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1.
LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China  相似文献   

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All normal subloops of a loopG form a modular latticeL n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.  相似文献   

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In this paper, we prove a sphere theorem for submanifolds in a Riemannian manifold with pinched positive curvature. This result generalizes a recent result of Leung.Supported by the Natural Science Foundation of China  相似文献   

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For n6 all sets of n points in the plane with three distinct distances are determined.Dedicated to Professor Dr. Laszlo Fejes Töth on the occasion of his eightieth birthday  相似文献   

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We generalize the Riesz potential of a compact domain in RmRm by introducing a renormalization of the rα−mrαm-potential for α?0α?0. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.  相似文献   

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For λ>√2 there exists a triangle-free graphG such that for nod canG be imbedded into thed-dimensional unit sphere with two points joined if and only if their distance is >λ.  相似文献   

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Summary It is shown that a finely superharmonic function in a planar fine domainU is greater than or equal to its lower integral with respect to harmonic measure associated with any bounded finely open setV with fine closure contained inU. Examples are given showing that this result does not extend to dimension 3 or more (unlessf is supposed to be, e.g., lower bounded onV) and also that the integral need not exist.  相似文献   

9.
Let f:M→Nf:MN be a smooth area decreasing map between two Riemannian manifolds (M,gM)(M,gM) and (N,gN)(N,gN). Under weak and natural assumptions on the curvatures of (M,gM)(M,gM) and (N,gN)(N,gN), we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.  相似文献   

10.
In a connected Finsler space Fn=(M,F) every ordered pair of points p,qM determines a distance ?F(p,q) as the infimum of the arc length of curves joining p to q. (M,?F) is a metric space if Fn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ?F(p,q)=?F(q,p) fails) if Fn is positively homogeneous only. It is known the Busemann-Mayer relation , for any differentiable curve p(t) in an Fn. This establishes a 1:1 relation between Finsler spaces Fn=(M,F) and (quasi-) metric spaces (M,?F).We show that a distance function ?(p,q) (with the differentiability property of ?F) needs not to be a ?F. This means that the family {(M,?)} is wider than {(M,?F)}. We give a necessary and sufficient condition in two versions for a ? to be a ?F, i.e. for a (quasi-) metric space (M,?) to be equivalent (with respect to the distance) to a Finsler space (M,F).  相似文献   

11.
We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form Mnλ. We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D0 has its q-centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D0 is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(nqd), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i>2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (nq)-plane orthogonal to P) around a d-dimensional geodesic plane.  相似文献   

12.
The perturbations of complex polynomials of one variable are considered in a wider class than the holomorphic one. It is proved that under certain conditions on a polynomial p   of the plane, the CrCr conjugacy class of a map f   in a C1C1 neighborhood of p depends only on the geometric structure of the critical set of f. This provides the first class of examples of structurally stable maps with critical points and nontrivial nonwandering set in dimension greater than one.  相似文献   

13.
We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning to such graphs. Furthermore, applications to the hyperbolic and Euclidean spaces are given.  相似文献   

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We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet  相似文献   

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David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain ΩΩ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of ΩΩ, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological   weak Finsler metric, in which the unit ball in each tangent space is naturally identified with the domain ΩΩ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure   on ΩΩ, and the unit ball of the Hilbert metric at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.  相似文献   

20.
The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5]. Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2]. In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite time. The proofs rely entirely on parabolic maximum principles. Received: 6 September 2006  相似文献   

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