Isoperimetric profile comparisons and Yamabe constants

Abstract

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 ).

The Yamabe Constant on Noncompact Manifolds

“…Thus the case we have proved already (applied to g,c instead of g, c) yields lower semicontinuity of Y M at g and hence, by conformal invariance of Y M , also at g. 22…”

Square-integrability of solutions of the Yamabe equation

“…where we used that H r is constant on F r . Differentiating this again and using Inequality (8) we get…”

Low-dimensional surgery and the Yamabe invariant

“…However it requires as input data a lower bound on the conformal Yamabe constant µ(R k+1 × S n−k−1 ). Such input data is provided in [17] and [18] in the cases (n, k) ∈ {(4, 1), (5, 1), (5, 2), (9, 1), (10, 1)}. Unfortunately their method has to be strongly modified for each pair of dimensions, and as a courtesy to us, Petean and Ruiz provided the above cases, as these are the ones which will lead to interesting applications in Section 5.…”

“…Together with (19) and (18), we obtain (17) which means that in the sense of distributions, equation (15) is satisfied on all of S 6 . By standard elliptic theory, v is C 2 (and even smooth outside its zero set).…”

“…(ii) (v, w) ∈ {(2, 2), (2, 3), (2, 7), (2,8), (3,2)}. In these cases bounds have been derived in [17,18] using isoperimetric profiles. (iii) v ≥ 3 and w ≥ 3.…”