Digital archive of theses discussed at the University of Pisa


Thesis etd-09252013-121934

Thesis type
Tesi di laurea magistrale
Thesis title
Fourth-order geometric flows and integral piching of the curvature
Course of study
relatore Dott. Mantegazza, Carlo
  • flussi geometrici
  • riemannian geometry
  • geometria riemanniana
  • integral pinching
  • geometric flows
  • pinching integrale
Graduation session start date

Geometric flows have recently become a very important tool for
studying the topology of smooth manifolds admitting Riemannian metrics
satisfying certain hypotheses. A geometric flow is the evolution of
the Riemannian metric $g_0$ of a smooth manifold according to a
differential rule of the form $\partial_t g(t) = P(g(t))$, where at each time
$g(t)$ is a positively defined $(2,0)$ tensor (such that $g(0)=g_0)$
on a fixed differentiable manifold $M$ and $P(g)$ is a smooth
differential operator depending on $g$ itself and on its space
derivatives, hopefully chosen in order to have the effect of
increasing the ``regularity'' of the Riemannian manifold. Once the
metric has been ``enhanced'' by the flow, one can study it more easily
and obtain topological results that, since the flow is smooth, must
also hold for the initial differentiable manifold.

The study of a geometric flow usually goes through some recurrent

1. The very first point is to show that, given the initial metric,
there is a (usually unique) smooth solution of the flow for at
least a short interval of time.

2. The maximal time for the existence of a smooth solution can be
finite or infinite: in the first case a singularity of the flow
develops, so its nature must be investigated in order to possibly
exclude it by a contradiction argument, or to classify it to get
topological information on the manifold, or finally to fully
understand its structure and possibly perform a smooth topological
``surgery'' in order to continue the flow after the singular
time. A very remarkable example of this last situation (which by
far is the most difficult case to deal with) is the success in the
study of Hamilton's Ricci flow, that is, the flow $\partial_t g(t)
= -2Ric_{g(t)}$, on the $3$--manifolds due to Perelman, leading
to the proof of the Poincaré conjecture.

In our work we will deal only with the first situation: assuming
that the flow of $g(t)$ is defined in the maximal time interval
$[0,T)$ with $0 < T < \infty$ and that at time $T$ a singularity
develops, we will try to exclude this scenario by a contradiction
argument (just to mention, another recent great success of the
application of Ricci flow to geometric problems, the proof of the
differentiable sphere theorem by Brendle and Schoen follows this
line). In this respect, a fundamental point of this program is to
show that the Riemann curvature tensor must be unbounded as $t\to

3. After obtaining the above result, the idea is to perform a
blow--up analysis: we take $t_i \nearrow T$ and dilate the metric
$g(t_i)$ so that the rescaled sequence of manifolds have uniformly
bounded curvatures; then, we prove that they stay within a
precompact class and take a limit of such sequence. At this point,
one has to study the properties of such possible limit manifold
(this may require a full classification result) in order to
proceed in one of the ways described above.

4. In our case, we actually want to find a contradiction in this
procedure by studying the limit manifold. This would imply that
the flow cannot actually be singular in finite time and the
maximal time of smooth existence has to be $+\infty$.

5. Then, once we know that the flow is defined for all times, we
prove again that there is a limit manifold as $t \to+ \infty$ and
we study its properties. For example, if the limit manifold turns
out to have constant positive sectional curvature, it must be the
quotient of the standard sphere. Hence, the initial manifold too
is topologically a quotient of the sphere, concluding the
geometric program.

Among the geometric flows, a special class is given by the ones
arising as gradients of geometric functionals of the metric and the
curvature. In such cases, because of the variational structure of the
flow, the natural energy (the value of the functional) is decreasing
in time and one can take advantage of this fact to carry out some of
the arguments mentioned above.

Our work, which fits in this context, is based upon the PhD thesis of
Vincent Bour, who studied a class of geometric gradient flows of the
fourth order. To briefly describe it, we recall that the Riemann
curvature tensor $Riem_g$ of a Riemannian manifold $(M^n,g)$ can be
orthogonally decomposed as

Riem_g = W_g + Z_g + S_g


S_g = R_g / (2n(n-1)) g . g
Z_g = 1 / (n-2) ( Ric_g - R_g / n * g ) . g

where $Ric_g$ is the Ricci tensor, $R_g$ the scalar curvature and the
remaining Weyl curvature $W_g$ is a fully traceless tensor (the
operation $.$ indicates the Kulkarni--Nomizu product, see the next
chapter for all the definitions).

Then, we define for $0 < \lambda < 1$

F^\lambda (g) = (1-\lambda) \int_{M^n} \abs{W_g}^2 \, dv_g + \lambda \int_{M^n} \abs{Z_g}^2 \, dv_g \]

and consider the gradient flow

\partial_t g(t) = -2 \nabla \F^\lambda(g(t)) . (1)

We will follow the steps outlined above in order to prove that if we
consider a compact manifold $M^4$ with an initial smooth metric $g_0$
such that $(M^4, g_0)$ has positive scalar curvature and initial
energy $\F^\lambda(g_0)$ sufficiently low, the flow (1) exists for all
times and converges in the $C^\infty$ topology to a smooth metric
$g_\infty$ on $M$ of positive constant sectional curvature. Thanks to
the Uniformization Theorem, we have that $M$ is diffeomorphic to a
quotient of the $4$--sphere, thus, it can only be either the
$4$--sphere or the $4$ dimensional real projective space.