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>