scholarly journals Conformal Metrics with Prescribed Fractional Scalar Curvature on Conformal Infinities with Positive Fractional Yamabe Constants

2020 ◽  
Author(s):  
Seunghyeok Kim
Keyword(s):  
Scalar Curvature ◽  
Conformal Metrics ◽  
Yamabe Constants

Topological tools for the prescribed scalar curvature problem on S n

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Dina Abuzaid ◽  
Randa Ben Mahmoud ◽  
Hichem Chtioui ◽  
Afef Rigane
Keyword(s):  
Scalar Curvature ◽  
Conformal Metrics ◽  
Standard Sphere ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Hopf Formula ◽  
Curvature Problem ◽  
Formula Type

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

scholarly journals Scalar curvatures of conformal metrics on Sn

1995 ◽  
Vol 140 ◽  
pp. 151-166
Author(s):  
Shigeo Kawai
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Smooth Function ◽  
Conformal Metrics ◽  
Conformal Metric ◽  
Canonical Metric ◽  
Dimensional Unit ◽  
Dimension 2

In this paper we consider the following problem: Given a smooth function K on the n-dimensional unit sphere Sn(n ≥ 3) with its canonical metric g0, is it possible to find a pointwise conformal metric which has K as its scalar curvature? This problem was presented by J. L. Kazdan and F. W. Warner. The associated problem for Gaussian curvature in dimension 2 had been presented by L. Nirenberg several years before.

scholarly journals Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

1993 ◽  
Vol 132 ◽  
pp. 155-173
Author(s):  
Shin Kato ◽  
Shin Nayatani
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Smooth Function ◽  
Compact Manifold ◽  
Second Order ◽  
Conformal Metrics ◽  
Derivatives Of

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equationwhere ▽u denotes the Laplacian of u, defined with respect to the metric g.

Conformal metrics with prescribed scalar curvature

1986 ◽  
Vol 86 (2) ◽  
pp. 243-254 ◽  
Author(s):  
Jose F. Escobar ◽  
Richard M. Schoen
Keyword(s):  
Scalar Curvature ◽  
Conformal Metrics

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].

Sign in / Sign up

Export Citation Format

Share Document