The conformal Yamabe constant of product manifolds

Abstract

Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and (W,h).Comment: 12 pages, to appear in Proc. AMS; v3: small changes, very last preprint version, close to published versio

The Yamabe problem on stratified spaces

“…We identify the local Yamabe invariants at all point p ∈ M as higher versions of the cylindrical/conic Yamabe invariants discussed above; these are simply the global Yamabe invariants for the model spaces R k × C(Z), or (conformally) equivalently, H k+1 × Z, where Z is a compact iterated edge space with lower singular 'depth' than the original space M . The special cases of these invariants when Z = S n−k−1 play an interesting role in the work of Ammann, Dahl and Humbert [5], [6], where quantitative estimates of the change of the σ-Yamabe invariant (which is the supremum of the Yamabe constants over all conformal classes) under surgeries are obtained. Finally, using the more specialized analytic tools available for the study of PDE on smoothly stratified spaces, we prove sharp regularity results about the behaviour of the minimizer u (or indeed any solution of the Yamabe equation) at the singular strata of M .…”

Fiberwise symmetrizations for variational problems on fibred manifolds and applications

“…which is the alternation 3 of a multi-linear map 3 The alternation Alt(T ) of a multi-linear map T :…”

“…He has also obtained an estimate of K when M = S 2 or S 3 , and m + n ≤ 5 in cowork with Ruiz [48,49]. The next significant progress toward the product formula is made by Ammann, Dahl, and Humbert : Theorem 1.8 (Ammann, Dahl, and Humbert [3]). Let (M m , g) and (N n , h) be smooth closed Riemannian manifolds satisfying m, n ≥ 3 and…”

Second Yamabe constant on Riemannian products

“…In the following proposition we use several known lower bounds for the Yamabe invariant to deduce lower bounds for the second Yamabe invariant of a Riemannian product. The statements in Proposition 4.12 are immediate consequence of apply Proposition 4.8 to the lower bounds for the Yamabe invariant obtained in [4], [17], [19], and [20]. In all the cases, in order to obtain the bounds, Theorem 2.3 (first equality) is used.…”