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A non-Standard Indefinite Einstein Solvmanifold 非标准无限爱因斯坦索曼菲尔德
arXiv - MATH - Differential Geometry Pub Date : 2024-08-31 DOI: arxiv-2409.00462
Federico A. Rossi
{"title":"A non-Standard Indefinite Einstein Solvmanifold","authors":"Federico A. Rossi","doi":"arxiv-2409.00462","DOIUrl":"https://doi.org/arxiv-2409.00462","url":null,"abstract":"We describe an example of an indefinite invariant Einstein metric on a\u0000solvmanifold which is not standard, and whose restriction on the nilradical is\u0000nondegenerate.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199074","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
Positivity properties of the vector bundle Monge-Ampère equation 向量束蒙日-安培方程的正向性质
arXiv - MATH - Differential Geometry Pub Date : 2024-08-31 DOI: arxiv-2409.00321
Aashirwad N. Ballal, Vamsi P. Pingali
{"title":"Positivity properties of the vector bundle Monge-Ampère equation","authors":"Aashirwad N. Ballal, Vamsi P. Pingali","doi":"arxiv-2409.00321","DOIUrl":"https://doi.org/arxiv-2409.00321","url":null,"abstract":"We study MA-positivity, a notion of positivity relevant to a vector bundle\u0000version of the complex Monge--Amp`ere equation introduced in an earlier work,\u0000and show that for rank-two holomorphic bundles over complex surfaces,\u0000MA-semi-positive solutions of the vector bundle Monge--Amp`ere (vbMA) equation\u0000are also MA-positive. For vector bundles of rank-three and higher, over complex\u0000manifolds of dimension greater than one, we show that this\u0000positivity-preservation property need not hold for an algebraic solution of the\u0000vbMA equation treated as a purely algebraic equation at a given point. Finally,\u0000we set up a continuity path for certain classes of highly symmetric rank-two\u0000vector bundles over complex three-folds and prove a restricted version of\u0000positivity preservation which is nevertheless sufficient to prove openness\u0000along this continuity path.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199071","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
$C^{1,1-ε}$ Isometric embeddings C^{1,1-ε}$ 等距嵌入
arXiv - MATH - Differential Geometry Pub Date : 2024-08-31 DOI: arxiv-2409.00440
Ángel D. Martínez
{"title":"$C^{1,1-ε}$ Isometric embeddings","authors":"Ángel D. Martínez","doi":"arxiv-2409.00440","DOIUrl":"https://doi.org/arxiv-2409.00440","url":null,"abstract":"In this paper we use the convex integration technique enhanced by an extra\u0000iteration originally due to K\"all'en and revisited by Kr\"oner to provide a\u0000local $h$-principle for isometric embeddings in the class $C^{1,1-epsilon}$\u0000for $n$-dimensional manifolds in codimension $frac{1}{2}n(n+1)$.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199069","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
K-semistability of log Fano cone singularities 对数法诺锥奇点的 K-semistability
arXiv - MATH - Differential Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.05189
Yuchen Liu, Yueqiao Wu
{"title":"K-semistability of log Fano cone singularities","authors":"Yuchen Liu, Yueqiao Wu","doi":"arxiv-2408.05189","DOIUrl":"https://doi.org/arxiv-2408.05189","url":null,"abstract":"We give a non-Archimedean characterization of K-semistability of log Fano\u0000cone singularities, and show that it agrees with the definition originally\u0000defined by Collins--Sz'ekelyhidi. As an application, we show that to test\u0000K-semistability, it suffices to test special test configurations. We also show\u0000that special test configurations give rise to lc places of torus equivariant\u0000bounded complements.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935208","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
Internal geometry and functors between sites 内部几何和场地之间的函数
arXiv - MATH - Differential Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.04989
Konrad Waldorf
{"title":"Internal geometry and functors between sites","authors":"Konrad Waldorf","doi":"arxiv-2408.04989","DOIUrl":"https://doi.org/arxiv-2408.04989","url":null,"abstract":"Locality is implemented in an arbitrary category using Grothendieck\u0000topologies. We explore how different Grothendieck topologies on one category\u0000can be related, and, more general, how functors between categories can preserve\u0000them. As applications of locality, we review geometric objects such as sheaves,\u0000groupoids, functors, bibundles, and anafunctors internal to an arbitrary\u0000Grothendieck site. We give definitions such that all these objects are\u0000invariant under equivalences of Grothendieck topologies and certain functors\u0000between sites. As examples of sites, we look at categories of smooth manifolds,\u0000diffeological spaces, topological spaces, and sheaves, and we study properties\u0000of various functors between those.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"113 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935207","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
Estimates of the gaps between consecutive eigenvalues for a class of elliptic differential operators in divergence form on Riemannian manifolds 黎曼流形上一类发散形式椭圆微分算子连续特征值间间隙的估计值
arXiv - MATH - Differential Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.05068
Cristiano S. Silva, Juliana F. R. Miranda, Marcio C. Araújo Filho
