IntroductionThis course is an introduction to the convergence and collapsing theory of Riemannian manifolds, which is an important tool in Riemannian geometry. We shall introduce the Gromov-Hausdorff distance and study the convergence theory of Riemannian manifolds with respect to this distance. We will discuss the collapsing theory of Riemannian manifolds with bounded sectional curvatures.Lect...
Abtract:We consider on arbitrary Riemannian manifolds the Leibenson equation:$\partial _{t}u=\Delta _{p}u^{q}.$This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium. Here $u(x,t)$ is the fraction of the volume that the liquid takes in porous medium at time $t$ at point $x$. The parameter $p$ characterizes the turbulence of a...