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On the texorpdfstring{$ν$}{nu}-invariant of two-step nilmanifolds with closed texorpdfstring{$mathrm G_2$}{G2}-structure 关于具有闭合 texorpdfstring{$ν$}{G2} 结构的两步无常域的texorpdfstring{$ν$}{nu}不变量
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06870
Anna Fino, Gueo Grantcharov, Giovanni Russo
{"title":"On the texorpdfstring{$ν$}{nu}-invariant of two-step nilmanifolds with closed texorpdfstring{$mathrm G_2$}{G2}-structure","authors":"Anna Fino, Gueo Grantcharov, Giovanni Russo","doi":"arxiv-2409.06870","DOIUrl":"https://doi.org/arxiv-2409.06870","url":null,"abstract":"For every non-vanishing spinor field on a Riemannian $7$-manifold, Crowley,\u0000Goette, and Nordstr\"om introduced the so-called $nu$-invariant. This is an\u0000integer modulo $48$, and can be defined in terms of Mathai--Quillen currents,\u0000harmonic spinors, and $eta$-invariants of spin Dirac and odd-signature\u0000operator. We compute these data for the compact two-step nilmanifolds admitting\u0000invariant closed $mathrm G_2$-structures, in particular determining the\u0000harmonic spinors and relevant symmetries of the spectrum of the spin Dirac\u0000operator. We then deduce the vanishing of the $nu$-invariants.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198977","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 non-Archimedean theory of complex spaces and the cscK problem 复数空间的非阿基米德理论和 cscK 问题
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06221
Pietro Mesquita-Piccione
{"title":"A non-Archimedean theory of complex spaces and the cscK problem","authors":"Pietro Mesquita-Piccione","doi":"arxiv-2409.06221","DOIUrl":"https://doi.org/arxiv-2409.06221","url":null,"abstract":"In this paper we develop an analogue of the Berkovich analytification for\u0000non-necessarily algebraic complex spaces. We apply this theory to generalize to\u0000arbitrary compact K\"ahler manifolds a result of Chi Li, proving that a\u0000stronger version of K-stability implies the existence of a unique constant\u0000scalar curvature K\"ahler metric.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198978","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
Curvature and local matchings of conference graphs and extensions 会议图的曲率和局部匹配及扩展
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06418
Kaizhe Chen, Shiping Liu, Heng Zhang
{"title":"Curvature and local matchings of conference graphs and extensions","authors":"Kaizhe Chen, Shiping Liu, Heng Zhang","doi":"arxiv-2409.06418","DOIUrl":"https://doi.org/arxiv-2409.06418","url":null,"abstract":"We confirm a conjecture of Bonini et. al. on the precise Lin-Lu-Yau curvature\u0000values of conference graphs, i.e., strongly regular graphs with parameters\u0000$(4gamma+1,2gamma,gamma-1,gamma)$, with $gammageq 2$. Our method only\u0000depends on the parameter relations and applies to more general classes of amply\u0000regular graphs. In particular, we develop a new combinatorial method for\u0000showing the existence of local perfect matchings. A key observation is that\u0000counting common neighbors leads to useful quadratic polynomials. Our result\u0000also leads to an interesting number theoretic consequence on quadratic\u0000residues.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198982","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 power series expansions of logarithmic Sobolev, $mathcal{W}$- functionals and scalar curvature rigidity 对数索波列夫、$mathcal{W}$- 函数的幂级数展开与标量曲率刚度
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06117
Liang Cheng
{"title":"The power series expansions of logarithmic Sobolev, $mathcal{W}$- functionals and scalar curvature rigidity","authors":"Liang Cheng","doi":"arxiv-2409.06117","DOIUrl":"https://doi.org/arxiv-2409.06117","url":null,"abstract":"In this paper, we obtain that logarithmic Sobolev and $mathcal{W}$-\u0000functionals have fantastic power series expansion formulas when we choose\u0000suitable test functions. By using these power series expansion formulas, we\u0000prove that if for some open subset $V$ in an $n$-dimensional manifold\u0000satisfying $$ frac{ int_V R dmu}{mathrm{Vol}(V)} ge n(n-1)K$$ and the\u0000isoperimetric profile of $V$ satisfying $$ operatorname{I}(V,beta)doteq\u0000inflimits_{Omegasubset V,mathrm{Vol}(Omega)=beta}mathrm{Area}(partial\u0000Omega) ge operatorname{I}(M^n_K,beta),$$ for all $beta<beta_0$ and some\u0000$beta_0>0$, where $R$ is the scalar curvature and $M^n_K$ is the space form of\u0000constant sectional curvature $K$,then $operatorname{Sec}(x)=K$ for all $xin\u0000V$. We also get several other new scalar curvature rigidity theorems regarding\u0000isoperimetric profile, logarithmic Sobolev inequality and Perelman's\u0000$boldsymbol{mu}$-functional.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198979","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
Real analyticity of the modified Laplacian coflow 修正拉普拉斯共流的实解析性
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06283
Chuanhuan Li, Yi Li
{"title":"Real analyticity of the modified Laplacian coflow","authors":"Chuanhuan Li, Yi Li","doi":"arxiv-2409.06283","DOIUrl":"https://doi.org/arxiv-2409.06283","url":null,"abstract":"Let (M,psi(t))_{tin[0, T]} be a solution of the modified Laplacian coflow\u0000(1.3) with coclosed G_{2}-structures on a compact 7-dimensional M. We improve\u0000Chen's Shi-type estimate [5] for this flow, and then show that\u0000(M,psi(t),g_{psi}(t)) is real analytic, where g_{psi}(t) is the associate\u0000Riemannian metric to psi(t), which answers a question proposed by Grigorian in\u0000[13]. Consequently, we obtain the unique-continuation results for this flow.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198976","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 Bogovskiǐ-type operator for Corvino-Schoen hyperbolic gluing 科尔维诺-肖恩双曲胶合的博戈夫斯基ǐ型算子
