## Thesis etd-09252013-121934 |

Thesis type

Tesi di laurea magistrale

Author

MASCELLANI, GIOVANNI

URN

etd-09252013-121934

Thesis title

Fourth-order geometric flows and integral piching of the curvature

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Dott. Mantegazza, Carlo

Keywords

- flussi geometrici
- riemannian geometry
- geometria riemanniana
- integral pinching
- geometric flows
- pinching integrale

Graduation session start date

18/10/2013

Availability

Full

Summary

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

steps:

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

T$.

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

with

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.

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