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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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
基于Hwang-Mok的无规射影流形的几何理论，该理论依赖于从代数几何和微分几何角度对极小有理切线(VMRTs)的变化的研究，Mok、Hong-Mok和Hong-Park研究了Picard数1的有理齐次空间X = G/P之间的标准嵌入。用S∧X表示一个复子流形的任意子形，它从X继承了一个几何结构，该几何结构是由vmrt与切子空间的相交定义的，并通过标准嵌入在X = G/P中的Picard数1的有理齐次空间X0 = G0/P0上建模为线性截面，如果总存在某些γ∈Aut(X)使得S是γ(X0)的开子集，则我们说(X0, X)是刚性的。我们证明了子图型对(X0, X)在X0为非线性时是刚性的，它在厄米对称情况下恢复了非线性光滑舒伯特循环的舒伯特刚性，并且在一般理性齐次情况下超越了先前处理全纯映射像的工作。我们的方法应用于无规投影流形(X,K)，为此我们引入了子vmrt结构π: C (S)→S的一般概念，证明了它们在相交C (S):= C (X)∩PT (S)的辅助条件下是合理饱和的，并证明了在vmrt上用第二基本形式表示的子结构的非简并性条件。在最小有理曲线为1次且由子vmrt张成的分布为夹角生成的附加假设下，我们证明S扩展到子变种Z∧x。为了证明它，我们从在某些标准有理曲线周围产生光滑项圈的“增厚引理”出发，证明子流形(S;x0)的子嗣以及(S;x0)上的子vmrt结构的相关子嗣可以沿着由x0发出的“增厚”曲线链传播。并通过解析延拓构造紧化后一族的有理曲线链的射影族，从而构造S的射影补全Z作为其像
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.
{"title":"Homogenization of random quasiconformal mappings and random Delauney triangulations","authors":"O. Ivrii, V. Marković","doi":"10.4310/jdg/1689262063","DOIUrl":"https://doi.org/10.4310/jdg/1689262063","url":null,"abstract":"In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43462021","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 develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Mobius energy. For the Mobius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the Mobius energy under Mobius transforms, which fails for all the other O'hara energies. We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.
{"title":"On O’Hara knot energies I: Regularity for critical knots","authors":"S. Blatt, P. Reiter, A. Schikorra","doi":"10.4310/jdg/1664378616","DOIUrl":"https://doi.org/10.4310/jdg/1664378616","url":null,"abstract":"We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Mobius energy. \u0000For the Mobius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the Mobius energy under Mobius transforms, which fails for all the other O'hara energies. \u0000We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44081165","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 consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$ we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.
{"title":"On the Index of Willmore spheres","authors":"J. Hirsch, E. Mader-Baumdicker","doi":"10.4310/jdg/1685121319","DOIUrl":"https://doi.org/10.4310/jdg/1685121319","url":null,"abstract":"We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$ we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44585087","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}
The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $#^n(S^1times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diffeomorphic to the standard 4-sphere in the special case that one of the 2-handles is attached along a knot of the form $Q_{p,q} = T_{p,q}#T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf. Along the way, we use thin position techniques from Heegaard theory to give a characterization of 2R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counterexamples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.
{"title":"Generalized square knots and homotopy $4$-spheres","authors":"J. Meier, Alexander Zupan","doi":"10.4310/jdg/1668186788","DOIUrl":"https://doi.org/10.4310/jdg/1668186788","url":null,"abstract":"The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $#^n(S^1times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diffeomorphic to the standard 4-sphere in the special case that one of the 2-handles is attached along a knot of the form $Q_{p,q} = T_{p,q}#T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf. \u0000Along the way, we use thin position techniques from Heegaard theory to give a characterization of 2R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counterexamples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48464413","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 prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs.
{"title":"Scherk-like translators for mean curvature flow","authors":"D. Hoffman, F. Mart'in, B. White","doi":"10.4310/jdg/1675712995","DOIUrl":"https://doi.org/10.4310/jdg/1675712995","url":null,"abstract":"We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43048590","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}
One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds X1 and X2 of dimension 2n−1, is there a non-constant CR morphism between them? In this paper, we use Kohn-Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specifically, if dimH KR(X1) < dimH p,q KR(X2) for any (p, q) with 1 ≤ q ≤ n− 2, then there is no non-constant CR morphism from X1 to X2.
{"title":"Kohn–Rossi cohomology and nonexistence of CR morphisms between compact strongly pseudoconvex CR manifolds","authors":"S. Yau, Huaiqing Zuo","doi":"10.4310/JDG/1552442610","DOIUrl":"https://doi.org/10.4310/JDG/1552442610","url":null,"abstract":"One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds X1 and X2 of dimension 2n−1, is there a non-constant CR morphism between them? In this paper, we use Kohn-Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specifically, if dimH KR(X1) < dimH p,q KR(X2) for any (p, q) with 1 ≤ q ≤ n− 2, then there is no non-constant CR morphism from X1 to X2.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1552442610","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41511164","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}
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