scholarly journals Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

2004 ◽  
Vol 211 (1) ◽  
pp. 71-152 ◽  
Author(s):  
José F. Escobar ◽  
Gonzalo Garcia
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Conformal Metrics ◽  
Prescribed Mean Curvature ◽  
Zero Scalar Curvature

scholarly journals About the Uniqueness of Conformal Metrics with Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundary

2011 ◽  
Vol 13 ◽  
pp. 71-79
Author(s):  
Gonzalo García ◽  
Jhovanny Muñoz
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Manifolds With Boundary ◽  
Conformal Metrics ◽  
Dirichlet Eigenvalue ◽  
Uniqueness Results ◽  
Manifold With Boundary ◽  
First Dirichlet Eigenvalue ◽  
Neumann Eigenvalue ◽  
Linear Elliptic Problem

Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

scholarly journals Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

2019 ◽  
Vol 21 (03) ◽  
pp. 1850021 ◽  
Author(s):  
Xuezhang Chen ◽  
Liming Sun
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Constant Scalar Curvature ◽  
Riemannian Metric ◽  
Conformal Metrics ◽  
Nonzero Constant ◽  
Dimension Formula ◽  
Yamabe Constant ◽  
Boundary Mean Curvature

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

PRESCRIBING MEAN CURVATURE ON 𝔹n

2010 ◽  
Vol 21 (09) ◽  
pp. 1157-1187 ◽  
Author(s):  
WAEL ABDELHEDI ◽  
HICHEM CHTIOUI
Keyword(s):  
Critical Points ◽  
Mean Curvature ◽  
Level Sets ◽  
Curvature Function ◽  
Number Of Solutions ◽  
Total Contribution ◽  
Prescribed Mean Curvature ◽  
The Difference ◽  
Zero Scalar Curvature ◽  
Lagrange Functional

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

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