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1.
Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology
class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and
Lie groupoids, and finally to gerbes.
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2.
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.
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3.
Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R
n+p
(c). Assume that r is even and , in this paper we introduce rth mean curvature function S
r
and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional
of , by calculation of the first variational formula of J
r
we show that x is a critical point of J
r
if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J
r
and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S
n+p
. When r = 0, noting S
0 = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit
sphere S
n+p
.
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4.
We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications of the Sasaki-Mok metric depending on one real parameter c ≠ 0. The metrics are “strongly invariant” in some special sense. In particular, we consider the case when (M, g) is a space of constant sectional curvature K. Then, for dim M > 2, we find always, among the metrics , two strongly invariant Einstein metrics on OM which are Riemannian for K > 0 and pseudo-Riemannian for K < 0. At least one of them is not locally symmetric. We also find, for dim M ≥ 2, two invariant metrics with vanishing scalar curvature.
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5.
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
6.
Let
be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian
and we prove that the limit of
when
is 2/d(M), where d(M) is the diameter of M. Moreover, if
is an oriented compact hypersurface of the Euclidean space
or
, we prove an upper bound of
in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of
then:
.
Mathematics Subject Classifications (2000): 53A07, 53C21. 相似文献
7.
<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations
Luis Sanguiao 《Annals of Global Analysis and Geometry》2008,33(3):271-292
For a Riemannian foliation on a closed manifold M, we define L
2-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L
2-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant
by foliated homotopy equivalences, and they are computed for several examples.
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8.
Ruben Jakob 《Annals of Global Analysis and Geometry》2008,33(3):231-244
In the present article, the author proves two generalizations of his “finiteness-result” (I.H.P. Anal. Non-lineaire, 2006,
accepted) which states for any extreme simple closed polygon that every immersed, stable disc-type minimal surface spanning Γ is an isolated point of the set of all disc-type minimal
surfaces spanning Γ w.r.t. the C
0-topology. First, it is proved that this statement holds true for any simple closed polygon in , provided it bounds only minimal surfaces without boundary branch points. Also requiring that the interior angles at the
vertices of such a polygon Γ have to be different from the author proves the existence of some neighborhood O of Γ in and of some integer , depending only on Γ, such that the number of immersed, stable disc-type minimal surfaces spanning any simple closed polygon
contained in O, with the same number of vertices as Γ, is bounded by .
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9.
A singular riemannian foliation on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of
Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely
generated if the sections are compact. We also prove that the leaves of coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.
Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq. 相似文献
10.
Jean-Christophe Bourgoin 《Annals of Global Analysis and Geometry》2007,32(1):1-13
In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E
f,p
among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst.
Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥ 7 for any real with , the map minimizes E
f,p
among the maps in which coincide with on .
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11.
We start with the universal covering space of a closed n-manifold and with a tree of fundamental domains which zips it . Our result is that, between T and , is an intermediary object, , obtained by zipping, such that each fiber of p is finite and admits a section.
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12.
Lars Schäfer 《Annals of Global Analysis and Geometry》2007,32(2):125-145
In this paper we study para-tt *-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt *-bundles with ${\nabla=D + S}In this paper we study para-tt
*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt
*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of
these solutions admit metric or symplectic para-tt
*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds
(M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt
*-bundles generalised para-pluriharmonic maps into , respectively, into SO
0(n,n)/U
π(C
n
), where U
π(C
n
) is the para-complex analogue of the unitary group.
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13.
We study minimal graphs in . First, we establish some relations between the geometry of the domain and the existence of certain minimal graphs. We then
discuss the problem of finding the maximal number of disjoint domains Ω ⊂ M that admit a minimal graph that vanishes on ∂Ω. When M is two-dimensional and has non-negative sectional curvature, we prove that this number is 3. This was proved by Tkachev in
.
Maria Fernanda Elbert was partially supported by CNPq and Faperj. 相似文献
14.
Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces 总被引:1,自引:0,他引:1
Joachim Toft 《Annals of Global Analysis and Geometry》2006,30(2):169-209
Let s w p be the set of all a ∈ ? such that a w (x, D) is Schatten p-operator on L 2. Then we prove the following:
- $S(m,g)\subseteq s_p^w
Let s w p be the set of all a ∈ ℓ such that a w (x, D) is Schatten p-operator on L 2. Then we prove the following:
Mathematics Subject Classifications (2000) Primary: 35S05, 47B10, 47L15 Secondary: 32F45, 16W80 相似文献• iff . Furthermore, when . Consequently, when ; • if , then is symplectically invariantly defined. Moreover, if and is slowly varying (and σ-temperate), then the same is true for G; • a generalization of sharp G?rding's inequality.
15.
Dusa McDuff 《Geometriae Dedicata》2008,132(1):1-29
We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also
obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on .
Partially supported by NSF grants DMS 0305939 and 0604769. 相似文献
16.
Let N ⊂ M be a finite Jones’ index inclusion of II1 factors and denote by U
N
⊂ U
M
their unitary groups. In this article, we study the homogeneous space U
M
/U
N
, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit of the Jones projection of the inclusion. We endow with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are
not complete); therefore, is a weak Riemannian manifold. We show that enjoys certain properties similar to classic Hilbert–Riemann manifolds. Among them are metric completeness of the geodesic
distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of
minimal geodesics. For instance, around each point p
1 of , there is a ball (of uniform radius r) of the usual norm of M, such that any point p
2 in the ball is joined to p
1 by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic
(algebraic) characterization of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and .
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17.
Oscar Mario Perdomo 《Geometriae Dedicata》2007,129(1):23-34
In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R
4, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S
3. Ann. Math. 92(2), 335–374).
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18.
Thomas Krainer 《Annals of Global Analysis and Geometry》2009,35(4):345-361
Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A
C
be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects
the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .
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19.
Dan Zaffran 《Mathematische Annalen》2008,341(4):717-733
We consider a family of holomorphic bundles constructed as follows:from any given , we associate a “multiplicative automorphism” of . Now let be a -invariant Stein Reinhardt domain. Then E
m
(D, M) is defined as the flat bundle over the annulus of modulus m > 0, with fiber D, and monodromy . We show that the function theory on E
m
(D, M) depends nontrivially on the parameters m, M and D. Our main result is that
where ρ(M) denotes the max of the spectral radii of M and M
−1. As corollaries, we: (1) obtain a classification result for Reinhardt domains in all dimensions; (2) establish a similarity
between two known counterexamples to a question of J.-P. Serre; and (3) suggest a potential reformulation of a disproved conjecture
of Siu Y.-T. 相似文献