Manifolds with pinched 2-positive curvature operator

Abstract

In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and δ-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by δ·scal > 0. If we relax the restriction of positivity of complex sectional curvature to nonnegativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.

Keywords

δ-pinched 2-positive curvature operator / complex sectional curvature / asymptotic volume ratio

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