Volume and macroscopic scalar curvature

We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

Some aspects of Ricci flow on the 4-sphere

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.

Scalar curvature on some open 3-manifolds

This article is a survey of recent results about the existence of Riemannian metrics of positive scalar curvature on some open 3-manifolds. This results culminate in a rigidity theorem obtained using the theory of stable minimal surfaces.

The action of the mapping class group on metrics of positive scalar curvature

We present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

Scalar curvature and the multiconformal class of a direct product Riemannian manifold

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