We study the strict stability of calibrated cones with an isolated singularity. For special Lagrangian cones and coassociative cones, we prove the strict stability. In the complex case, we give non-strictly stable examples.
{"title":"Strict stability of calibrated cones","authors":"Bryan Dimler, Jooho Lee","doi":"arxiv-2409.06094","DOIUrl":"https://doi.org/arxiv-2409.06094","url":null,"abstract":"We study the strict stability of calibrated cones with an isolated\u0000singularity. For special Lagrangian cones and coassociative cones, we prove the\u0000strict stability. In the complex case, we give non-strictly stable examples.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198980","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 conclude the classification of isoparametric (or equivalently, polar) foliations of complex and quaternionic projective spaces. This is done by investigating the projections of certain inhomogeneous isoparametric foliations of the 31-sphere under the respective Hopf fibrations, thereby solving the last remaining open cases.
{"title":"On isoparametric foliations of complex and quaternionic projective spaces","authors":"Miguel Dominguez-Vazquez, Andreas Kollross","doi":"arxiv-2409.06032","DOIUrl":"https://doi.org/arxiv-2409.06032","url":null,"abstract":"We conclude the classification of isoparametric (or equivalently, polar)\u0000foliations of complex and quaternionic projective spaces. This is done by\u0000investigating the projections of certain inhomogeneous isoparametric foliations\u0000of the 31-sphere under the respective Hopf fibrations, thereby solving the last\u0000remaining open cases.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198981","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 $mathcal{F}_d(mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $mathbb{P}^n$ ($n geq 2$) with degree $d geq 1.$ We show that there is subset $mathcal{S}_d(mathbb{P}^n)$ of $mathcal{F}_d(mathbb{P}^n)$ with full Lebesgue measure with the following properties: 1. for every $mathcal{F} in mathcal{S}_d(mathbb{P}^n),$ all singular points of $mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d geq 2,$ then every $mathcal{F}$ does not possess any invariant algebraic curve.
让 $mathcal{F}_d(mathbb{P}^n)$ 是 $mathbb{P}^n$ ($n geq 2$)上所有度数为 $d geq 1 的曲线的奇异全形变换空间。$ Weshow that there is subset $mathcal{S}_d(mathbb{P}^n)$ of$mathcal{F}_d(mathbb{P}^n)$ with full Lebesgue measure with the followingproperties：1. 对于每一个 $mathcal{F}在 mathcal{S}_d(mathbb{P}^n)中，$ $mathcal{F}$的所有奇点都是可线性化双曲的。2.此外，如果 $d geq 2, $ 那么每个 $mathcal{F}$ 都不具有任何不变的代数曲线。
{"title":"Generic singularities of holomorphic foliations by curves on $mathbb{P}^n$","authors":"Sahil Gehlawat, Viêt-Anh Nguyên","doi":"arxiv-2409.06052","DOIUrl":"https://doi.org/arxiv-2409.06052","url":null,"abstract":"Let $mathcal{F}_d(mathbb{P}^n)$ be the space of all singular holomorphic\u0000foliations by curves on $mathbb{P}^n$ ($n geq 2$) with degree $d geq 1.$ We\u0000show that there is subset $mathcal{S}_d(mathbb{P}^n)$ of\u0000$mathcal{F}_d(mathbb{P}^n)$ with full Lebesgue measure with the following\u0000properties: 1. for every $mathcal{F} in mathcal{S}_d(mathbb{P}^n),$ all singular\u0000points of $mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d geq 2,$ then every $mathcal{F}$ does not possess any\u0000invariant algebraic curve.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199100","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 the universal central extension of the Lie algebra of exact divergence-free vector fields, proving a conjecture by Claude Roger from 1995. The proof relies on the analysis of a Leibniz algebra that underlies these vector fields. As an application, we construct the universal central extension of the (infinite-dimensional) Lie group of exact divergence-free diffeomorphisms of a compact 3-dimensional manifold.
{"title":"Universal central extension of the Lie algebra of exact divergence-free vector fields","authors":"Bas Janssens, Leonid Ryvkin, Cornelia Vizman","doi":"arxiv-2409.05182","DOIUrl":"https://doi.org/arxiv-2409.05182","url":null,"abstract":"We construct the universal central extension of the Lie algebra of exact\u0000divergence-free vector fields, proving a conjecture by Claude Roger from 1995.\u0000The proof relies on the analysis of a Leibniz algebra that underlies these\u0000vector fields. As an application, we construct the universal central extension\u0000of the (infinite-dimensional) Lie group of exact divergence-free\u0000diffeomorphisms of a compact 3-dimensional manifold.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198984","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 the sharp $mathrm{L}^infty$ estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of K"ahler manifolds, we prove that the oscillation of any admissible solution to a degenerate fully non-linear elliptic equation satisfying several structural conditions can be controlled by the $mathrm{L}^1(logmathrm{L})^n(loglogmathrm{L})^r(r>n)$ norm of the right-hand function (in a regularized form). This result improves that of Guo-Phong-Tong. In addition to their method of comparison with auxiliary complex Monge-Amp`ere equations, our proof relies on an inequality of H"older-Young type and an iteration lemma of De Giorgi type. For the case of Hermitian manifolds with non-degenerate background metrics, we prove a similar $mathrm{L}^infty$ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the $mathrm{L}^infty$ estimates given here may fail when $rleqslant n-1$. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions.
