# zbMATH — the first resource for mathematics

Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature. (English) Zbl 0881.53036
Let $$M$$ be a compact smooth manifold of dimension $$n\geq 3$$. The Yamabe functional $$I$$ on the conformal class $$C$$ of $$M$$ is defined by $$I(g)= {\int_MS_g dv_g \over V_g^{(n-2)/n}}$$ for $$g\in C$$, where $$S_g$$, $$dv_g$$ and $$V_g$$ denote the scalar curvature, the volume element and the volume of $$(M,g)$$, respectively. The Yamabe invariant of $$(M,C)$$ is the infimum of this functional and is denoted by $$\mu(M,C)$$. A metric $$g$$ which minimizes the Yamabe functional $$I$$ on the conformal class $$C$$ is called a Yamabe metric.
Let $${\mathcal Y}_1 (n,\mu_0)$$ be the class of compact connected smooth $$n$$-manifolds $$M(n\geq 3)$$ with Yamabe metrics $$g$$ of unit volume which satisfy $$\mu(M,[g]) \geq\mu_0>0$$, where $$[g]$$ denotes the conformal class of $$g$$. In the paper under review, the author proves several convergence theorems for Riemannian manifolds in $${\mathcal Y}_1 (n,\mu_0)$$ with integral bounds on curvature. One of them includes a pinching theorem for flat conformal structures of positive Yamabe invariant on compact 3-manifolds.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text: