The second Yamabe invariant with boundary

Abstract

Abstract Let ( M , ∂ M , g ) \left(M,\partial M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y ( M , ∂ M , g ) Y\left(M,\partial M,g) , we define the kth Yamabe invariant with boundary Y k ( M , ∂ M , g ) {Y}_{k}\left(M,\partial M,g) . We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on ( M , ∂ M , g ) \left(M,\partial M,g) by using the Yamabe flow with boundary.