Uniqueness of Conformal Metrics with Prescribed Scalar and mean Curvatures on Compact Manifolds with Boundary

Departamento de Matemáticas, Universidad del Valle, Calle 13 No. 100 - 00, Cali, Colombia

Let $(M^n, g)$ be a compact manifold with boundary and $n \geq 2$. In this paper we prove the variational characterization of the Neumann eigen\-values of an elliptic operator associated to the problem of conformal deformation of metrics and we study the uniqueness of metrics in the conformal class of the metric $g$ having the same scalar curvature of the manifold and the same mean curvature of its boundary.

Uniqueness of Conformal Metrics with Prescribed Scalar and mean Curvatures on Compact Manifolds with Boundary Unicidad de métricas conformes con curvatura escalar y media prescritas sobre variedades compactas con frontera GONZALO GARCÍA1, JHOVANNY MUÑOZ2

1Universidad del Valle, Cali, Colombia. Email: ggarcia@univalle.edu.co 
2Universidad del Valle, Cali, Colombia. Email: jhovamu@univalle.edu.co 

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Abstract

Let (Mn, g) be a compact manifold with boundary and n ≥ 2. In this paper we prove the variational characterization of the Neumann eigenvalues of an elliptic operator associated to the problem of conformal deformation of metrics and we study the uniqueness of metrics in the conformal class of the metric g having the same scalar curvature of the manifold and the same mean curvature of its boundary.

Key words: Uniqueness, Conformal metrics, Curvature.

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2000 Mathematics Subject Classification: 53A30, 53C21, 58J32.

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Resumen

Sea (Mn, g) una variedad riemanniana compacta con frontera de dimensión n ≥ 2. En este artículo demostramos la caracterización variacional de los valores propios de Neumann de un operador elíptico asociado al problema de deformación conforme de métricas y estudiamos la unicidad de métricas en la clase conforme de la métrica g que tienen la misma curvatura escalar de la variedad y la misma curvatura media de su frontera.

Palabras clave: Unicidad, métricas conformes, curvatura.

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Texto completo disponible en PDF

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References

[1] Cherrier, `Problémes de Neumann non linéaires sur variétés riemanniennes´, J. Funct. Anal 57, (1984), 154-206.

[2] J. F. Escobar, `Addendum: Conformal Deformation of a Riemannian Metric to a Scalar Flat Metric with Constant Mean Curvature´, The Annals of Mathematics 139, 3 (1994), 749-750. Second Series

[3] J. F. Escobar, `Uniqueness and Non-Uniqueness of Metrics with Prescribed Scalar and Mean Curvature on Compact Manifolds with Boundary´, J. of functional analysis 202, (2003), 424-442.

(Recibido en julio de 2009. Aceptado en agosto de 2010)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n2a02, 
    AUTHOR  = {García, Gonzalo and Muñoz, Jhovanny}, 
    TITLE   = {{Uniqueness of Conformal Metrics with Prescribed Scalar and mean Curvatures on Compact Manifolds with Boundary}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {2}, 
    pages   = {91-101} 
}