

Preprint 75/2003
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
Pages: 39
MSC-Numbers: 58
Keywords and phrases: non-negative curvature, cheeger-gromoll soul conjecture, oka lemma and takeuchi theorem, distance non-increasing retraction
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Abstract:
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.