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 An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by \(k\)-forms with \(k \geq 3\).
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Received: 16 May 1996
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Llarull, M. Sharp estimates and the Dirac operator. Math Ann 310, 55–71 (1998). https://doi.org/10.1007/s002080050136
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DOI: https://doi.org/10.1007/s002080050136