A new proof of Cheeger-Gromoll soul conjecture and Takeuchi Theorem
Jianguo Cao and Mei-Chi Shaw
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Submission date: 04. Aug. 2003
Keywords and phrases: non-negative curvature, cheeger-gromoll soul conjecture, oka lemma and takeuchi theorem, distance non-increasing retraction
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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 is a complete, non-compact -smooth Riemannian manifold with nonnegative sectional curvature, then any distance non-increasing retraction from to its soul must be a -smooth Riemannian submersion, a result obtained independently by B. Wilking.