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Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85. 对“非负截面曲率和非唯一灵魂的模空间”的修正，J.Diff.Geom。89（2011），第1号，49–85。
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-09-01 DOI: 10.4310/jdg/1631124246
I. Belegradek, S. Kwasik, R. Schultz
{"title":"Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85.","authors":"I. Belegradek, S. Kwasik, R. Schultz","doi":"10.4310/jdg/1631124246","DOIUrl":"https://doi.org/10.4310/jdg/1631124246","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42694039","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: 0
Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori 矩形环面上多重格林函数临界点的精确个数与非退化性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-07-01 DOI: 10.4310/JDG/1625860623
Zhijie Chen, Changshou Lin
{"title":"Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori","authors":"Zhijie Chen, Changshou Lin","doi":"10.4310/JDG/1625860623","DOIUrl":"https://doi.org/10.4310/JDG/1625860623","url":null,"abstract":"Let $E_{tau}:= mathbb{C}/(mathbb{Z}+ mathbb{Z} tau)$ be a flat torus and $G(z; tau)$ be the Green function on $E_{tau}$. Consider the multiple Green function $G_{n}$ on$(E_{tau})^{n}$: [ G_n (z_1, cdots ,z_n ; tau) := sum_{i lt j} G(z_i - z_j ; tau) - n sum_{i=1}^n G(z_i ; tau). ] We prove that for $ tau in i mathbb{R}_{gt 0}$, i.e. $E_tau$ is a rectangular torus, $G_n$ has exactly $2n + 1$ critical points modulo the permutation group $S_n$ and all critical points are non-degenerate. More precisely, there are exactly $n$ (resp. $n+1$) critical points $ boldsymbol{a}$’s with the Hessian satisfying $(-1)^n det D^2 G_n (boldsymbol{a} ; tau) lt 0$ (resp. $gt 0$). This confirms a conjecture in [4]. Our proof is based on the connection between $G_n$ and the classical Lame equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of $G_{n}$ in terms of the monodromy data of the Lame equation. As an application, we show that the mean field equation [ Delta_u + e^u = rho delta_0 textrm{ on } E_tau ] has exactly $n$ solutions for $8 pi n - rho gt 0$ small, and exactly $n+1$ solutions for $rho - 8 pi n gt 0$ small.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47882033","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: 7
Harmonic quasi-isometric maps into Gromov hyperbolic $operatorname{CAT}(0)$-spaces Gromov双曲$operatorname{CAT}(0)$-空间中的调和拟等距映射
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-07-01 DOI: 10.4310/JDG/1625860625
H. Sidler, S. Wenger
{"title":"Harmonic quasi-isometric maps into Gromov hyperbolic $operatorname{CAT}(0)$-spaces","authors":"H. Sidler, S. Wenger","doi":"10.4310/JDG/1625860625","DOIUrl":"https://doi.org/10.4310/JDG/1625860625","url":null,"abstract":"We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a proper, Gromov hyperbolic, $operatorname{CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist–Hulin.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47343349","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: 1
Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds 紧流形上Ginzburg-Landau方程最小-极大解的存在性及极限行为
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-06-01 DOI: 10.4310/JDG/1622743143
Daniel Stern
{"title":"Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds","authors":"Daniel Stern","doi":"10.4310/JDG/1622743143","DOIUrl":"https://doi.org/10.4310/JDG/1622743143","url":null,"abstract":"We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $geq 2$. We investigate the limiting behavior of these critical points as $varepsilon to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $varepsilon to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"118 1","pages":"335-371"},"PeriodicalIF":2.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43631625","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: 22
On the existence of harmonic $Z_2$ spinors 关于调和Z_2旋量的存在性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-03-01 DOI: 10.4310/JDG/1615487003
Aleksander Doan, Thomas Walpuski
{"title":"On the existence of harmonic $Z_2$ spinors","authors":"Aleksander Doan, Thomas Walpuski","doi":"10.4310/JDG/1615487003","DOIUrl":"https://doi.org/10.4310/JDG/1615487003","url":null,"abstract":"We prove the existence of singular harmonic Z2 spinors on 3–manifolds with b1 > 1. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic Z2 spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce [Joy17] regarding Donaldson and Segal’s proposal for counting G2–instantons [DS11].","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"117 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42370502","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: 4
