Sharp estimates and the Dirac operator

Abstract.

This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric $g$ on $S^n$ that strictly dominates the standard metric $g_0$ must have somewhere scalar curvature strictly less than that of $g_0$. More generally, if $M$ is any compact spin manifold of dimension $n$ which admits a distance decreasing map $f:M \rightarrow S^n$ of non-zero degree, then either there is a point $x \in M$ with normalized scalar curvature $\tilde{\kappa}(x)< 1$, or $M$ is isometric to $S^n$. The distance decreasing hypothesis can be replaced by the weaker assumption $f$ is contracting on $2$-forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by $k$-forms with $k \geq 3$.