We describe an example of an indefinite invariant Einstein metric on a solvmanifold which is not standard, and whose restriction on the nilradical is nondegenerate.
{"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":null,"pages":null},"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}
We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge--Amp`ere equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive solutions of the vector bundle Monge--Amp`ere (vbMA) equation are also MA-positive. For vector bundles of rank-three and higher, over complex manifolds of dimension greater than one, we show that this positivity-preservation property need not hold for an algebraic solution of the vbMA equation treated as a purely algebraic equation at a given point. Finally, we set up a continuity path for certain classes of highly symmetric rank-two vector bundles over complex three-folds and prove a restricted version of positivity preservation which is nevertheless sufficient to prove openness along this continuity path.
{"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":null,"pages":null},"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}
In this paper we use the convex integration technique enhanced by an extra iteration originally due to K"all'en and revisited by Kr"oner to provide a local $h$-principle for isometric embeddings in the class $C^{1,1-epsilon}$ for $n$-dimensional manifolds in codimension $frac{1}{2}n(n+1)$.
{"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":null,"pages":null},"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}
Locality is implemented in an arbitrary category using Grothendieck topologies. We explore how different Grothendieck topologies on one category can be related, and, more general, how functors between categories can preserve them. As applications of locality, we review geometric objects such as sheaves, groupoids, functors, bibundles, and anafunctors internal to an arbitrary Grothendieck site. We give definitions such that all these objects are invariant under equivalences of Grothendieck topologies and certain functors between sites. As examples of sites, we look at categories of smooth manifolds, diffeological spaces, topological spaces, and sheaves, and we study properties of various functors between those.
{"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":null,"pages":null},"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}
We give a non-Archimedean characterization of K-semistability of log Fano cone singularities, and show that it agrees with the definition originally defined by Collins--Sz'ekelyhidi. As an application, we show that to test K-semistability, it suffices to test special test configurations. We also show that special test configurations give rise to lc places of torus equivariant bounded complements.
{"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":null,"pages":null},"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}
Cristiano S. Silva, Juliana F. R. Miranda, Marcio C. Araújo Filho
In this work, we obtain estimates for the upper bound of gaps between consecutive eigenvalues for the eigenvalue problem of a class of second-order elliptic differential operators in divergent form, with Dirichlet boundary conditions, in a limited domain of n-dimensional Euclidean space. This class of operators includes the well-known Laplacian and the square Cheng-Yau operator. For the Laplacian case, our estimate coincides with that obtained by D. Chen, T. Zheng, and H. Yang, which is the best possible in terms of the order of the eigenvalues. For pinched Cartan-Hadamard manifolds the estimates were made in particular cases of this operator.
在这项工作中，我们获得了在 n 维欧几里得空间有限域中，一类具有迪里夏特边界条件的发散形式二阶椭圆微分算子的特征值问题的连续特征值之间间隙的上界估计值。对于拉普拉斯算子，我们的估计与 D. Chen、T. Zheng 和 H. Yang 的估计不谋而合。对于拉普拉斯算子，我们的估计值与陈德强、郑俊涛和杨海峰的估计值不谋而合。对于捏合 Cartan-Hadamard 流形，我们是在该算子的特殊情况下进行估计的。
{"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":null,"pages":null},"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}
The main aim of this paper is to simplify and popularise the construction from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other things) derives the Plebanski-Demianski family of solutions of GR using ideas of complex geometry. The starting point of this construction is the observation that the Euclidean versions of these metrics should have two different commuting complex structures, as well as two commuting Killing vector fields. After some linear algebra, this leads to an ansatz for the metrics, which is half-way to their complete determination. Kerr metric is a special 2-parameter subfamily in this class, which makes these considerations directly relevant to Kerr as well. This results in a derivation of the Kerr metric that is self-contained and elementary.
本文的主要目的是简化和普及阿波斯托洛夫、卡尔德班克和高杜松在 2013 年发表的论文中的构造，该论文（除其他外）利用复几何学的思想推导出了 GR 的普莱班斯基-德米安斯基解族。这一构造的出发点是观察到这些度量的欧几里得版本应该有两个不同的换元复数结构，以及两个换元基林向量场。克尔公度量是该类中一个特殊的 2 参数子族，这使得这些考虑也与克尔公度量直接相关。因此，对克尔公设的推导是自足和基本的。
{"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":null,"pages":null},"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}
This is a continuation of our previous work (Advances in Mathematics 450 (2024), Paper No. 109768). In this paper, we characterize complete metrics with finite total Q-curvature as normal metrics for all dimensional cases. Secondly, we introduce another volume entropy to provide geometric information regarding complete non-normal metrics with finite total Q-curvature. In particular, we show that if the scalar curvature is bounded from below, the volume growth of such complete metrics is controlled.
{"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":null,"pages":null},"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}
In the theory of flat chains with coefficients in a normed abelian group, we give a simple necessary and sufficient condition on a group element $g$ in order for the following fundamental regularity principle to hold: if a mass-minimizing chain is, in a ball disjoint from the boundary, sufficiently weakly close to a multiplicity $g$ disk, then, in a smaller ball, it is a $C^{1,alpha}$ perturbation with multiplicity $g$ of that disk.
{"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":null,"pages":null},"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}
Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/pi$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function.
{"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":null,"pages":null},"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}
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