2012/11/28 by Sayed, Safaa El
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1211.6617
Let (M,g) be a compact Riemannian manifold of dimension n≥ 3. For a metric g on M, we let \la2(g) be the second eigenvalue of the Yamabe operator Lg:= (4(n-1))/(n-2) Δg + \scalg. Then, the second Yamabe invariant is defined as \si2(M) \definedas sup infh ∈ [g] \la2(h) \Vol(M,h)2/n. where the supremum is taken over all metrics g and the infimum is taken over the metrics in the conformal class [g]. Assume that \si2(M)>0. In the spirit of \citeammann.dahl.humbert:08, we prove that if N is obtained from M by a k-dimensional surgery (0 ≤ k ≤ n-3), there exists a positive constant Λn depending only on n such that \si2(N) ≥ min(σ2(M), Λn). We then give some topological conclusions of this result.