MiS Preprint Repository

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 $C^{\mathrm{\infty }}$-smooth Riemannian manifold with nonnegative sectional curvature, then any distance non-increasing retraction from $M^n$ to its soul $\mathcal{S}$ must be a $C^{\mathrm{\infty }}$-smooth Riemannian submersion, a result obtained independently by B. Wilking.