Journal of Differential Geometry最新文献

Conical Calabi–Yau metrics on toric affine varieties and convex cones 环仿射变异和凸锥上的圆锥Calabi-Yau度量

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432924

Robert J. Berman

引用次数: 6

The index formula for families of Dirac type operators on pseudomanifolds 伪流形上狄拉克型算子族的指数公式

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432923

Pierre Albin, Jesse Gell-Redman

引用次数: 13

Existence of multiple closed CMC hypersurfaces with small mean curvature 具有小平均曲率的多个闭合CMC超曲面的存在性

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432925

Akashdeep Dey

{"title":"Existence of multiple closed CMC hypersurfaces with small mean curvature","authors":"Akashdeep Dey","doi":"10.4310/jdg/1696432925","DOIUrl":"https://doi.org/10.4310/jdg/1696432925","url":null,"abstract":"Min-max theory for constant mean curvature (CMC) hypersurfaces has been developed by Zhou–Zhu $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$ and Zhou $[href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$. In particular, in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$, Zhou and Zhu proved that for any $c gt 0$, every closed Riemannian manifold $(M^{n+1}, g), 3 leq n + 1 leq 7$, contains a closed $c$-CMC hypersurface. In this article we will show that the min-max theory for CMC hypersurfaces in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}, href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$ can be extended in higher dimensions using the regularity theory of stable CMC hypersurfaces, developed by Bellettini–Wickramasekera $[href{https://doi.org/10.48550/arXiv.1802.00377}{4}, href{https://doi.org/10.48550/arXiv.1902.09669}{5}]$ and Bellettini–Chodosh–Wickramasekera $[href{https://doi.org/10.1016/j.aim.2019.05.023}{3}]$. Furthermore, we will prove that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g), n+1 geq 3$, tends to infinity as $c to 0^+$. More quantitatively, there exists a constant $gamma_0$, depending on $g$, such that for all $c gt 0$, there exist at least $gamma_0 c^{-frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 9

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties 退化Calabi-Yau变型附近Maurer-Cartan方程的几何性质

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236591

Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma

{"title":"Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties","authors":"Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma","doi":"10.4310/jdg/1695236591","DOIUrl":"https://doi.org/10.4310/jdg/1695236591","url":null,"abstract":"Given a degenerate Calabi–Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin–Vilkovisky algebra $PV^{ast,ast}(X)$, producing a singular version of the extended Kodaira–Spencer differential graded Lie algebra in the Calabi–Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov–Tian–Todorov–type unobstructedness theorem for smoothing of singular Calabi–Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi–Yau varieties (as studied by Friedman [$href{https://doi.org/10.2307/2006955}{22}$] and Kawamata–Namikawa [$href{ https://doi.org/10.1007/BF01231538}{41}$]) and maximally degenerate Calabi–Yau varieties (as studied by Kontsevich–Soibelman $[href{ https://link.springer.com/chapter/10.1007/0-8176-4467-9_9}{45}$] and Gross–Siebert [$href{ http://doi.org/10.4007/annals.2011.174.3.1}{30}$]). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov–Kontsevich [$href{https://doi.org/10.1155/S1073792898000166}{2}$, $href{https://doi.org/10.48550/arXiv.math/9903124}{1}$].","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 18

Harmonic maps from $S^3$ into $S^2$ with low Morse index 低摩尔斯指数从$S^3$到$S^2$的谐波映射

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236594

Tristan Rivière

引用次数: 1

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters 非关键参数平面环面上$SU(3)$ Toda系统解的数量和偶数性

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236592

Zhijie Chen, Chang-Shou Lin

{"title":"On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters","authors":"Zhijie Chen, Chang-Shou Lin","doi":"10.4310/jdg/1695236592","DOIUrl":"https://doi.org/10.4310/jdg/1695236592","url":null,"abstract":"We study the $SU(3)$ Toda system with singular sources begin{equation*} begin{cases} Delta u+2e^{u}-e^{v}=4pi sum _{k=0}^{m} n_{1,k}delta _{p_{k}} quad text{ on }; E_{tau}, Delta v+2e^{v}-e^{u}=4pi sum _{k=0}^{m} n_{2,k}delta _{p_{k}} quad text{ on }; E_{tau}, end{cases} end{equation*} where $E_{tau}:=mathbb{C}/(mathbb{Z}+mathbb{Z}tau )$ with $operatorname{Im}tau gt 0$ is a flat torus, $delta _{p_{k}}$ is the Dirac measure at $p_{k}$, and $n_{i,k}in mathbb{Z}_{geq 0}$ satisfy $sum _{k}n_{1,k}not equiv sum _{k} n_{2,k} mod 3$. This is known as the non-critical case and it follows from a general existence result of [$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$] that solutions always exist. In this paper we prove that (i) The system has at most begin{equation*} frac{1}{3times 2^{m+1}}prod _{k=0}^{m}(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2) in mathbb{N} end{equation*} solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_{0}=0$, the system has even solutions if and only if at least one of ${n_{1,0}, n_{2,0}}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}lt n_{2,0}$, then except for finitely many $tau $’s modulo $SL(2,mathbb{Z})$ action, the system has exactly $frac{n_{1,0}+1}{2}$ even solutions. Differently from [$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Creating Stein surfaces by topological isotopy 用拓扑同位素法创建斯坦因表面

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236593

Robert E. Gompf

引用次数: 2

Isoperimetry and volume preserving stability in real projective spaces 实射影空间中的等距性和保体积稳定性

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236595

Celso Viana

引用次数: 3

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $mathrm{SU}(m)$ or $mathrm{U}(m)$ 具有规范群$ mathm {SU}(m)$或$ mathm {U}(m)$的$G_2$-实例的模空间的正则取向

IF 2.5 1区数学

Journal of Differential Geometry Pub Date : 2023-06-01 DOI: 10.4310/jdg/1686931600

D. joyce, M. Upmeier

引用次数: 0

The renormalized volume of a $4$-dimensional Ricci-flat ALE space 4维ricci -平坦ALE空间的重整化体积

1区数学

Journal of Differential Geometry Pub Date : 2023-03-01 DOI: 10.4310/jdg/1683307004

Olivier Biquard, Hans-Joachim Hein

引用次数: 0