共查询到20条相似文献,搜索用时 46 毫秒
1.
Vladimir Tkachev 《Complex Analysis and Operator Theory》2010,4(3):685-700
In this paper, we construct two infinite families of algebraic minimal cones in
^n{\mathbb{R}^{n}}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes
of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a
byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in
\mathbbRn{\mathbb{R}^{n}} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in
\mathbbRm2{\mathbb{R}^{m^2}}, m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in
\mathbbRn{\mathbb{R}^{n}} posed by Wu-Yi Hsiang in the 1960s. 相似文献
2.
We consider the Radon transform on the (flat) torus
\mathbbTn = \mathbbRn/\mathbbZn{\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^n} defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give
a characterization of the image of the space of smooth functions on
\mathbbTn{\mathbb{T}^{n}} . 相似文献
3.
Denote by
\mathbbHn{\mathbb{H}^n} the 2n + 1 dimensional Heisenberg group. We show that the pairs
(\mathbbRk ,\mathbbHn){(\mathbb{R}^k ,\mathbb{H}^n)} and
(\mathbbHk ,\mathbbHn){(\mathbb{H}^k ,\mathbb{H}^n)} do not have the Lipschitz extension property for k > n. 相似文献
4.
Erwan Le Gruyer 《Geometric And Functional Analysis》2009,19(4):1101-1118
We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain
\mathbbRn{\mathbb{R}^n} with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun’s extension theorem for
Lipschitz functions and, therefore, establishes a link between Kirszbraun’s theorem and Whitney’s theorem. In fact this result
holds not only in Euclidean
\mathbbRn{\mathbb{R}^n} but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable
extension problem in Euclidean spaces and we show that no Brudnyi–Shvartsman-type theorem holds for this last problem. We
conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was
originally studied by Aronsson in the continuous case. 相似文献
5.
Juan A. Aledo Victorino Lozano José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270
We prove that the only compact surfaces of positive constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant
angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive
constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds
are attained, the surface is again a piece of a rotational complete surface. 相似文献
6.
We study hypersurfaces in the Lorentz-Minkowski space
\mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L
k
ψ = Aψ + b, where L
k
is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1,
A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and
b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces
\mathbbSn1(r){\mathbb{S}^n_1(r)} or
\mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders
\mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}},
\mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or
\mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in
\mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006). 相似文献
7.
8.
We study maximal L
p
-regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder
\mathbbRn ×\mathbbR{\mathbb{R}^n \times \mathbb{R}} or
X ×\mathbbR{X \times \mathbb{R}} , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize
the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms
between suitable L
p
-Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse
of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems
and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction. 相似文献
9.
The field of quaternions, denoted by
\mathbbH{\mathbb{H}} can be represented as an isomorphic four dimensional subspace of
\mathbbR4×4{\mathbb{R}^{4\times 4}}, the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional
subspace in
\mathbbR4×4{\mathbb{R}^{4\times 4}} which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called
field of pseudoquaternions. It exists in
\mathbbR4×4{\mathbb{R}^{4\times 4}} but not in
\mathbbH{\mathbb{H}}. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in
\mathbbR4{\mathbb{R}^4}. And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b. 相似文献
10.
We study diophantine properties of a typical point with respect to measures on
\mathbbRn .\mathbb{R}^n . Namely, we identify geometric conditions on a measure μ on
\mathbbRn \mathbb{R}^n guaranteeing that μ-almost every
y ? \mathbbRn {\bf y}\,\in\,\mathbb{R}^n is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples
include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of
examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products
and pushforwards by certain smooth maps. 相似文献
11.
Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475
Let
\mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1, j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let
[`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in
\mathbb Cn{\mathbb {C}^n}. Consider the ring
Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f is continuous and f(z)=[`(f([`(z)]))] (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\} 相似文献
12.
Peng Zhu 《Archiv der Mathematik》2011,97(3):271-279
We prove that a complete noncompact orientable stable minimal hypersurface in
\mathbbSn+1{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L
2-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in
\mathbbRn+1{\mathbb{R}^{n+1}} or
\mathbbSn+1{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L
2-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in
\mathbbRn+1{\mathbb{R}^{n+1}}. 相似文献
13.
Stefan Nemirovski 《Geometric And Functional Analysis》2009,19(3):902-909
It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into
\mathbbR2n{\mathbb{R}^{2n}} if and only if n is odd. 相似文献
14.
We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber
\mathbbRn{\mathbb{R}^n} . This amounts to an alternative proof of Novikov’s theorem on the topological invariance of the rational Pontryagin classes
of vector bundles. Transversality arguments and torus tricks are avoided. 相似文献
15.
For n = 1, the space of ${\mathbb{R}}
16.
In this article we consider variable coefficient time dependent wave equations in \mathbb R ×\mathbb Rn{\mathbb {R} \times \mathbb {R}^n} . Using phase space methods we construct outgoing parametrices and prove Strichartz type estimates globally in time. This is done in the context of C 2 metrics which satisfy a weak asymptotic flatness condition at infinity. 相似文献
17.
Mahmoud Baroun Lahcen Maniar Roland Schnaubelt 《Integral Equations and Operator Theory》2009,65(2):169-193
We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with
inhomogeneous boundary values on
\mathbbR{\mathbb{R}} and
\mathbbR±\mathbb{R}_{\pm}, if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the ‘Acquistapace–Terreni’
conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals ( − ∞, − T] and [T, ∞), then we obtain a Fredholm alternative of the equation on
\mathbbR{\mathbb{R}} in the space of functions being asymptotically almost periodic on
\mathbbR+{\mathbb{R}}_{+} and
\mathbbR-\mathbb{R}_{-}. 相似文献
18.
In this paper we study certain groups of bilipschitz maps of the boundary minus a point of a negatively curved space of the
form
\mathbbR \ltimesM \mathbbRn{\mathbb{R} \ltimes_{M} \mathbb{R}^{n}}, where M is a matrix whose eigenvalues all lie outside of the unit circle. The case where M is diagonal was previously studied by Dymarz (Geom Funct Anal (GAFA) 19:1650–1687, 2009). As an application, combined with work of Eskin-Fisher-Whyte and Peng, we provide the last steps in the proof of quasi-isometric
rigidity for a class of lattices in solvable Lie groups. 相似文献
19.
The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}
20.
We study the limiting behavior of the K?hler–Ricci flow on
\mathbbP(O\mathbbPn ?O\mathbbPn(-1)?(m+1)){{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}} for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses
to
\mathbbPn{{\mathbb{P}^n}} or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the K?hler–Ricci flow resolves a certain type of cone singularities
in the Gromov–Hausdorff sense. 相似文献
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