共查询到20条相似文献,搜索用时 944 毫秒
1.
J.-M. Schlenker 《Geometric And Functional Analysis》2002,12(2):395-435
A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S
2
with curvature K > —1 is induced on a unique convex surface in H
3
. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon
happens with yet another metric on immersed surfaces, which we call the horospherical metric.?This result extends in higher
dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics
obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced
metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a
good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.?The results concerning the
third fundamental form are obtained using a duality between H
3
and the de Sitter space . In the same way, the results concerning the horospherical metric are proved through a duality between H
n
and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.
Submitted: March 2001, Revised: November 2001. 相似文献
2.
Michael T. Anderson 《Selecta Mathematica, New Series》2010,16(3):343-375
We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on compact manifolds M with boundary ∂M. Given a metric γ on ∂M, this is the issue of whether any Killing field X of (∂M, γ) extends to a Killing field of any Einstein metric (M, g) bounding (∂M, γ). Under a mild condition on the fundamental group, this is proved to be the case at least when X preserves the mean curvature of ∂M in (M, g). 相似文献
3.
Pengzi Miao Luen-Fai Tam 《Calculus of Variations and Partial Differential Equations》2009,36(2):141-171
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive
a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on
which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary
can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of
a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two
volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation
formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional. 相似文献
4.
We consider the measure of points, the measure of lines and the measure of planes intersecting a given convex body K in a space form. We obtain some integral formulas involving the width of K and the curvature of its boundary ∂K. Also we study the special case of constant width. Moreover we obtain a generalisation of the Heintze–Karcher inequality
to space forms.
Work partially supported by grant number ACI2003-44 Joint action Catalonia–Baden-Württemberg and by FEDER/MEC grant number MTM2006-04353. The third author was also supported by the program Ramón y Cajal, MEC. 相似文献
5.
For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l. 相似文献
6.
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.
相似文献
7.
Ralph Howard 《manuscripta mathematica》1999,99(4):471-483
For a complete Riemannian manifold M with compact boundary ∂M denote by $\Cut$ the cut locus of $\f M$ in M. The rolling radius of M is roll(M)≔ dist(∂M, ?∂
M
). Let Focal(∂M) be the focal distance of ∂M in M. Then conditions are given that imply the equality roll(M)= Focal(∂M). This generalizes Blaschke's rolling theorem from bounded convex domains in Euclidean space to more general Euclidean domains
and to Riemannian manifolds with boundary.
Received: 28 August 1998 / Revised version: 8 February 1999 相似文献
8.
Bao Fu Wang 《数学学报(英文版)》2009,25(8):1353-1362
Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞(δΩ), Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer. 相似文献
9.
Roberto Frigerio 《Geometriae Dedicata》2006,118(1):105-131
Suppose
, let M
1, M
2 be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π1 (M
1) is quasi-isometric to π1 (M
2) (with respect to the word metric). Also suppose that if n=3, then ∂M
1 and ∂M
2 are compact. We show that M
1 is commensurable with M
2. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M
1 above and G is a finitely generated group which is quasi-isometric to π1 (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π1 (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16 相似文献
10.
Keomkyo Seo 《Archiv der Mathematik》2008,90(2):173-180
Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point , then we prove a sharp relative isoperimetric inequality
where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove the relative isoperimetric inequalities for minimal submanifolds outside a closed convex set in a higher-dimensional
Riemannian manifold.
Received: 3 February 2007 相似文献
11.
Gaiane Panina 《Central European Journal of Mathematics》2006,4(2):270-293
Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness
conjecture: Let K ⊂ ℝ3
be a smooth convex body. If for a constant C, at every point of ∂K, we have R
1 ≤ C ≤ R
2
then K is a ball. (R
1
and R
2
stand for the principal curvature radii of ∂K.)
This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples
to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number
of horns is constructed.
Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed. 相似文献
12.
Yves Benoist 《Inventiones Mathematicae》2006,164(2):249-278
Divisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient
M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:
The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.
Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these
triangles is dense in the boundary of Ω (see Figs. 1 to 4).
Moreover, we construct examples of such divisible convex open sets Ω.
相似文献
13.
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n − 1)-submanifold S. The main result is that the fundamental group of M\ S{M{\setminus} S} is relatively hyperbolic, relative to fundamental groups of the ends of M\ S{M{\setminus} S} , and M\ S{M{\setminus} S} admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n > 1 the fundamental group of M\ S{M{\setminus} S} satisfies Mostow-type Rigidity, has solvable word and conjugacy problems, has finite asymptotic dimension and rapid decay
property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with
property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M\ S{M{\setminus} S} is biautomatic and satisfies Strong Tits Alternative. 相似文献
14.
The kernelK of a convex polyhedronP
0, as defined by L. Fejes Tóth, is the limit of the sequence (P
n), whereP
n is the convex hull of the midpoints of the edges ofP
n−1. The boundary ∂K of the convex bodyK is investigated. It is shown that ∂K contains no two-dimensional faces and that ∂K need not belong toC
2. The connection with similar algorithms from CAD (computer aided design) is explained and utilized.
R. J. Gardner was supported in part by a von Humboldt fellowship. 相似文献
15.
Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S3 is singled out. Using the YasudaShimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S3. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by YasudaShimada. It isexplained why this family of Finslerian space forms is not projectivelyflat. 相似文献
16.
Consider a compact manifold M with boundary ∂
M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point
and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂
M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined
up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result
in Riemannian geometry that is more general than related results in the literature. 相似文献
17.
Lixin Liu Athanase Papadopoulos Weixu Su Guillaume Théret 《Monatshefte für Mathematik》2010,31(1):295-311
We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary,
that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare
these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call
“ε
0-relative e{\epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε
0 and e{\epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families
of arcs. 相似文献
18.
19.
DaGang Yang 《Inventiones Mathematicae》2000,142(2):435-450
An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category
of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε0, ε0≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S
4, RP
4 with constant sectional curvature K=1/3, or CP
2 with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy
either one of the above two conditions on S
4 and CP
2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S
4, RP
4, or CP
2 but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies
either one of the conditions.
Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000 相似文献
20.
Estate Khmaladze Wolfgang Weil 《Annals of the Institute of Statistical Mathematics》2008,60(4):813-842
We investigate the behaviour of Poisson point processes in the neighbourhood of the boundary ∂K of a convex body K in ,d ≥ 2. Making use of the geometry of K, we show various limit results as the intensity of the Poisson process increases and the neighbourhood shrinks to ∂K. As we shall see, the limit processes live on a cylinder generated by the normal bundle of K and have intensity measures expressed in terms of the support measures of K. We apply our limit results to a spatial version of the classical change-point problem, in which random point patterns are
considered which have different distributions inside and outside a fixed, but unknown convex body K. 相似文献