For every non-vanishing spinor field on a Riemannian $7$-manifold, Crowley, Goette, and Nordstr"om introduced the so-called $nu$-invariant. This is an integer modulo $48$, and can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $eta$-invariants of spin Dirac and odd-signature operator. We compute these data for the compact two-step nilmanifolds admitting invariant closed $mathrm G_2$-structures, in particular determining the harmonic spinors and relevant symmetries of the spectrum of the spin Dirac operator. We then deduce the vanishing of the $nu$-invariants.
{"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}
In this paper we develop an analogue of the Berkovich analytification for non-necessarily algebraic complex spaces. We apply this theory to generalize to arbitrary compact K"ahler manifolds a result of Chi Li, proving that a stronger version of K-stability implies the existence of a unique constant scalar curvature K"ahler metric.
{"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}
We confirm a conjecture of Bonini et. al. on the precise Lin-Lu-Yau curvature values of conference graphs, i.e., strongly regular graphs with parameters $(4gamma+1,2gamma,gamma-1,gamma)$, with $gammageq 2$. Our method only depends on the parameter relations and applies to more general classes of amply regular graphs. In particular, we develop a new combinatorial method for showing the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. Our result also leads to an interesting number theoretic consequence on quadratic residues.
{"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}
In this paper, we obtain that logarithmic Sobolev and $mathcal{W}$- functionals have fantastic power series expansion formulas when we choose suitable test functions. By using these power series expansion formulas, we prove that if for some open subset $V$ in an $n$-dimensional manifold satisfying $$ frac{ int_V R dmu}{mathrm{Vol}(V)} ge n(n-1)K$$ and the isoperimetric profile of $V$ satisfying $$ operatorname{I}(V,beta)doteq inflimits_{Omegasubset V,mathrm{Vol}(Omega)=beta}mathrm{Area}(partial Omega) ge operatorname{I}(M^n_K,beta),$$ for all $beta0$, where $R$ is the scalar curvature and $M^n_K$ is the space form of constant sectional curvature $K$,then $operatorname{Sec}(x)=K$ for all $xin V$. We also get several other new scalar curvature rigidity theorems regarding isoperimetric profile, logarithmic Sobolev inequality and Perelman's $boldsymbol{mu}$-functional.
在本文中，当我们选择合适的检验函数时，我们得到对数Sobolev和$mathcal{W}$函数具有奇妙的幂级数展开公式。通过使用这些幂级数展开公式，我们证明，如果在一个 $n$ 维流形中，对于某个开放子集 $V$ 满足 $$ frac{ int_V R dmu}{mathrm{Vol}(V)} ge n(n-1)K$$ 且 $V$ 的等距轮廓满足 $$ operatorname{I}(V,beta)doteqinflimits_{Omegasubset V、mathrm{Vol}(Omega)=beta}mathrm{Area}(partialOmega) ge operatorname{I}(M^n_K,beta),$$ for all $beta0$、其中 $R$ 是标量曲率，$M^n_K$ 是恒定截面曲率 $K$ 的空间形式，那么对于所有 $xinV$ 来说，$operatorname{Sec}(x)=K$。我们还得到了关于等周剖面、对数索波列夫不等式和佩雷尔曼函数的其他几个新的标量曲率刚性定理。
{"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}
Let (M,psi(t))_{tin[0, T]} be a solution of the modified Laplacian coflow (1.3) with coclosed G_{2}-structures on a compact 7-dimensional M. We improve Chen's Shi-type estimate [5] for this flow, and then show that (M,psi(t),g_{psi}(t)) is real analytic, where g_{psi}(t) is the associate Riemannian metric to psi(t), which answers a question proposed by Grigorian in [13]. Consequently, we obtain the unique-continuation results for this flow.
设(M,psi(t))_{t/in[0, T]} 是紧凑 7 维 M 上具有可闭 G_{2} 结构的修正拉普拉斯共流（1.3）的解。我们改进了陈氏对此流的 Shi 型估计[5]，然后证明（M,psi(t),g_{psi}(t)）是实解析的，其中 g_{psi}(t) 是 psi(t)的关联黎曼度量，这回答了格里高利安在[13]中提出的一个问题。因此，我们得到了该流的唯一延续结果。
{"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}
We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino-Schoen-type hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing method to the hyperbolic setting.
{"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}
We introduce the Darboux-Lie derivative for fiber-bundle maps from natural bundles to associated fiber bundles and study its properties.
我们介绍了从自然束到相关纤维束的纤维束映射的达布-李导数，并研究了它的性质。
{"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}
We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space $bbh^dm$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space $bbh^dm$. To prove our results we derived several Hardy-Littlewood-Sobolev (HLS)-type inequalities on general Cartan-Hadamard manifolds of bounded curvature, which have an interest in their own.
{"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}
The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Krist'aly and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish surprising analytic aspects of Finsler manifolds: if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role.
芬斯勒度量流形上函数不等式的有效性基于三个非黎曼量，即度量引起的可逆性、旗曲率和$S$曲率。在对可逆性和旗曲率的温和假设下，结果发现著名的函数不等式--如Hardy不等式、Heisenberg--Pauli--Weyltyprinciple和Caffarelli--Kohn--Nirenberg不等式--通常在具有非正$S$曲率的前向完全Finsler流形上成立，参见Huang, Krist'aly and Zhao [Trans. Amer. Math. Soc., 2020]。然而，在本文中，我们证明--在与之前相似的可逆性和旗曲率假设下--只要$S$曲率为正，上述函数不等式就失效。因此，我们的结果清楚地揭示了函数不等式对$S$曲率的深刻依赖性。由于这些结果，我们建立了令人惊讶的芬斯勒流形的分析方面：如果旗曲率是非正的，里奇曲率从下往上是有界的，而$S$曲率是正的，那么可逆性就会变成无限的。在一般丰克度量空间中，$S$曲率也起着决定性的作用。
{"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}
The goal of the paper is to show that the event horizons of the spacetimes constructed in cite{KS}, see also cite{KS-Schw}, in the proof of the nonlinear stability of slowly rotating Kerr spacetimes $mathcal{K}(a_0,m_0)$, are necessarily smooth null hypersurfaces. Moreover we show that the result remains true for the entire range of $|a_0|/m_0$ for which stability can be established.
{"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}
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