Results 1  10
of
308
On some conformally invariant fully nonlinear equations, Part II: Liouville, . . .
, 2005
"... ..."
(Show Context)
A fully nonlinear conformal flow on locally conformally flat manifolds
 JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
, 2001
"... We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2. ..."
Abstract

Cited by 64 (14 self)
 Add to MetaCart
(Show Context)
We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2.
Local Estimates for a Class of Fully Nonlinear Equations Arising From Conformal Geometry
 Int. Math. Res. Not
, 2001
"... this paper, we are interested in a class of fully nonlinear differential equations related to the deformation of conformal metrics. Let (M; g 0 ) be a compact connected smooth Riemannian manifold of dimension n 3, and let [g 0 ] denote the conformal class of g 0 . The Schouten tensor of the metric ..."
Abstract

Cited by 52 (10 self)
 Add to MetaCart
(Show Context)
this paper, we are interested in a class of fully nonlinear differential equations related to the deformation of conformal metrics. Let (M; g 0 ) be a compact connected smooth Riemannian manifold of dimension n 3, and let [g 0 ] denote the conformal class of g 0 . The Schouten tensor of the metric g is defined as S g = 1 n \Gamma 2 ` Ric g \Gamma R g 2(n \Gamma 1) \Delta g ' ; where Ric g and R g are the Ricci tensor and scalar curvature of g respectively. This tensor is connected to the study of conformal invariants, in particular conformally invariant tensors and differential operators (e.g., see [6] and references therein). In [16], The following oe k scalar curvatures of g were considered by Viaclovsky in [16]: oe k (g) := oe k (g \Gamma1 \Delta S g ); where oe k is the kth elementary symmetric function, g \Gamma1 \DeltaS g is locally defined by (g \Gamma1 \DeltaS g ) i j = g ik (S g ) kj . When k = 1, oe 1 scalar curvature is just the scalar curvature R (upto a constant multiple). oe k can also be viewed as a function of the eigenvalues of symmetric matrices, that is a function in R n . According to G arding [7], \Gamma + k = f = ( 1 ; 2 ; \Delta \Delta \Delta ; n ) 2 R n j oe j () ? 0; 8j kg; is a natural class for oe k . A metric g is said to be in \Gamma + k if oe j (g)(x) ? 0 for j k and x 2 M . The case of k = 1, deforming scalar curvature R to a constant in its conformal class is known as the Yamabe problem, the final solution was obtained by Schoen in [12] (see also [1] and [15]). We refer [10] for the literature on Yamabe problem. There is a recent interest in deforming oe k scalar curvature in its conformal class. This type of problem was Date: August, 2001. 1991 Mathematics Subject Classification. [. Key words a...
Global existence and convergence of Yamabe flow
 Centre for Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Weimin Sheng: Department of Mathematics, Zhejiang University, Hangzhou
, 1994
"... Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)= ..."
Abstract

Cited by 50 (0 self)
 Add to MetaCart
(Show Context)
Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)=
Moduli spaces of critical Riemannian metrics in dimension four
"... Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can ..."
Abstract

Cited by 43 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. 1.
Moduli spaces of singular Yamabe metrics
, 1994
"... Complete, conformally flat metrics of constant positive scalar curvature on the complement of k points in the nsphere, k ≥ 2, n ≥ 3, were constructed by R. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and w ..."
Abstract

Cited by 43 (13 self)
 Add to MetaCart
(Show Context)
Complete, conformally flat metrics of constant positive scalar curvature on the complement of k points in the nsphere, k ≥ 2, n ≥ 3, were constructed by R. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension k. For a generic set of nearby conformal classes the moduli space is shown to be a k−dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.
Blowup phenomena for the Yamabe equation
 J. Amer. Math. Soc
, 2008
"... Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to ..."
Abstract

Cited by 43 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n ≥ 52. 1.
Yamabe constants and the perturbed SeibergWitten equations
 Comm. Anal. Geom
, 1997
"... Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1 ..."
Abstract

Cited by 42 (9 self)
 Add to MetaCart
(Show Context)
Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1
Killing Spinor Equations In Dimension 7 And Geometry Of Integrable G_2Manifolds
, 2008
"... We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the ..."
Abstract

Cited by 41 (0 self)
 Add to MetaCart
We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3form field. In dimension n = 7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G2structure into a cocalibrated one of pure type W3.