115 research outputs found
Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part I
We give a generalized curvature-dimension inequality connecting the geometry
of sub-Riemannian manifolds with the properties of its sub-Laplacian. This
inequality is valid on a large class of sub-Riemannian manifolds obtained from
Riemannian foliations. We give a geometric interpretation of the invariants
involved in the inequality. Using this inequality, we obtain a lower bound for
the eigenvalues of the sub-Laplacian. This inequality also lays the foundation
for proving several powerful results in Part~II.Comment: 31 pages, Part 1 of 2. To appear in Mathematische Zeitschrif
The differentiation of hypoelliptic diffusion semigroups
Basic derivative formulas are presented for hypoelliptic heat semigroups and
harmonic functions extending earlier work in the elliptic case. Emphasis is
placed on developing integration by parts formulas at the level of local
martingales. Combined with the optional sampling theorem, this turns out to be
an efficient way of dealing with boundary conditions, as well as with finite
lifetime of the underlying diffusion. Our formulas require hypoellipticity of
the diffusion in the sense of Malliavin calculus (integrability of the inverse
Malliavin covariance) and are formulated in terms of the derivative flow, the
Malliavin covariance and its inverse. Finally some extensions to the nonlinear
setting of harmonic mappings are discussed
Li-Yau Type Gradient Estimates and Harnack Inequalities by Stochastic Analysis
In this paper we use methods from Stochastic Analysis to establish Li-Yau
type estimates for positive solutions of the heat equation. In particular, we
want to emphasize that Stochastic Analysis provides natural tools to derive
local estimates in the sense that the gradient bound at given point depends
only on universal constants and the geometry of the Riemannian manifold locally
about this point
Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations
We develop a variational theory of geodesics for the canonical variation of
the metric of a totally geodesic foliation. As a consequence, we obtain
comparison theorems for the horizontal and vertical Laplacians. In the case of
Sasakian foliations, we show that sharp horizontal and vertical comparison
theorems for the sub-Riemannian distance may be obtained as a limit of
horizontal and vertical comparison theorems for the Riemannian distances
approximations.Comment: Typos corrected, some improved bound
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