{"title":"Sharp $mathrm{L}^infty$ estimates for fully non-linear elliptic equations on compact complex manifolds","authors":"Yuxiang Qiao","doi":"arxiv-2409.05157","DOIUrl":"https://doi.org/arxiv-2409.05157","url":null,"abstract":"We study the sharp $mathrm{L}^infty$ estimates for fully non-linear\u0000elliptic equations on compact complex manifolds. For the case of K\"ahler\u0000manifolds, we prove that the oscillation of any admissible solution to a\u0000degenerate fully non-linear elliptic equation satisfying several structural\u0000conditions can be controlled by the\u0000$mathrm{L}^1(logmathrm{L})^n(loglogmathrm{L})^r(r>n)$ norm of the\u0000right-hand function (in a regularized form). This result improves that of\u0000Guo-Phong-Tong. In addition to their method of comparison with auxiliary\u0000complex Monge-Amp`ere equations, our proof relies on an inequality of\u0000H\"older-Young type and an iteration lemma of De Giorgi type. For the case of\u0000Hermitian manifolds with non-degenerate background metrics, we prove a similar\u0000$mathrm{L}^infty$ estimate which improves that of Guo-Phong. An explicit\u0000example is constucted to show that the $mathrm{L}^infty$ estimates given here\u0000may fail when $rleqslant n-1$. The construction relies on a gluing lemma of\u0000smooth, radial, strictly plurisubharmonic functions.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199025","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 derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3le nle 5$ by Chodosh-Li, Chodosh-Li-Minter-Stryker and Mazet. In particular, the estimates indicate that the methods in their proofs may not work for $n=6$, which is observed also by Antonelli-Xu. The method of derivation in this work might also be applied to other problems.
{"title":"Some estimates on stable minimal hypersurfaces in Euclidean space","authors":"Luen-Fai Tam","doi":"arxiv-2409.04947","DOIUrl":"https://doi.org/arxiv-2409.04947","url":null,"abstract":"We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The\u0000estimates are related to recent proofs of Bernstein theorems for complete\u0000stable minimal hypersurfaces in $R^{n+1}$ for $3le nle 5$ by Chodosh-Li,\u0000Chodosh-Li-Minter-Stryker and Mazet. In particular, the estimates indicate that\u0000the methods in their proofs may not work for $n=6$, which is observed also by\u0000Antonelli-Xu. The method of derivation in this work might also be applied to\u0000other problems.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199022","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}
Luca Benatti, Carlo Mantegazza, Francesca Oronzio, Alessandra Pluda
Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $mathrm{Ric}geqvarepsilonmathrm{R} g$ for some $varepsilon>0$, where $mathrm{Ric}$ and $mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and by Huisken-K"{o}rber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called Hamilton's pinching conjecture.
假设$(M,g)$满足里奇夹角条件$mathrm{Ric}geqvarepsilonmathrm{R} g$ for some $varepsilon>0$, 其中$mathrm{Ric}$ 和$mathrm{R}$ 分别是里奇张量和标量曲率。在这篇短文中，我们基于势论给出了另一种证明，即如果 $(M,g)$ 具有欧几里得体积增长，那么它就是平坦的。Deruelle-Schulze-Simon和Huisken-K"{o}rber已经证明了这一结果，再加上Lott和Lee-Topping的贡献，导致了所谓汉密尔顿捏合猜想的证明。
{"title":"A Note on Ricci-pinched three-manifolds","authors":"Luca Benatti, Carlo Mantegazza, Francesca Oronzio, Alessandra Pluda","doi":"arxiv-2409.05078","DOIUrl":"https://doi.org/arxiv-2409.05078","url":null,"abstract":"Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold.\u0000Suppose that $(M,g)$ satisfies the Ricci--pinching condition\u0000$mathrm{Ric}geqvarepsilonmathrm{R} g$ for some $varepsilon>0$, where\u0000$mathrm{Ric}$ and $mathrm{R}$ are the Ricci tensor and scalar curvature,\u0000respectively. In this short note, we give an alternative proof based on\u0000potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then\u0000it is flat. Deruelle-Schulze-Simon and by Huisken-K\"{o}rber have already shown\u0000this result and together with the contributions by Lott and Lee-Topping led to\u0000a proof of the so-called Hamilton's pinching conjecture.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199020","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}
Non-trivial examples of generalized paracomplex structures (in the sense of the generalized geometry `a la Hitchin) are constructed applying the twistor space construction scheme.
