567216955
R. Petrides
{"title":"Maximizing Steklov eigenvalues on surfaces","authors":"R. Petrides","doi":"10.4310/jdg/1567216955","DOIUrl":"https://doi.org/10.4310/jdg/1567216955","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/jdg/1567216955","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48470351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 21
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