{"title":"Estimates of the gaps between consecutive eigenvalues for a class of elliptic differential operators in divergence form on Riemannian manifolds","authors":"Cristiano S. Silva, Juliana F. R. Miranda, Marcio C. Araújo Filho","doi":"arxiv-2408.05068","DOIUrl":"https://doi.org/arxiv-2408.05068","url":null,"abstract":"In this work, we obtain estimates for the upper bound of gaps between\u0000consecutive eigenvalues for the eigenvalue problem of a class of second-order\u0000elliptic differential operators in divergent form, with Dirichlet boundary\u0000conditions, in a limited domain of n-dimensional Euclidean space. This class of\u0000operators includes the well-known Laplacian and the square Cheng-Yau operator.\u0000For the Laplacian case, our estimate coincides with that obtained by D. Chen,\u0000T. Zheng, and H. Yang, which is the best possible in terms of the order of the\u0000eigenvalues. For pinched Cartan-Hadamard manifolds the estimates were made in\u0000particular cases of this operator.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935206","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
Elementary derivation of the Kerr metric 克尔公设的基本推导
arXiv - MATH - Differential Geometry Pub Date : 2024-08-08 DOI: arxiv-2408.04389
Kirill Krasnov, Adam Shaw
{"title":"Elementary derivation of the Kerr metric","authors":"Kirill Krasnov, Adam Shaw","doi":"arxiv-2408.04389","DOIUrl":"https://doi.org/arxiv-2408.04389","url":null,"abstract":"The main aim of this paper is to simplify and popularise the construction\u0000from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other\u0000things) derives the Plebanski-Demianski family of solutions of GR using ideas\u0000of complex geometry. The starting point of this construction is the observation\u0000that the Euclidean versions of these metrics should have two different\u0000commuting complex structures, as well as two commuting Killing vector fields.\u0000After some linear algebra, this leads to an ansatz for the metrics, which is\u0000half-way to their complete determination. Kerr metric is a special 2-parameter\u0000subfamily in this class, which makes these considerations directly relevant to\u0000Kerr as well. This results in a derivation of the Kerr metric that is\u0000self-contained and elementary.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935214","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 total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II) 总 Q 曲率、体积熵和多项式增长多谐函数 (II)
arXiv - MATH - Differential Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.03640
Mingxiang Li
{"title":"The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II)","authors":"Mingxiang Li","doi":"arxiv-2408.03640","DOIUrl":"https://doi.org/arxiv-2408.03640","url":null,"abstract":"This is a continuation of our previous work (Advances in Mathematics 450\u0000(2024), Paper No. 109768). In this paper, we characterize complete metrics with\u0000finite total Q-curvature as normal metrics for all dimensional cases. Secondly,\u0000we introduce another volume entropy to provide geometric information regarding\u0000complete non-normal metrics with finite total Q-curvature. In particular, we\u0000show that if the scalar curvature is bounded from below, the volume growth of\u0000such complete metrics is controlled.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935217","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 fundamental regularity theorem for mass-minimizing flat chains 论质量最小化平面链的基本正则定理
arXiv - MATH - Differential Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.04083
Brian White
{"title":"On the fundamental regularity theorem for mass-minimizing flat chains","authors":"Brian White","doi":"arxiv-2408.04083","DOIUrl":"https://doi.org/arxiv-2408.04083","url":null,"abstract":"In the theory of flat chains with coefficients in a normed abelian group, we\u0000give a simple necessary and sufficient condition on a group element $g$ in\u0000order for the following fundamental regularity principle to hold: if a\u0000mass-minimizing chain is, in a ball disjoint from the boundary, sufficiently\u0000weakly close to a multiplicity $g$ disk, then, in a smaller ball, it is a\u0000$C^{1,alpha}$ perturbation with multiplicity $g$ of that disk.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935209","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
Delayed parabolic regularity for curve shortening flow 曲线缩短流的延迟抛物线正则性
arXiv - MATH - Differential Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.04049
Arjun Sobnack, Peter M. Topping
{"title":"Delayed parabolic regularity for curve shortening flow","authors":"Arjun Sobnack, Peter M. Topping","doi":"arxiv-2408.04049","DOIUrl":"https://doi.org/arxiv-2408.04049","url":null,"abstract":"Given two curves bounding a region of area $A$ that evolve under curve\u0000shortening flow, we propose the principle that the regularity of one should be\u0000controllable in terms of the regularity of the other, starting from time\u0000$A/pi$. We prove several results of this form and demonstrate that no estimate\u0000can hold before that time. As an example application, we construct solutions to\u0000graphical curve shortening flow starting with initial data that is merely an\u0000$L^1$ function.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935215","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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