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1.
Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S
n
, and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC. 相似文献
2.
Olivier Druet Emmanuel Hebey 《Calculus of Variations and Partial Differential Equations》2008,31(2):205-230
Let (M, g) be a smooth compact Riemannian n-manifold, n ≥ 3. Let also p ≥ 1 be an integer, and be the vector space of symmetrical p × p real matrix. We consider critical elliptic systems of equations which we write in condensed form as
where , is a p-map, is the Laplace–Beltrami operator acting on p-maps, and 2* is the critical Sobolev exponent. We fully answer the question of getting sharp asymptotics for local minimal
type solutions of such systems. As an application, we prove compactness of minimal type solutions and prove that the result
is sharp by constructing explicit examples where blow-up occurs when the compactness assumptions are not fulfilled. 相似文献
3.
Oscar Perdomo 《Journal of Geometry》2006,84(1-2):100-105
In this paper we prove that if
is a closed minimal surface, then,
, for any homogeneous polynomial f of degree 3 with 0 a regular value of the function
. 相似文献
4.
We solve Blaschke’s problem for hypersurfaces of dimension . Namely, we determine all pairs of Euclidean hypersurfaces that induce conformal metrics on M
n
and envelop a common sphere congruence in . 相似文献
5.
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that
This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem.
The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with , where denotes the first positive eigenvalue of the Laplace operator. 相似文献
6.
7.
Let (M,
, g) be a sub-Riemannian manifold (i.e. M is a smooth manifold,
is a smooth distribution on M and g is a smooth metric defined on
) such that the dimension of M is either 3 or 4 and
is a contact or odd-contact distribution, respectively. We construct an adapted connection on M and use it to study the equivalence problem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric. 相似文献
8.
A submanifold M
n
r
of Minkowski space
is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of
to the tangent space of M
n
r
at every point of M
n
r
. In this paper we completely classify hypersurfaces of restricted type in
. More precisely, we prove that a hypersurface of
is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S
k
×
, S
k
1
×
, H
k
×
, S
n
1
, H
n
, with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium. 相似文献
9.
William Arveson 《Mathematische Annalen》2009,343(4):757-771
Let M and N be full matrix algebras. A unital completely positive (UCP) map is said to preserve entanglement if its inflation has the following property: for every maximally entangled pure state ρ of , is an entangled state of . We show that there is a dichotomy in that every UCP map that is not entanglement breaking in the sense of Horodecki–Shor–Ruskai
must preserve entanglement, and that entanglement preserving maps of every possible rank exist in abundance. We also show
that with probability 1, all UCP maps of relatively small rank preserve entanglement, but that this is not so for UCP maps of maximum rank. 相似文献
10.
11.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature , provided M is asymptotically harmonic of constant h > 0.
Received: 4 October 2007 相似文献
12.
For a bounded function defined on
, let
be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries
are equal to
These matrices can be thought of as variable-coefficient Toeplitz matrices or
generalized Toeplitz matrices. Matrices of the above form can be also thought
of as the discrete analogue of pseudodifferential operators. Under a certain
smoothness assumption on the function , we prove that
where the constant c1 and a part of c2 are shown to have explicit integral
representations. The other part of c2 turns out to have a resemblance to the
Toeplitz case. This asymptotic formula can be viewed as a generalization of
the classical theory on singular values of Toeplitz matrices. 相似文献
13.
Florent Schaffhauser 《Mathematische Annalen》2008,342(2):405-447
The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ?M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π1(Σ = ?M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations ${\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and
Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians,
vol.1, 2 (Zürich, 1994), pp. 526–536. Birkh?user, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer
homology of the representation space Hom(π1(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the
3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations of the fundamental group π
g,l
of a punctured Riemann surface Σ
g,l
into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on . We show that the involution is induced by a form-reversing involution β defined on the quasi-Hamiltonian space . The fact that has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser
(A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. ). The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.
Supported by the Japanese Society for Promotion of Science (JSPS). 相似文献
14.
Let be the classical kernel density estimator based on a kernel K and n independent random vectors X
i
each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h
n
that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error,
thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of
Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical
processes, needed in the proofs, are also included.
相似文献
15.
Let M be a compact oriented minimal
hypersurface of the unit n-dimensional sphere
Sn.
It is known that if the norm squared of the second fundamental form,
, satisfies that
for all
, then M is isometric to a Clifford
minimal hypersurface ([2], [5]). In this paper we will generalize this result
for minimal hypersurfaces with two principal curvatures and dimension greater
than 2. For these hypersurfaces we will show that if the average of the function
is n - 1, then M
must be a Clifford hypersurface.
Received: 24 December 2002 相似文献
16.
Bing Ye Wu 《Annals of Global Analysis and Geometry》2007,31(4):375-384
Let
be a Minkowski 3-space of Randers type with
, where
is the Euclidean metric and
. We consider minimal surfaces in
and prove that if a connected surface M in
is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel
translation of
, M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x
3-axis.
相似文献
17.
18.
In this paper, we give an Lp-Lq-version of Morgans theorem for the Dunkl-Bessel transform
on
More precisely, we prove that for all
and
then for all measurable function f on
the conditions
and
imply f = 0, if and only if
where
are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004 相似文献
19.
In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression
are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space
with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to
with n ≥ 3, where
is the scale of Hilbert spaces associated with L in
相似文献
20.
A class of manifolds which admit an f-structure with s-dimensional parallelizable kernel is introduced and studied. Such manifolds are Kenmotsu manifolds if s = 1, and carry a locally conformal K?hler structure of Kashiwada type when s = 2. The existence of several foliations allows to state some local decomposition theorems. The Ricci tensor together with
Einstein-type conditions and f-sectional curvatures are also considered. Furthermore, each manifold carries a homogeneous Riemannian structure belonging
to the class
of the classification stated by Tricerri and Vanhecke, provided that it is a locally symmetric space.
Dedicated to the memory of Professor Aldo Cossu 相似文献