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.07502
Piotr T. Chruściel, Albachiara Cogo, Andrea Nützi
{"title":"A Bogovskiǐ-type operator for Corvino-Schoen hyperbolic gluing","authors":"Piotr T. Chruściel, Albachiara Cogo, Andrea Nützi","doi":"arxiv-2409.07502","DOIUrl":"https://doi.org/arxiv-2409.07502","url":null,"abstract":"We construct a solution operator for the linearized constant scalar curvature\u0000equation at hyperbolic space in space dimension larger than or equal to two.\u0000The solution operator has good support propagation properties and gains two\u0000derivatives relative to standard norms. It can be used for Corvino-Schoen-type\u0000hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing\u0000method to the hyperbolic setting.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198975","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
Darboux-Lie derivatives 达布-里派生词
arXiv - MATH - Differential Geometry Pub Date : 2024-09-10 DOI: arxiv-2409.06596
Antonio De Nicola, Ivan Yudin
{"title":"Darboux-Lie derivatives","authors":"Antonio De Nicola, Ivan Yudin","doi":"arxiv-2409.06596","DOIUrl":"https://doi.org/arxiv-2409.06596","url":null,"abstract":"We introduce the Darboux-Lie derivative for fiber-bundle maps from natural\u0000bundles to associated fiber bundles and study its properties.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198974","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
Existence of ground states for free energies on the hyperbolic space 双曲空间自由能的基态存在性
arXiv - MATH - Differential Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.06022
José A. Carrillo, Razvan C. Fetecau, Hansol Park
{"title":"Existence of ground states for free energies on the hyperbolic space","authors":"José A. Carrillo, Razvan C. Fetecau, Hansol Park","doi":"arxiv-2409.06022","DOIUrl":"https://doi.org/arxiv-2409.06022","url":null,"abstract":"We investigate a free energy functional that arises in aggregation-diffusion\u0000phenomena modelled by nonlocal interactions and local repulsion on the\u0000hyperbolic space $bbh^dm$. The free energy consists of two competing terms:\u0000an entropy, corresponding to slow nonlinear diffusion, that favours spreading,\u0000and an attractive interaction potential energy that favours aggregation. We\u0000establish necessary and sufficient conditions on the interaction potential for\u0000ground states to exist on the hyperbolic space $bbh^dm$. To prove our results\u0000we derived several Hardy-Littlewood-Sobolev (HLS)-type inequalities on general\u0000Cartan-Hadamard manifolds of bounded curvature, which have an interest in their\u0000own.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199021","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
Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature 芬斯勒流形上著名函数不等式的失效：$S$曲率的影响
arXiv - MATH - Differential Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.05497
Alexandru Kristály, Benling Li, Wei Zhao
{"title":"Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature","authors":"Alexandru Kristály, Benling Li, Wei Zhao","doi":"arxiv-2409.05497","DOIUrl":"https://doi.org/arxiv-2409.05497","url":null,"abstract":"The validity of functional inequalities on Finsler metric measure manifolds\u0000is based on three non-Riemannian quantities, namely, the reversibility, flag\u0000curvature and $S$-curvature induced by the measure. Under mild assumptions on\u0000the reversibility and flag curvature, it turned out that famous functional\u0000inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty\u0000principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward\u0000complete Finsler manifolds with non-positive $S$-curvature, cf. Huang,\u0000Krist'aly and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we\u0000prove that -- under similar assumptions on the reversibility and flag curvature\u0000as before -- the aforementioned functional inequalities fail whenever the\u0000$S$-curvature is positive. Accordingly, our results clearly reveal the deep\u0000dependence of functional inequalities on the $S$-curvature. As a consequence of\u0000these results, we establish surprising analytic aspects of Finsler manifolds:\u0000if the flag curvature is non-positive, the Ricci curvature is bounded from\u0000below and the $S$-curvature is positive, then the reversibility turns out to be\u0000infinite. Examples are presented on general Funk metric spaces, where the\u0000$S$-curvature plays again a decisive role.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198983","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
Regularity of the Future Event Horizon in Perturbations of Kerr 克尔扰动中未来事件视界的规律性
arXiv - MATH - Differential Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.05700
Xuantao Chen, Sergiu Klainerman
{"title":"Regularity of the Future Event Horizon in Perturbations of Kerr","authors":"Xuantao Chen, Sergiu Klainerman","doi":"arxiv-2409.05700","DOIUrl":"https://doi.org/arxiv-2409.05700","url":null,"abstract":"The goal of the paper is to show that the event horizons of the spacetimes\u0000constructed in cite{KS}, see also cite{KS-Schw}, in the proof of the\u0000nonlinear stability of slowly rotating Kerr spacetimes $mathcal{K}(a_0,m_0)$,\u0000are necessarily smooth null hypersurfaces. Moreover we show that the result\u0000remains true for the entire range of $|a_0|/m_0$ for which stability can be\u0000established.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199024","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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