On the Yamabe constants of S2×R3S2×R3 and S3×R2S3×R2

We compare the isoperimetric profiles of S2×R3S2×R3 and of S3×R2S3×R2 with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of S2×R3S2×R3 and S3×R2S3×R2. Explicitly we show that Y(S3×R2,[g03+dx2])>(3/4)Y(S5) and Y(S2×R3,[g02+dx2])>0.63Y(S5). We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.