Abstract

For a closed, connected direct product Riemannian manifold (M, g)=(M_1, g_1) times cdots times (M_l, g_l), we define its multiconformal class [![ g ]!] as the totality {f_1^2g_1oplus cdots oplus f_l^2g_l} of all Riemannian metrics obtained from multiplying the metric g_i of each factor M_i by a positive function f_i on the total space M. A multiconformal class [![ g ]!] contains not only all warped product type deformations of g but also the whole conformal class [tilde{g}] of every tilde{g}in [![ g ]!]. In this article, we prove that [![ g ]!] contains a metric of positive scalar curvature if and only if the conformal class of some factor (M_i, g_i) does, under the technical assumption dim M_ige 2. We also show that, even in the case where every factor (M_i, g_i) has positive scalar curvature, [![ g ]!] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided lge 2 and dim Mge 3.

Highlights

  • Let M be a closed connected m-dimensional manifold and let [g] be the conformal class of a Riemannian metric g

  • By the resolution of the Yamabe problem this infimum is always attained by some metric gof constant scalar curvature

  • We show that within the multiconformal class [[g]] we can always find a metric with constant scalar curvature equal to −1 but with arbitrarily large volume

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Summary

Introduction

Let M be a closed connected m-dimensional manifold and let [g] be the conformal class of a Riemannian metric g. Each Riemannian metric g on a closed manifold M can be conformally changed into a metric of constant scalar curvature. The sign of the resulting scalar curvature is in a direct relation with the signs of the conformal Yamabe invariants μ(Mi , [gi ]) for i = 1, 2 Another common family of product manifolds are warped products (M1 × M2, g1 ⊕ f 2g2), where f : M1 → R+ is a positive function. We show that within the multiconformal class [[g]] we can always find a metric with constant scalar curvature equal to −1 but with arbitrarily large volume This is even the case when μ(Mi , [gi ]) is nonnegative for each 1 ≤ i ≤ l.

The normalized Einstein–Hilbert functional
The scalar curvature of a multiconformal class
The sign of a multiconformal class
The infimum of the conformal Yamabe invariants
Multiconformal metric of permutation type
A Computation of the scalar curvature
Results

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