A NEW PROOF OF CHEEGER-GROMOLL SOUL CONJECTURE AND THE TAKEUCHI THEOREM

JIANGUO CAO MEI-CHI SHAW

Abstract. shortabstract: In this paper, we study the geometry for the evolution of (possibly non-smooth) equi-distant hypersurfaces in real and complex manifolds. First we use the matrix-valued Riccati equation to provide a new proof of the Takeuchi Theorem for peudo-convex Kähler domains with positive curvature. We derive a new monotone principle for both smooth and non-smooth portions of equi-distant hypersurfaces in manifolds with nonnegative curvature. Such a new monotone principle leads to a new proof of the Cheeger-Gromoll soul conjecture without using Perelman’s flat strip theorem.

In addition, we show that if M^{n} is a complete, non-compact C^{∞}-smooth Riemannian manifold with nonnegative sectional curvature, then any distance nonincreasing retraction from M^{n} to its soul S must be a C^{∞}-smooth Riemannian submersion, a result obtained independently by B. Wilking.