Abstract. We estimate from below the isoperimetric profile of S 2 × R 2 and use this information to obtain lower bounds for the Yamabe constant of S 2 × R 2 . This provides a lower bound for the Yamabe invariants of products S 2 × M 2 for any closed Riemann surface M . Explicitly we show that Y (S 2 × M 2 ) > (2/3)Y (S 4 ).
Let (M, g) be a closed Riemannian manifold of dimension n ≥ 3 and x 0 ∈ M be an isolated local minimum of the scalar curvature s g of g. For any positive integer k we prove that for ǫ > 0 small enough the subcritical Yamabe equation −ǫ 2 ∆u + (1 + c N ǫ 2 s g )u = u q has a positive k-peaks solution which concentrate around x 0 , assuming that a constant β is non-zero. In the equation c N = N −2 4(N −1)for an integer N > n and q = N +2 N −2 . The constant β depends on n and N , and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products (M × X, g + ǫ 2 h), where (X, h) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.
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