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.
{"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}
{"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}
{"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}
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).
{"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}
{"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}
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.
{"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}
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：