Pengzi Miao

Monash University (Australia)

A note on 3-dimensional positive Ricci curvature metrics with relatively large volume

Let g be a Riemannian metric with positive Ricci curvature on some closed manifold Mn. 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 (Mn,g) satisfies V ol(g) V ol(𝕊n), where 𝕊n is the standard n-dimensional unit sphere. On the other hand, it is a theorem of Colding that V ol(g) is close to V ol(𝕊n) if and only if (Mn,g) is close to 𝕊n in the Gromov-Hausdorff distance. In this talk, we consider a positive Ricci curvature metric g on the 3-dimensional sphere S3 from a rather different point of view. We show that, if g satisfies Ric(g) 2 and V ol(g) 12V ol(𝕊3), then the stereographic projection of (S3,g) contains no closed minimal surfaces, hence generalizing a well known fact that the Euclidean space R3 has no closed minimal surfaces. Our consideration is motivated by the problem of existence of apparent horizons in general relativity.