Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

“…We consider the restriction of Rm to Λ Using minimal surface theory, Micallef and Moore [88] proved that any compact simply connected n-dimensional manifold with positive curvature operator is homeomorphic to the sphere S n . But it is still an open question whether a compact simply connected n-dimensional manifold with positive curvature operator is diffeomorphic to the sphere S n .…”

Topological and differentiable sphere theorems for complete submanifolds

“…In 1988, using minimal surface techniques, Micallef and Moore [8] investigated relations between curvature and topology of a manifold, and proved the topological sphere theorem for point-wise 1/4-pinched manifolds. Further more, they proved the famous topological sphere theorem for compact and simply connected manifolds with positive isotropic curvature.…”

“…We say that M has positive isotropic curvature if K(σ) > 0 for all totally isotropic two-planes at any point in M . It was shown in [8] that M has positive isotropic curvature if and only if for every orthonormal four-frame {e 1 , e 2 , e 3 , e 4 } at any point in M the inequality for curvature tensor of M…”

“…, e n } is a local orthonormal frame of M . In [8], Micallef and Moore proved that any n(≥ 4)-dimensional compact and simply connected manifold with positive isotropic curvature must be homeomorphic to a sphere. Moreover, they observed that any n(≥ 4)-manifold with point-wise 1/4-pinched curvature must have positive isotropic curvature.…”

An approach to the regularity for stable-stationary harmonic maps

“…To get the extent of the manifold N which does not carry stable harmonic S 2 , let us recall a famous index estimate result by Micallef and Moore [12]. For the case of a stable-stationary harmonic map, there are many contributions to Liouville-type theorems along these lines, for example, the condition 2-superstrongly unstable manifold introduced by Wei and Yau [17].…”