Pengzi Miao Monash University (Australia)
A note on 3dimensional positive Ricci curvature metrics with relatively large volume
Let g be a Riemannian metric with positive Ricci curvature on some closed manifold M^{n}. If the Ricci curvature
of g is normalized to satisfy Ric(g) ≥ n  1, then one knows by Bishop Volume Comparison theorem that the
volume of (M^{n},g) satisfies V ol(g) ≤ V ol(𝕊^{n}), where 𝕊^{n} is the standard ndimensional unit sphere. On the other
hand, it is a theorem of Colding that V ol(g) is close to V ol(𝕊^{n}) if and only if (M^{n},g) is close to 𝕊^{n} in the
GromovHausdorff distance. In this talk, we consider a positive Ricci curvature metric g on the
3dimensional sphere S^{3} from a rather different point of view. We show that, if g satisfies Ric(g) ≥ 2 and
V ol(g) ≥V ol(𝕊^{3}), then the stereographic projection of (S^{3},g) contains no closed minimal surfaces, hence
generalizing a well known fact that the Euclidean space R^{3} has no closed minimal surfaces. Our
consideration is motivated by the problem of existence of apparent horizons in general relativity.
