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1.
Let M n , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C s , where C s is the Sobolev constant for M, then there are no L 2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.  相似文献   

2.
A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C n :n = 1,2,..}, allows for constructing measures nxCx ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M
f(x)=?g ? Gf(g)nCx(g). f(x)=\sum_{\gamma\in\Gamma}f(\gamma)\nu^{\mathbf{C}}_x(\gamma).  相似文献   

3.

It is well known that the harmonic functions u on the Euclidean upper (n + 1)-dimensional half-space E+ n+1 = {(x, y) = (x 1,…,xn,y) ? E n+1: y > 0} satisfying sup y>0 ||u(·,y)||1 < ∞ are precisely the Poisson-integrals u(x,y) = ∫ En P(x - t,y)(t) with respect to a measure μ of finite variation on En , and that (Fatou's theorem in E+ n+1) in almost every point x ? En the non-tangential boundary limit of u exists and coincides with du/dλ. While this is a special case of a general assertion in potential theory, it is shown that the proof of Fatou's theorem for harmonic functions on a ball may readily be transferred to the given setup and that the influence of a singular component of μ on the boundary behaviour of u may also be established without recourse to the existence of the derivative dμ/dλ. Finally the C 0-property of u is characterized by suitable conditions on μ.  相似文献   

4.
Let f be a bi-Lipschitz mapping of the Euclidean ball B n into ℓ2 with both Lipschitz constants close to one. We investigate the shape of f(B n). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of n.  相似文献   

5.
Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P *-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R 01 or R 02-functions to keep the property of bounded level sets. This work is partially supported by National Science Council of Taiwan.  相似文献   

6.
An immersed surface M in N n ×ℝ is a helix if its tangent planes make constant angle with t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N n ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.  相似文献   

7.
《Quaestiones Mathematicae》2013,36(4):527-530
In this paper, we prove that there exists an infinite dimensional closed vector space M of harmonic functions in R n such that each v ? M \{0} is a universal harmonic function.  相似文献   

8.
We sharpen the remainder estimate of the asymptotic formula for the eigenvalue distribution for the degenerate operator -div{(1−|x|2)grad·} in the unit ball of the Euclidean spaceR n . In particular we find the second term whenn=2.  相似文献   

9.
On the setting of the half-spaceR n–1×R +, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.  相似文献   

10.
We consider a finite subgroup n of the group O(N) of orthogonal matrices, where N = 2 n , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 n -dimensional Euclidean space R N . We prove that representations of the group n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit n,2 x t of any initial point x on the sphere S N – 1 is a 7-design in the Euclidean space of dimension 2 n .  相似文献   

11.
We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS n ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM 1#M 2 with certain metrics is close to the union of the spectra ofM 1 and ofM 2.Partially supported by SFB 256 and by the GADGET program of the EU  相似文献   

12.
For a convex body K ⊂ ℝn and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K 0 ⊂ ℝn be smooth convex bodies with boundaries of class C 2 and positive Gauss-Kronecker curvature and assume K 0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K 0 are homothetic if their i-th and j-th projection functions are proportional. When K 0 is a Euclidean ball this shows that a convex body with C 2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.  相似文献   

13.

The authors prove a version, in utmost generality, of the Bun Wong-Rosay theorem on a complex manifold M. The essence of the result is that a domain Ω?M with non-compact automorphism group and boundary orbit accumulation point that is strongly pseudoconvex must be biholomorphic to the unit ball in C n .  相似文献   

14.
For a submanifoldM n of a Riemannian manifoldM q, the concept of a torsion bivector at the point x M n for given one- and two-dimensional directions fromT x M n is introduced using only the first and second fundamental forms ofM n. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM n is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM n into some totally geodesicM n+1 inM q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM 2 inM q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.  相似文献   

15.
The tensor structure of spaces L p (R n ) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L p (R 1) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated.  相似文献   

16.

In this paper we generalize a number of known integral inequalities for analytic functions defined on the unit ball B ? C n or on the polydisk Un .  相似文献   

17.
We consider the problem of monotonicity testing over graph products. Monotonicity testing is one of the central problems studied in the field of property testing. We present a testing approach that enables us to use known monotonicity testers for given graphs G1, G2, to test monotonicity over their product G1 × G2. Such an approach of reducing monotonicity testing over a graph product to monotonicity testing over the original graphs, has been previously used in the special case of monotonicity testing over [n]d for a limited type of testers; in this article, we show that this approach can be applied to allow modular design of testers in many interesting cases: this approach works whenever the functions are boolean, and also in certain cases for functions with a general range. We demonstrate the usefulness of our results by showing how a careful use of this approach improves the query complexity of known testers. Specifically, based on our results, we provide a new analysis for the known tester for [n]d which significantly improves its query complexity analysis in the low‐dimensional case. For example, when d = O(1), we reduce the best known query complexity from O(log 2n/ε) to O(log n/ε). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008  相似文献   

18.
Ricci curvature and the topology of open manifolds   总被引:6,自引:0,他引:6  
In this paper, we prove that an open Riemannian n-manifold with Ricci curvature and for some is diffeomorphic to a Euclidean n-space if the volume growth of geodesic balls around p is not too far from that of the balls in . We also prove that a complete n-manifold M with is diffeomorphic to if , where is the volume of unit ball in . Received 5 May, 1997  相似文献   

19.
We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) n (R n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG n (R n+p). As a consequence of this formula we get a rigidity theorem.  相似文献   

20.
We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \mathbbRn{{\mathbb{R}}^n}, n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \mathbbRn{{\mathbb{R}}^n}, n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B n of \mathbbRn{{\mathbb{R}}^n}, the M{{\mathcal{M}}}-invariant measure on the unit ball B 2n of \mathbbCn{{\mathbb{C}}^n}, n ≥ 1, and the quasihyperbolic measure on any domain D ì \mathbbRn{D\subset {\mathbb{R}}^n}, D 1 \mathbbRn{D\ne {\mathbb{R}}^n}. Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also u p is quasi-nearly subharmonic for all p > 0.  相似文献   

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