应用扭转空间构造方案构建了广义准复数结构（在广义几何（the generalized geometry a la Hitchin）的意义上）的非微观实例。
{"title":"Generalized paracomplex structures on generalized reflector spaces","authors":"Johann Davidov","doi":"arxiv-2409.04835","DOIUrl":"https://doi.org/arxiv-2409.04835","url":null,"abstract":"Non-trivial examples of generalized paracomplex structures (in the sense of\u0000the generalized geometry `a la Hitchin) are constructed applying the twistor\u0000space construction scheme.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199023","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 the cohomology of an elliptic differential complex arising from the infinitesimal moduli of heterotic string theory. We compute these cohomology groups at the standard embedding, and show that they decompose into a direct sum of cohomologies. While this is often assumed in the literature, it had not been explicitly demonstrated. Given a stable gauge bundle over a complex threefold with trivial canonical bundle and no holomorphic vector fields, we also show that the Euler characteristic of this differential complex is zero. This points towards a perfect obstruction theory for the heterotic moduli problem, at least for the most physically relevant compactifications.
{"title":"The decoupling of moduli about the standard embedding","authors":"Beatrice Chisamanga, Jock McOrist, Sebastien Picard, Eirik Eik Svanes","doi":"arxiv-2409.04350","DOIUrl":"https://doi.org/arxiv-2409.04350","url":null,"abstract":"We study the cohomology of an elliptic differential complex arising from the\u0000infinitesimal moduli of heterotic string theory. We compute these cohomology\u0000groups at the standard embedding, and show that they decompose into a direct\u0000sum of cohomologies. While this is often assumed in the literature, it had not\u0000been explicitly demonstrated. Given a stable gauge bundle over a complex\u0000threefold with trivial canonical bundle and no holomorphic vector fields, we\u0000also show that the Euler characteristic of this differential complex is zero.\u0000This points towards a perfect obstruction theory for the heterotic moduli\u0000problem, at least for the most physically relevant compactifications.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199033","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}
Hannah de Lázari, Jason D. Lotay, Henrique Sá Earp, Eirik Eik Svanes
The heterotic $SU(3)$ system, also known as the Hull--Strominger system, arises from compactifications of heterotic string theory to six dimensions. This paper investigates the local structure of the moduli space of solutions to this system on a compact 6-manifold $X$, using a vector bundle $Q=(T^{1,0}X)^* oplus {End}(E) oplus T^{1,0}X$, where $Eto X$ is the classical gauge bundle arising in the system. We establish that the moduli space has an expected dimension of zero. We achieve this by studying the deformation complex associated to a differential operator $bar{D}$, which emulates a holomorphic structure on $Q$, and demonstrating an isomorphism between the two cohomology groups which govern the infinitesimal deformations and obstructions in the deformation theory for the system. We also provide a Dolbeault-type theorem linking these cohomology groups to v{C}ech cohomology, a result which might be of independent interest, as well as potentially valuable for future research.
{"title":"Local descriptions of the heterotic SU(3) moduli space","authors":"Hannah de Lázari, Jason D. Lotay, Henrique Sá Earp, Eirik Eik Svanes","doi":"arxiv-2409.04382","DOIUrl":"https://doi.org/arxiv-2409.04382","url":null,"abstract":"The heterotic $SU(3)$ system, also known as the Hull--Strominger system,\u0000arises from compactifications of heterotic string theory to six dimensions.\u0000This paper investigates the local structure of the moduli space of solutions to\u0000this system on a compact 6-manifold $X$, using a vector bundle $Q=(T^{1,0}X)^*\u0000oplus {End}(E) oplus T^{1,0}X$, where $Eto X$ is the classical gauge bundle\u0000arising in the system. We establish that the moduli space has an expected\u0000dimension of zero. We achieve this by studying the deformation complex\u0000associated to a differential operator $bar{D}$, which emulates a holomorphic\u0000structure on $Q$, and demonstrating an isomorphism between the two cohomology\u0000groups which govern the infinitesimal deformations and obstructions in the\u0000deformation theory for the system. We also provide a Dolbeault-type theorem\u0000linking these cohomology groups to v{C}ech cohomology, a result which might be\u0000of independent interest, as well as potentially valuable for future research.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199027","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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