Equivariant Yamabe problem with boundary
Let \((M,g)\) be a closed \(n\)-dimensional manifold with Riemannian metric \(g\) and \(n\ge 3\). The question of whether there is a metric on \(M\) conformal to \(g\) with constant scalar curvature is known as the Yamabe problem. \textit{E. Hebey} and \textit{M. Vaugon} [Bull. Sci. Math., II. Sér. 117, No. 2, 241--286 (1993; Zbl 0786.53024)] studied an equivariant extension of this problem. Namely, given a compact subgroup \(G\), of the isometry group of \(M\), does there exist a \(G\)-invariant metric conformal to \(g\) with constant scalar curvature? They were able to state conditions which imply an affirmative answer to this question. There are versions of the Yamabe problem that can be stated for manifolds with boundary as well as equivariant generalizations. Similar to the results of Hebey and Vaugon, for this equivariant generalization, the authors prove a sufficient condition that implies that the equivariant Yamabe problem for manifolds with boundary is solvable. As corollaries, they prove that this condition holds for certain manifolds with umbilic boundary which are locally conformally flat or have dimension \(3\le n\le 5\).
scientific article; zbMATH DE number 7474269
scientific article