scholarly journals A fully nonlinear conformal flow on locally conformally flat manifolds

2003 ◽  
Vol 2003 (557) ◽  
Author(s):  
Pengfei Guan ◽  
Guofang Wang
Keyword(s):  
Conformally Flat ◽  
Fully Nonlinear ◽  
Conformally Flat Manifolds ◽  
Locally Conformally Flat ◽  
Flat Manifolds ◽  
Conformal Flow

scholarly journals Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019 ◽  
Vol 74 (4) ◽  
Author(s):  
Marcos Tulio Carvalho ◽  
Mauricio Pieterzack ◽  
Romildo Pina
Keyword(s):  
Sufficient Conditions ◽  
Conformally Flat ◽  
Symmetric Tensors ◽  
Conformally Flat Manifolds ◽  
Dimensional Group ◽  
Schouten Tensor ◽  
Locally Conformally Flat ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Complete Metrics

Abstract We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

COMPACTNESS AND GLOBAL ESTIMATES FOR A FOURTH ORDER EQUATION OF CRITICAL SOBOLEV GROWTH ARISING FROM CONFORMAL GEOMETRY

2006 ◽  
Vol 08 (01) ◽  
pp. 9-65 ◽  
Author(s):  
EMMANUEL HEBEY ◽  
FRÉDÉRIC ROBERT ◽  
YULIANG WEN
Keyword(s):  
Fourth Order ◽  
Compact Riemannian Manifold ◽  
Conformal Geometry ◽  
Conformally Flat ◽  
Conformally Flat Manifolds ◽  
Locally Conformally Flat ◽  
Constant Coefficients ◽  
Global Estimates ◽  
Definition Of ◽  
Flat Manifolds

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like Pgu = u2♯-1, where [Formula: see text] is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and [Formula: see text] is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

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