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1.
Seongtag Kim 《Geometriae Dedicata》1997,64(3):373-381
We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu
(n+2)/(n–2) on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity. 相似文献
2.
José F. Escobar 《Journal of Functional Analysis》2003,202(2):424-442
Let (Mn,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M. 相似文献
3.
Sharief Deshmukh 《Annali dell'Universita di Ferrara》2010,56(2):231-236
In this short note we consider an n-dimensional compact Riemannian manifold (M, g) of constant scalar curvature S = n(n − 1)c and show that the presence of a nontrivial conformal vector field ξ on M forces S to be positive. Then we show that an appropriate control on the energy of ξ makes M to be isometric to the n-sphere S
n
(c). 相似文献
4.
An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the
full Riemannian curvature tensor. In this article, supposing (M
n
, g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that
L\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C
0 bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds.
Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M
n
× [0, T), the curvature tensor stays uniformly bounded on M
n
× [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented. 相似文献
5.
Jimmy Petean 《Annals of Global Analysis and Geometry》2001,20(3):231-242
We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N
n
,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M
m
, the Yamabe invariantof M
m
× N
n
is no less than K times the invariant ofS
n + m
. We will find some estimates for the constant K in the case N =S
n
. 相似文献
6.
In this paper, we prove the following Myers type theorem: If (M n ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M n must be compact. 相似文献
7.
Ezequiel R. Barbosa 《Bulletin des Sciences Mathématiques》2010,134(7):677
Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition
8.
In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rn+1. If Mn−1 is a compact hypersurface of the sphere Sn(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature.
To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an ε for which the truncate cone C(M)ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mn−1 of the sphere with zero scalar curvature and S3 different from zero, for which all truncated cones based on M are stable.
Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70. 相似文献
9.
Peng Zhu 《Annals of Global Analysis and Geometry》2011,40(4):427-434
We prove that L
2 harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L
2 harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic
manifold can not admit any almost K?hler structure of positive isotropic curvature. 相似文献
10.
Hongxin Guo 《Mathematische Annalen》2009,345(4):883-894
Assume Mn{\mathcal{M}^n} is a complete noncompact steady gradient Ricci soliton with positive Ricci curvature. First, by deriving a useful formula
we characterize the condition of the scalar curvature and the potential function having a same level surface. Then, we assume
the dimension n = 3 and characterize the rotational symmetry geometrically. Finally, for all dimensions n ≥ 3, we prove a dimension reduction result at spatial infinity under additional assumptions that Mn{\mathcal M^n} is a κ-solution and the scalar curvature is
O(\frac1r),{O\left(\frac{1}{r}\right),} where r is the distance function. 相似文献
11.
We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient
manifold (M, g) has non-degenerate critical points. 相似文献
12.
Lorenzo Mazzieri 《manuscripta mathematica》2009,129(2):137-168
In this paper we construct constant scalar curvature metrics on the generalized connected sum
M = M1 \sharpK M2{M = M_1 \, \sharp_K \, M_2} of two compact Riemannian scalar flat manifolds (M
1, g
1) and (M
2, g
2) along a common Riemannian submanifold (K, g
K
) whose codimension is ≥3. Here we present two constructions: the first one produces a family of “small” (in general nonzero)
constant scalar curvature metrics on the generalized connected sum of M
1 and M
2. It yields an extension of Joyce’s result for point-wise connected sums in the spirit of our previous issues for nonzero
constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1+) class in the Kazdan–Warner classification, we refine the first construction in order to produce a family of scalar flat
metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact
time symmetric initial data sets, extending the Isenberg–Mazzeo–Pollack gluing construction. 相似文献
13.
For any 3-manifold M3 and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form we construct such a metric with positive scalar curvature. More generally, we construct such a metric with Scal>0 (and such surfaces) on any 3-manifold which carries a metric with Scal>0. 相似文献
14.
By topological arguments, we set forth sufficient hypotheses for a given function K, on a compact Riemannian manifold (M, g), to be the scalar curvature of a metric conformal to g. 相似文献
15.
Kazuo Akutagawa 《Mathematische Zeitschrift》2003,243(1):85-98
For a supergroup , we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact
n-dimensional -manifold W with nonempty boundary M, , in terms of the bordism class [M] in the Stolz obstruction group associated to [St2]. In par ticular, when W is a 5-dimensional spin manifold and the -invariant of a connected component of M is nonzero, we prove that W does not admit a positive scalar curvature metric with minimal boundary condition.
Received: 4 July 2001; in final form: 5 February 2002 / Published online: 8 November 2002
RID="*"
ID="*" Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No.
11640070. 相似文献
16.
On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S3 if ker s′*g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
A. L. Yampol'skií 《Journal of Mathematical Sciences》1990,48(1):108-117
We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T
1(M
n, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT
1 (M
n, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT
1
S
n. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145. 相似文献
18.
Abdênago Barros Ernani Ribeiro Jr 《Bulletin of the Brazilian Mathematical Society》2014,45(2):325-341
The aim of this paper is to present some structural equations for generalized m-quasi-Einstein metrics (M n , g, ? f, λ), which was defined recently by Catino in [11]. In addition, supposing that M n is an Einstein manifold we shall show that it is a space form with a well defined potential f. Finally, we shall derive a formula for the Laplacian of its scalar curvature which will give some integral formulae for such a class of compact manifolds that permit to obtain some rigidity results. 相似文献
19.
ZhangJianfeng 《高校应用数学学报(英文版)》2005,20(2):183-196
Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c), Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated. 相似文献
20.
Ryszard Deszcz Ma?gorzata G?ogowska Miroslava Petrovi?-Torga?ev Leopold Verstraelen 《Results in Mathematics》2011,59(3-4):401-413
Chen ideal submanifolds M n in Euclidean ambient spaces E n+m (of arbitrary dimensions n ?? 2 and codimensions m??? 1) at each of their points do realise an optimal equality between their squared mean curvature, which is their main extrinsic scalar valued curvature invariant, and their ???C(= ??(2)?C) curvature of Chen, which is one of their main intrinsic scalar valued curvature invariants. From a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i.e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann?CChristoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces (M n , g) for which the (0, 4) tensor R is proportional to the Nomizu?CKulkarni square of their (0, 2) metric tensor g. In the present article, for the class of the Chen ideal submanifolds M n of Euclidean spaces E n+m , we study the relationship between these geometric and algebraic generalisations of the real space forms. 相似文献