*preprint*

**Inserted:** 12 jun 2018

**Last Updated:** 12 jun 2018

**Year:** 2009

Published under previous name Reto Müller (please cite as Müller). Name changed here in order to import to author page correctly.

**Abstract:**

We investigate a new geometric flow which consists of a coupled system of the
Ricci flow on a closed manifold M with the harmonic map flow of a map phi from
M to some closed target manifold N with a (possibly time-dependent) positive
coupling constant alpha. This system can be interpreted as the gradient flow of
an energy functional F_{alpha} which is a modification of Perelman's energy F for
the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly,
the coupled system may be less singular than the Ricci flow or the harmonic map
flow alone. In particular, we can always rule out energy concentration of phi
a-priori - without any assumptions on the curvature of the target manifold N -
by choosing alpha large enough. Moreover, if alpha is bounded away from zero it
suffices to bound the curvature of (M,g(t)) to also obtain control of phi and
all its derivatives - a result which is clearly not true for alpha = 0. Besides
these new phenomena, the flow shares many good properties with the Ricci flow.
In particular, we can derive the monotonicity of an entropy functional W_{alpha
}
similar to Perelman's Ricci flow entropy W and of so-called reduced volume
functionals. We then apply these monotonicity results to rule out non-trivial
breathers and geometric collapsing at finite times.