Fukaya $A_infty$-structures associated to Lefschetz fibrations. III Fukaya $A_infty$ -与Lefschetz纤维相关的结构。3。
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-03-01 DOI: 10.4310/jdg/1615487005
Paul Seidel
{"title":"Fukaya $A_infty$-structures associated to Lefschetz fibrations. III","authors":"Paul Seidel","doi":"10.4310/jdg/1615487005","DOIUrl":"https://doi.org/10.4310/jdg/1615487005","url":null,"abstract":"Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"19 4","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503391","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: 0
From the Hitchin section to opers through nonabelian Hodge 从希钦部分到非贝利式霍奇的歌剧
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-02-01 DOI: 10.4310/JDG/1612975016
Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, R. Mazzeo, M. Mulase, A. Neitzke
{"title":"From the Hitchin section to opers through nonabelian Hodge","authors":"Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, R. Mazzeo, M. Mulase, A. Neitzke","doi":"10.4310/JDG/1612975016","DOIUrl":"https://doi.org/10.4310/JDG/1612975016","url":null,"abstract":"For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point $mathbf{u}$ of the base of Hitchin’s integrable system for $(G,C)$. One family $nabla_{hbar ,mathbf{u}}$ consists of $G$-opers, and depends on $hbar in mathbb{C}^times$. The other family $nabla_{R, zeta,mathbf{u}}$ is built from solutions of Hitchin’s equations, and depends on $zeta in mathbb{C}^times , R in mathbb{R}^+$. We show that in the scaling limit $R to 0, zeta = hbar R$, we have $nabla_{R,zeta,mathbf{u}} to nabla_{hbar,mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"117 1","pages":"223-253"},"PeriodicalIF":2.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45535930","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: 13
Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces K3表面上全纯盘与热带盘的对应定理
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-01-06 DOI: 10.4310/jdg/1609902017
Yu-Shen Lin
{"title":"Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces","authors":"Yu-Shen Lin","doi":"10.4310/jdg/1609902017","DOIUrl":"https://doi.org/10.4310/jdg/1609902017","url":null,"abstract":"In this paper, we prove that the open Gromov–Witten invariants defined in [20] on K3 surfaces satisfy the Kontsevich–Soibelman wall-crossing formula. One hand, this gives a geometric interpretation of the slab functions in Gross–Siebert program. On the other hands, the open Gromov–Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties [26][27] but on compact Calabi–Yau surfaces.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"20 4","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503390","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: 0
Limit of Weierstrass measure on stable curves 稳定曲线上Weierstrass测度的极限
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2020-12-17 DOI: 10.4310/jdg/1668186789
Ngai-fung Ng, Sai-Kee Yeung
{"title":"Limit of Weierstrass measure on stable curves","authors":"Ngai-fung Ng, Sai-Kee Yeung","doi":"10.4310/jdg/1668186789","DOIUrl":"https://doi.org/10.4310/jdg/1668186789","url":null,"abstract":"The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $ggeqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne-Mumford compactification $overline{mathcal M}_g$ of the moduli $mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42515865","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: 0
Index to Volume 126 第126卷索引
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2020-12-03 DOI: 10.4310/jdg/1606964419
{"title":"Index to Volume 126","authors":"","doi":"10.4310/jdg/1606964419","DOIUrl":"https://doi.org/10.4310/jdg/1606964419","url":null,"abstract":"<p><strong>Source: </strong>Journal of Differential Geometry, Volume 116, Number 3</p>","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"21 3","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503389","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: 0
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