# American Institute of Mathematical Sciences

November  2009, 8(6): 1759-1777. doi: 10.3934/cpaa.2009.8.1759

## On the geometric dependence of Riemannian Sobolev best constants

 1 Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil, Brazil

Received  September 2008 Revised  February 2009 Published  August 2009

We concerns here with the continuity on the geometry of the second Riemannian $L^p$-Sobolev best constant $B_0(p,g)$ associated to the AB program. Precisely, for $1 \leq p \leq 2$, we prove that $B_0(p,g)$ depends continuously on $g$ in the $C^2$-topology. Moreover, this topology is sharp for $p = 2$. From this discussion, we deduce some existence and $C^0$-compactness results on extremal functions.
Citation: Ezequiel R. Barbosa, Marcos Montenegro. On the geometric dependence of Riemannian Sobolev best constants. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1759-1777. doi: 10.3934/cpaa.2009.8.1759
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