Bar-Ilan University
 
ANALYSIS SEMINAR
 
(1st of two talks. See separate announcement of talk at 15:05)
 
Speaker: Prof. Rovenski Vladimir
 
University of Haifa
 
Date: Monday, April 16, 2012
 
Time: 14:00
 
Place: 2nd floor Colloquium Room, Building 216
 
Title: Flows of metrics on a fiber bundle
 
Abstract:
 
Let $(M^{n+p},g)$ be a closed Riemannian manifold, and $\pi: M\to B$
a smooth fiber bundle with compact and orientable $p$-dimensional
fiber $F$. Denote by $D_F$ ($D$) the distribution tangent
(orthogonal, resp.) to fibers.
 
We discuss conformal flows of the metric restricted to $D$ with the
speed proportional to
 
(i) the divergence of the mean curvature vector $H$ of $D$,
 
(ii) the mixed scalar curvature $Sc_{mix}$ of the distributions.
     (If $M$ is a surface, then $Sc_{mix}$ is the gaussian curvature
$K$).
 
For (i), we show that the flow is equivalent to the heat flow of the
1-form dual to $H$, provided the initial 1-form is $D_F$-closed. We
use known long-time existence results for the heat flow to show that
our flow has a global solution $g_t$. It converges to a limiting
metric, for which $D$ is harmonic (i.e., $H=0$); actually under some
topological assumptions we can prescribe $H$.
 
For (ii) on a twisted product, we observe that $H$ satisfies the
Burgers type PDE, while the warping function satisfies the heat
equation; in this case the metrics $g_t$ converge to the product.
 
We consider illustrative examples of flows similar to (i) and (ii)
on a surface (of revolution), they yield convection-diffusion PDEs
for curvature of $D$-curves (parallels) and solutions -- non-linear
waves.
 
For $M$ with general $D$, we modify the flow (ii) with the help of a
measure of ``non-umbilicity" of $D_F$, and the integrability tensor
of $D$, while the fibers are totally geodesic. Let $\lambda_0$ be
the smallest eigenvalue of certain Schrödinger operator on the
fibers. We assume $H$ to be $D_F$-potential and show that
 
-- $H$ satisfies the forced Burgers type PDE;
 
-- the flow has a unique solution converging to a metric, for which
$Sc_{mix}\ge-n\lambda_0$,
   and $H$ depends only on the $D$-conformal class of the initial
metric.
 
-- if $D$ had constant rate of ``non-umbilicity" on fibers, then the
limiting metric has the properties: $Sc_{mix}$ is quasi-positive, and
$D$ is harmonic.
 
---------------------------------------------------------
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Elijah Liflyand   <liflyand@gmail.com>