__Lecture
course__

Professor** Klaus
Ecker**, Free University of Berlin

*«Entropies,
Logarithmic
Sobolev inequalities, *

*Differential
Harnack
inequalities and*

*Applications
in
Geometry»*

every
**Wednesday**,
starting from Wednesday 12-6-2013

at **11:15-12:45
B214**
(Mathematics building)

Abstract

In this lecture course we
present an entropy
functional for evolving domains which is essentially the one
used by Perelman
for Ricci flow as part of his solution of the Poincare
conjecture, but we augment
it by a boundary integral which incorporates the speed
function for the
boundary of the evolving domain. This entropy more or less
arises as the
time-derivative of the more standard entropy quantity also featuring in
thermodynamics where we
integrate the function u log u over a (potentially
evolving) domain. Here, u is a
solution of the heat
equation or the backward heat equation on this domain and
satisfies suitable
Neumann boundary conditions related to the velocity of the
boundary of the
domain. We prove basic properties of this functional, in
particular show its
relation to logarithmic Sobolev inequalities. We also explain
logarithmic
Sobolev inequalities and show how they can be derived from ordinary Sobolev
inequalities. Furthermore, we
show a connection between the time-derivative of the entropy
and differential
Harnack inequalities, the latter in particular for the
standard heat equation.
We will also mention potential applications of this to
geometric evolution equations
such as the mean curvature ow. Moreover, we shall explain
Perelman's original
arguments pertaining to this entropy in the context of Ricci
flow, which is both
interesting from the nonlinear PDE and the differential
geometry point of view.
We assume basic familiarity with linear PDE but background on
differential geometry
will be provided in lectures whenever required.

Lecture Notes: Part1