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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
{"title":"Creating Stein surfaces by topological isotopy","authors":"Robert E. Gompf","doi":"10.4310/jdg/1695236593","DOIUrl":"https://doi.org/10.4310/jdg/1695236593","url":null,"abstract":"We combine Freedman’s topology with Eliashberg’s holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded $2$-complex $K$ in a complex surface, after $C^0$-small topological ambient isotopy, is the intersection of an uncountable nested family of Stein regular neighborhoods that are all topologically ambiently isotopic rel $K$, but frequently realize uncountably many diffeomorphism types. These arise from the Cantor set levels of a topological mapping cylinder. The boundaries of the neighborhoods are $3$-manifolds that are only topologically embedded, but still satisfy a notion of pseudoconvexity. Such $3$-manifolds share some basic properties of hypersurfaces that are strictly pseudoconvex in the usual smooth sense, but they are far more common. The complementary notion of topological pseudoconcavity is realized by uncountably many diffeomorphism types homeomorphic to $mathbb{R}^4$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"22 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":"135637914","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
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
{"title":"Isoperimetry and volume preserving stability in real projective spaces","authors":"Celso Viana","doi":"10.4310/jdg/1695236595","DOIUrl":"https://doi.org/10.4310/jdg/1695236595","url":null,"abstract":"We classify the volume preserving stable hypersurfaces in the real projective space $mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $mathbb{RP}^k subset mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on $mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $mathbb{S}^n$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"13 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":"135685689","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
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
{"title":"Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $mathrm{SU}(m)$ or $mathrm{U}(m)$","authors":"D. joyce, M. Upmeier","doi":"10.4310/jdg/1686931600","DOIUrl":"https://doi.org/10.4310/jdg/1686931600","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47224374","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
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
{"title":"The renormalized volume of a $4$-dimensional Ricci-flat ALE space","authors":"Olivier Biquard, Hans-Joachim Hein","doi":"10.4310/jdg/1683307004","DOIUrl":"https://doi.org/10.4310/jdg/1683307004","url":null,"abstract":"We introduce a natural definition of the renormalized volume of a $4$-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of $4$-dimensional Ricci-flat ALE spaces are Kronheimer’s gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer’s period map.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"55 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":"136096819","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
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
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