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On the topology and index of minimal surfaces II 关于极小曲面的拓扑和索引2
1区 数学
Journal of Differential Geometry Pub Date : 2023-03-01 DOI: 10.4310/jdg/1683307005
Otis Chodosh, Davi Maximo
{"title":"On the topology and index of minimal surfaces II","authors":"Otis Chodosh, Davi Maximo","doi":"10.4310/jdg/1683307005","DOIUrl":"https://doi.org/10.4310/jdg/1683307005","url":null,"abstract":"For an immersed minimal surface in $mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in $mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134996012","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
Morse homotopy for the $SU(2)$-Chern–Simons perturbation theory $SU（2）$-Chern-Simons微扰理论的Morse同胚性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2023-02-01 DOI: 10.4310/jdg/1680883580
Tatsuro Shimizu
{"title":"Morse homotopy for the $SU(2)$-Chern–Simons perturbation theory","authors":"Tatsuro Shimizu","doi":"10.4310/jdg/1680883580","DOIUrl":"https://doi.org/10.4310/jdg/1680883580","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42400811","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
Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles 具有充分正则或反正则线束的非奇异射影$n$-折叠上Chern类的不等式
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-11-01 DOI: 10.4310/jdg/1675712992
Rong Du, Hao Sun
{"title":"Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles","authors":"Rong Du, Hao Sun","doi":"10.4310/jdg/1675712992","DOIUrl":"https://doi.org/10.4310/jdg/1675712992","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49150549","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
The stable cohomology of moduli spaces of sheaves on surfaces 曲面上轮轴模空间的稳定上同调
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-06-01 DOI: 10.4310/jdg/1659987893
Izzet Coskun, Matthew Woolf
{"title":"The stable cohomology of moduli spaces of sheaves on surfaces","authors":"Izzet Coskun, Matthew Woolf","doi":"10.4310/jdg/1659987893","DOIUrl":"https://doi.org/10.4310/jdg/1659987893","url":null,"abstract":"Let X be a smooth, irreducible, complex projective surface, H a polarization on X. Let γ = (r, c,∆) be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves MX,H(γ). When the rank r = 1, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank r ≥ 1 and the first Chern class c, then the Betti numbers (and more generally the Hodge numbers) of MX,H(r, c,∆) stabilize as the discriminant ∆ tends to infinity and that the stable Betti numbers are independent of r and c. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when X is a rational surface and KX · H < 0, then the classes [MX,H(γ)] stabilize in an appropriate completion of the Grothendieck ring of varieties as ∆ tends to ∞. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when X is a rational surface, H · KX < 0 and there are no strictly semistable objects in MX,H(γ).","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44619126","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: 10
Visible actions of compact Lie groups on complex spherical varieties 复球面上紧李群的可见作用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-02-01 DOI: 10.4310/jdg/1645207534
Yu-ichi Tanaka
{"title":"Visible actions of compact Lie groups on complex spherical varieties","authors":"Yu-ichi Tanaka","doi":"10.4310/jdg/1645207534","DOIUrl":"https://doi.org/10.4310/jdg/1645207534","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45735296","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: 3
Collapsing ancient solutions of mean curvature flow 塌缩平均曲率流的古老解
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-10-01 DOI: 10.4310/jdg/1632506300
T. Bourni, Mathew T. Langford, G. Tinaglia
{"title":"Collapsing ancient solutions of mean curvature flow","authors":"T. Bourni, Mathew T. Langford, G. Tinaglia","doi":"10.4310/jdg/1632506300","DOIUrl":"https://doi.org/10.4310/jdg/1632506300","url":null,"abstract":"We construct a compact, convex ancient solution of mean curvature flow in $mathbb{R}^{n+1}$ with $O(1) times O(n)$ symmetry that lies in a slab of width $pi$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $pi$ and in no smaller slab.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665165","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: 33
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
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