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1.
HE Yijun & WANG Changping LMAM School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2005,48(3):341-349
Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(?) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(?)RPn(?) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (?) RPn(?) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space. 相似文献
2.
There are examples of complete spacelike surfaces in the Lorentzian product ℍ2 × ℝ1 with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ2 × ℝ1 with constant Gaussian curvature K > −1. 相似文献
3.
4.
V. Yaskin 《Journal of Geometric Analysis》2006,16(4):735-745
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝn with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer
to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve
the lower dimensional Busemann-Petty problem in the hyperbolic space ℍn. 相似文献
5.
O. D. Frolkina 《Moscow University Mathematics Bulletin》2009,64(6):253-258
In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ.
In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x
n
)
n∈ℤ ∈ [0, 1]ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case
of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings
X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s
results are obtained. 相似文献
6.
It has been recently conjectured that, in the context of the Heisenberg group ℍn endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the
class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties
of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by
the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg
bubbles are actually isoperimetric among the whole class of measurable subsets of ℍn, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn–Minkowski
inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F⊂ℍn of class C2 can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed.
Mathematics Subject Classification (2000) 28A75, 22E25, 49Q20 相似文献
7.
Let G be a discrete subgroup of PU(1,n). Then G acts on ℙℂ
n
preserving the unit ball ℍℂ
n
, where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙℂ
n
and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙℂ
n
which are tangent to ∂ℍℂ
n
at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍℂ
n
which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper
k
-chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space. 相似文献
8.
Sigmundur Gudmundsson 《manuscripta mathematica》1997,93(1):421-433
Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces
ℍH
n, ℂH
n, ℝH
2
t+1, ℍP
n, ℂP
n and ℝP
2n+1 of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases.
This work was supported by The Swedish Natural Science Research Council.
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. 相似文献
9.
We classify all surfaces in ℍ2 × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ2 is the hyperbolic plane.
The author was supported by grants CEEX ET 5883/2006-2008 and PNII ID_ 398/2007-2010 ANCS (Romania). 相似文献
10.
M. V. Korobkov 《Siberian Advances in Mathematics》2010,20(4):256-284
We say that a domain U ⊂ ℝ
n
is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain
on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝ
n
such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination
of a domain by the relative metric of its Hausdorff boundary. 相似文献
11.
This note proves that, forF = ℝ, ℂ or ℍ, the bordism classes of all non-bounding Grassmannian manifoldsG
k(F
n+k), withk <n and having real dimensiond, constitute a linearly independent set in the unoriented bordism group N
d
regarded as a ℤ2-vector space. 相似文献
12.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
13.
Maciej Paluszyński Hrvoje Šikić Guido Weiss Shaoliang Xiao 《Journal of Geometric Analysis》2001,11(2):311-342
A tight frame wavelet ψ is an L
2(ℝ) function such that {ψ jk(x)} = {2j/2
ψ(2
j
x −k), j, k ∈ ℤ},is a tight frame for L
2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight
frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained
from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions
and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained
by other authors. 相似文献
14.
In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ
n
which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal
vector fields in ℍ
n
.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean
curvature (CMC) hypersurface. Our definition coincides with previous ones.
Our main result describes which are the CMC hypersurfaces of revolution in ℍ
n
.The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential
equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart
in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean
space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ
n
. 相似文献
15.
In this article we extend the notion of constant angle surfaces in $
\mathbb{S}^2
$
\mathbb{S}^2
× ℝ and ℍ2 × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give
a complete local classification in the Heisenberg group. 相似文献
16.
Antonio J. Di Scala 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2009,79(1):37-46
Let M⊂ℝ
n
be a submanifold of a euclidean space. A vector d∈ℝ
n
is called a helix direction of M if the angle between d and any tangent space T
p
M is constant. Let ℋ(M) be the set of helix directions of M. If the set ℋ(M) contains r linearly independent vectors we say that M is a weak r-helix. We say that M is a strong r-helix if ℋ(M) is a r-dimensional linear subspace of ℝ
n
. For curves and hypersurfaces both definitions agree. The object of this article is to show that these definitions are not
equivalent. Namely, we construct (non strong) weak 2-helix surfaces of ℝ4.
The author is supported by the Project M.I.U.R. “Riemann Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.,
Italy. 相似文献
17.
DunYanYAN GuoEnHU 《数学学报(英文版)》2005,21(1):209-214
In this paper, the authors discuss a class of multilinear singular integrals and obtain that the operators are bounded from H^1(R^n) to weak L^1(R^n). Using this result, we can directly prove a main theorem in [5]. 相似文献
18.
Károly J. BöröczkyJr Lars Michael Hoffmann Daniel Hug 《Periodica Mathematica Hungarica》2008,57(2):143-164
Let K be a convex body in ℝ
d
, let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C
+2, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C
+3) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K.
Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953). 相似文献
19.
By a “reproducing” method forH =L
2(ℝ
n
) we mean the use of two countable families {e
α : α ∈A}, {f
α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e
α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e
α >:f
α.
A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature
in common: they are generated by a single or a finite collection of functions by applying to the generators two countable
families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor
systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety
of wavelets) involve translations and dilations.
A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article
we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities.
Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach
that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need
not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ
n
. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations
for different kinds of dilation matrices. 相似文献
20.
Marty Ross 《Journal of Geometric Analysis》1998,8(2):313-317
Let S ⊂ ℝn be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x1 (M1) ∪ … ∪ xr (Mr) where each Mj is connected, xj: Mj → Vj is a classical minimal immersion into an affine subspace Vj of ℝn, and the subspaces V1,…, Vr are pairwise orthogonal. Here we prove that if Mj is orientable, then xj (Mj) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola. 相似文献