Journal of Differential Geometry最新文献

{"title":"Scalar curvature and harmonic maps to $S^1$","authors":"Daniel Stern","doi":"10.4310/jdg/1669998185","DOIUrl":"https://doi.org/10.4310/jdg/1669998185","url":null,"abstract":"For a harmonic map $u:M^3to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2pi int_{thetain S^1}chi(Sigma_{theta})geq frac{1}{2}int_{thetain S^1}int_{Sigma_{theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Sigma_{theta}=u^{-1}{theta}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;mathbb{Z})$ in terms of $|R_M^-|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(min R_M)sys_2(M)leq 8pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42654624","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":"Riemann moduli spaces are quantum ergodic","authors":"Dean Baskin, Jesse Gell-Redman, X. Han","doi":"10.4310/jdg/1683307003","DOIUrl":"https://doi.org/10.4310/jdg/1683307003","url":null,"abstract":"In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48304673","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":"Bounded multiplicity for eigenvalues of a circular vibrating clamped plate","authors":"Y. Lvovsky, D. Mangoubi","doi":"10.4310/jdg/1659987895","DOIUrl":"https://doi.org/10.4310/jdg/1659987895","url":null,"abstract":"We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44488050","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":"On the geometric structure of currents tangent to smooth distributions","authors":"G. Alberti, A. Massaccesi, E. Stepanov","doi":"10.4310/jdg/1668186786","DOIUrl":"https://doi.org/10.4310/jdg/1668186786","url":null,"abstract":"It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47939071","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":"Area estimates for high genus Lawson surfaces via DPW","authors":"Lynn Heller, Sebastian Heller, M. Traizet","doi":"10.4310/jdg/1685121318","DOIUrl":"https://doi.org/10.4310/jdg/1685121318","url":null,"abstract":"Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $xi_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ xi_{m,k}$ in terms of their genus $g=m k gg1$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48615380","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":"Index to Volume 112","authors":"","doi":"10.4310/jdg/1573787080","DOIUrl":"https://doi.org/10.4310/jdg/1573787080","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45392349","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":"Nonclassical minimizing surfaces with smooth boundary","authors":"Camillo De Lellis, G. Philippis, J. Hirsch","doi":"10.4310/jdg/1669998183","DOIUrl":"https://doi.org/10.4310/jdg/1669998183","url":null,"abstract":"We construct a Riemannian metric $g$ on $mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $Gammasubset mathbb R^4$ such that the unique area minimizing surface spanned by $Gamma$ has infinite topology. Furthermore the metric is almost Kahler and the area minimizing surface is calibrated.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41918502","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":"Global Nash–Kuiper theorem for compact manifolds","authors":"Wentao Cao, L. Sz'ekelyhidi","doi":"10.4310/jdg/1668186787","DOIUrl":"https://doi.org/10.4310/jdg/1668186787","url":null,"abstract":"We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,theta}$ isometric immersions of compact manifolds with optimal Holder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $theta<1/5$. This extends previous results on embedding 2-discs as well as higher dimensional analogues.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46973800","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":"Rigidity of pairs of rational homogeneous spaces of Picard number $1$ and analytic continuation of geometric substructures on uniruled projective manifolds","authors":"N. Mok, Yunxin Zhang","doi":"10.4310/JDG/1559786425","DOIUrl":"https://doi.org/10.4310/JDG/1559786425","url":null,"abstract":"Building on the geometric theory of uniruled projective manifolds by Hwang-Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong-Mok and Hong-Park have studied standard embeddings between rational homogeneous spaces X = G/P of Picard number 1. Denoting by S ⊂ X an arbitrary germ of complex submanifold which inherits from X a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space X0 = G0/P0 of Picard number 1 embedded in X = G/P as a linear section through a standard embedding, we say that (X0, X) is rigid if there always exists some γ ∈ Aut(X) such that S is an open subset of γ(X0). We prove that a pair (X0, X) of sub-diagram type is rigid whenever X0 is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds (X,K), for which we introduce a general notion of sub-VMRT structures π : C (S) → S, proving that they are rationally saturated under an auxiliary condition on the intersection C (S) := C (X) ∩ PT (S) and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree 1 and that distributions spanned by sub-VMRTs are bracket generating, we prove that S extends to a subvariety Z ⊂ X. For its proof, starting with a “Thickening Lemma ” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold (S;x0) and hence the associated germ of sub-VMRT structure on (S;x0) can be propagated along chains of “thickening ” curves issuing from x0, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion Z of S as its image under","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1559786425","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43673951","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":"Band width estimates via the Dirac operator","authors":"Rudolf Zeidler","doi":"10.4310/jdg/1668186790","DOIUrl":"https://doi.org/10.4310/jdg/1668186790","url":null,"abstract":"Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M times [-1,1]$ with scalar curvature bounded below by $sigma > 0$, the distance between the boundary components of $V$ is at most $C_n/sqrt{sigma}$, where $C_n = sqrt{(n-1)/{n}} cdot C$ with $C < 8(1+sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M times mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the \"$mathcal{KO}$-width\" of a closed manifold and deduce that infinite $mathcal{KO}$-width is an obstruction to positive scalar curvature.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49192743","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}