Pengzi Miao

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) ≥V 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.