The rate of convergence on fractional power dissipative operator on compact manifolds

On a compact connected manifold M $\mathbb{M}$ , we concern the fractional power dissipative operator e − t L α $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}$ , and obtain the almost-everywhere convergence rate (as t → 0+) of e − t L α f $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}\left( f\right)$ when f is in some Sobolev type Hardy spaces. The main result is a non-trivial extension of a recent result on ℝ n by Cao and Wang in 2.

Homotopy type of manifolds with partially horoconvex boundary

Let [Formula: see text] be an [Formula: see text]-dimensional compact connected manifold with boundary, [Formula: see text] a constant and [Formula: see text] an integer. We prove that [Formula: see text] supports a Riemannian metric with the interior [Formula: see text]-curvature [Formula: see text] and the boundary [Formula: see text]-curvature [Formula: see text], if and only if [Formula: see text] has the homotopy type of a CW complex with a finite number of cells with dimension [Formula: see text]. Moreover, any Riemannian manifold [Formula: see text] with sectional curvature [Formula: see text] and boundary principal curvature [Formula: see text] is diffeomorphic to the standard closed [Formula: see text]-ball.

A global invertibility theorem for manifolds with boundary

SynopsisA local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.