Abstract
On a Riemannian manifold X, we consider aK+s, where a is a nonnegative constant, K is the sectional curvature and s is the scalar curvature. It is shown that if X admits a metric with aK+s > 0, then so does any manifold obtained from X by surgeries of codimension ≥3. This implies the existence of such metrics on certain compact simply connected manifolds of dimension ≥5 by using the cobordism argument. We also study the corresponding minimal volume problem. As a corollary, we derive that every compact simply connected manifold of dimension ≥5 and every compact complex surface of Kodaira dimension ≤1 whose minimal model is not of Class VII collapse with aK+s bounded below.
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Sung, C. Surgery, Curvature, and Minimal Volume. Annals of Global Analysis and Geometry 26, 209–229 (2004). https://doi.org/10.1023/B:AGAG.0000042898.55061.ef
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DOI: https://doi.org/10.1023/B:AGAG.0000042898.55061.ef