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1.
Let M be an orientable connected and compact real hypersurfaceof the complex space form C(n + 1)/2. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequalityn22 (n2 – 1) + f2, where is the characteristic vectorfield, A is the shape operator and (n – 1) is the infimumof the Ricci curvatures of hypersurface M, then it is shownthat is a constant and M is the sphere Sn(2) in C(n + 1)/2.  相似文献   

2.
Let (R,m) be a local ring with prime ideals p and q such that. If R is regular and containsa field, and dim(R/p)+dim(R/q)=dim(R), then it is proved thatp(m) q(n) mm+n for all positive integers m and n. This isproved using a generalization of Serre's Intersection Theoremwhich is applied to a hypersurface R/fR. The generalizationgives conditions that guarantee that Serre's bound on the intersectiondimension (R/p)+(R/q)dim(R) holds when R is nonregular.  相似文献   

3.
Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature   总被引:3,自引:0,他引:3  
The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere Sn–1(1). The hypersurfaceSk(c1)xSnk(c2) in a unit sphere Sn+1(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sn+1(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sn+1(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S1xSn–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sn+1(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and   相似文献   

4.
On the number of homotopy types of fibres of a definable map   总被引:1,自引:0,他引:1  
In this paper we prove a single exponential upper bound on thenumber of possible homotopy types of the fibres of a Pfaffianmap in terms of the format of its graph. In particular, we showthat if a semi-algebraic set SRm+n, where R is a real closedfield, is defined by a Boolean formula with s polynomials ofdegree less than d, and : Rm+nRn is the projection on a subspace,then the number of different homotopy types of fibres of doesnot exceed s2(m+1)n(2m nd)O(nm). As applications of our mainresults we prove single exponential bounds on the number ofhomotopy types of semi-algebraic sets defined by fewnomials,and by polynomials with bounded additive complexity. We alsoprove single exponential upper bounds on the radii of ballsguaranteeing local contractibility for semi-algebraic sets definedby polynomials with integer coefficients.  相似文献   

5.
Many new universal relations are obtained between the Eulernumbers of manifolds of singular supporting hyperplanes of anarbitrary generic smooth closed k-dimensional submanifold inRn where n 7 or k = 1. These relations are applied to Barner-convexcurves in an odd-dimensional space Rn. A universal (nontrivial)linear relation is established between the numbers of singularsupporting hyperplanes of various types but of the same totalmultiplicity n of tangency with a given generic smooth closedconnected Barner-convex curve in Rn. The coefficients of thisrelation are defined by Catalan numbers.  相似文献   

6.
Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(IT)Tn || cn–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tn)is dominated by the continuous semigroup (et(IT))t 0 in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(IT)Tn || c n for all n, in terms ofestimates on the semigroup et(IT).  相似文献   

7.
The invariantly harmonic functions in the unit ball Bn in Cnare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sn),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sn,is the unique invariantly harmonic function u in Bn, continuousup to the boundary, such that u=f on Sn. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sn and invariantly harmonic functiontheory in Bn. When n 2, it is natural to consider on Sn functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.  相似文献   

8.
9.
A Topological Criterion for the Existence of Half-Bound States   总被引:1,自引:0,他引:1  
The following theorem is proved: if (M4n+1,g) is a completeRiemannian manifold and M is an oriented hypersurface partitioningM and with non-zero signature, then the spectrum of the Hodge–deRhamLaplacian is [0,]. This result is obtained by a new Callias-typeindex. This new formula links half-bound harmonic forms (thatis, nearly L2 but not in L2) with the signature of .  相似文献   

10.
McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t2, ..., tn), 0 < t< 1, maps Lp(Rn) boundedly into Lp'(Rn) if and only if 2n(n+1)/(n2+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rn maps Lp(Rn) boundedly into Lp'(Rn) if p = 2n(n+1)/(n2+n+2).  相似文献   

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