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1.
We study a particular class of open manifolds. In the category of
Riemannian manifolds these are complete manifolds with cylindrical
ends. We give a natural setting for the conformal geometry on such
manifolds including an appropriate notion of the cylindrical Yamabe
constant/invariant. This leads to a corresponding version of the Yamabe
problem on cylindrical manifolds. We find a positive solution to
this Yamabe problem: we prove the existence of minimizing metrics
and analyze their singularities near infinity. These singularities turn
out to be of very particular type: either almost conical or almost cuspsingularities. We describe the supremum case, i.e., when the cylindrical
Yamabe constant is equal to the Yamabe invariant of the sphere.
We prove that in this case such a cylindrical manifold coincides conformally
with the standard sphere punctured at a finite number of
points. In the course of studying the supremum case, we establish a
Positive Mass Theorem for specific asymptotically flat manifolds with
two almost conical singularities. As a by-product, we revisit known
results on surgery and the Yamabe invariant.
Submitted: Submitted: August 2001. Revision: January 2003
RID="*"
ID="*"Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 14540072. 相似文献
2.
B. Ammann 《Journal of Functional Analysis》2006,235(2):377-412
Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation. 相似文献
3.
We generalize the discrete Yamabe flow to α order. This Yamabe flow deforms the α-order curvature to a constant. Using this new flow, we manage to find discrete α-quasi-Einstein metrics on the triangulations of S3. 相似文献
4.
本文研究完备的局部共形平坦的Riemannian 流形Mn. 证明了在Yamabe 流下, 流形在无穷远处曲率趋向于零的性质是随时间保持的. 作为应用, 可以得到这个流形的渐近体积比是一个常数. 相似文献
5.
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
6.
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
7.
We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem. 相似文献
8.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.supported by the National Natural Science Foundation of China (10371138). 相似文献
9.
The conformal class of a Hermitian metric g on a compact almost complex manifold (M2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted
version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature sJ. We ask if the conformal class of g carries a metric with constant sJ, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the
affirmative. In fact, we show that (2m−1)sJ−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution
of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant
is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal
bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that
when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian
Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere
equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the
conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well.
We discuss some applications. 相似文献
10.
Chanyoung Sung 《Differential Geometry and its Applications》2006,24(3):271-287
We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants. 相似文献
11.
We prove some existence results for the fractional Yamabe problem in the case that the boundary manifold is umbilic, thus covering some of the cases not considered by González and Qing. These are inspired by the work of Coda-Marques on the boundary Yamabe problem but, in addition, a careful understanding of the behavior at infinity for asymptotically hyperbolic metrics is required. 相似文献
12.
Brian Clarke 《Annals of Global Analysis and Geometry》2011,39(2):131-163
We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate
the topology on the manifold of metrics induced by the distance function of the L
2 Riemannian metric—so-called because it induces an L
2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L
1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically
with the completion of the L
2 metric. We also give a user-friendly criterion for convergence (with respect to the L
2 metric) in the manifold of metrics. 相似文献
13.
Brian Clarke Yanir A. Rubinstein 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric. 相似文献
14.
We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S 3 with Ric = 2F 2, Ric = 0 and Ric = -2F 2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not. 相似文献
15.
Daniel Champion David Glickenstein Andrea Young 《Differential Geometry and its Applications》2011,29(1):108-124
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge?s Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics. 相似文献
16.
Let be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder‐type estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting. 相似文献
17.
The fractional Yamabe problem, proposed by González and Qing (Analysis PDE 6:1535–1576, 2013), is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal conformally invariant operators. We investigate a non-compactness property of the fractional Yamabe problem by constructing bubbling solutions to its small perturbations. 相似文献
18.
We introduce two versions of the Yamabe flow which preserve negative scalar-curvature bounds. First we show existence and
smooth convergence of solutions to these flows. We then show that a metric with negative scalar curvature is controlled by
the Yamabe metrics in the same conformal class with constant extremal scalar curvatures. This implies that the volume entropy
of our original metric is controlled by the entropies of these Yamabe metrics. We eventually use these Yamabe flows to prove
an entropy-rigidity result: when the Yamabe metric has negative sectional curvature, the entropy of a metric in the same conformal
class is extremal if and only if the metric has constant extremal scalar curvature. 相似文献
19.
20.
Giovanni Calvaruso 《Differential Geometry and its Applications》2008,26(4):419-433
We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17-28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R3, having the prescribed principal Ricci curvatures. 相似文献