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Tesi etd-09252013-121934

Thesis type
Tesi di laurea magistrale
Fourth-order geometric flows and integral piching of the curvature
Corso di studi
relatore Dott. Mantegazza, Carlo
Parole chiave
  • flussi geometrici
  • riemannian geometry
  • geometria riemanniana
  • integral pinching
  • geometric flows
  • pinching integrale
Data inizio appello
Riassunto analitico
<br>Geometric flows have recently become a very important tool for<br>studying the topology of smooth manifolds admitting Riemannian metrics<br>satisfying certain hypotheses. A geometric flow is the evolution of<br>the Riemannian metric $g_0$ of a smooth manifold according to a<br>differential rule of the form $\partial_t g(t) = P(g(t))$, where at each time<br>$g(t)$ is a positively defined $(2,0)$ tensor (such that $g(0)=g_0)$<br>on a fixed differentiable manifold $M$ and $P(g)$ is a smooth<br>differential operator depending on $g$ itself and on its space<br>derivatives, hopefully chosen in order to have the effect of<br>increasing the ``regularity&#39;&#39; of the Riemannian manifold. Once the<br>metric has been ``enhanced&#39;&#39; by the flow, one can study it more easily<br>and obtain topological results that, since the flow is smooth, must<br>also hold for the initial differentiable manifold.<br><br>The study of a geometric flow usually goes through some recurrent<br>steps:<br><br> 1. The very first point is to show that, given the initial metric,<br> there is a (usually unique) smooth solution of the flow for at<br> least a short interval of time.<br><br> 2. The maximal time for the existence of a smooth solution can be<br> finite or infinite: in the first case a singularity of the flow<br> develops, so its nature must be investigated in order to possibly<br> exclude it by a contradiction argument, or to classify it to get<br> topological information on the manifold, or finally to fully<br> understand its structure and possibly perform a smooth topological<br> ``surgery&#39;&#39; in order to continue the flow after the singular<br> time. A very remarkable example of this last situation (which by<br> far is the most difficult case to deal with) is the success in the<br> study of Hamilton&#39;s Ricci flow, that is, the flow $\partial_t g(t)<br> = -2Ric_{g(t)}$, on the $3$--manifolds due to Perelman, leading<br> to the proof of the Poincaré conjecture.<br><br> In our work we will deal only with the first situation: assuming<br> that the flow of $g(t)$ is defined in the maximal time interval<br> $[0,T)$ with $0 &lt; T &lt; \infty$ and that at time $T$ a singularity<br> develops, we will try to exclude this scenario by a contradiction<br> argument (just to mention, another recent great success of the<br> application of Ricci flow to geometric problems, the proof of the<br> differentiable sphere theorem by Brendle and Schoen follows this<br> line). In this respect, a fundamental point of this program is to<br> show that the Riemann curvature tensor must be unbounded as $t\to<br> T$.<br><br> 3. After obtaining the above result, the idea is to perform a<br> blow--up analysis: we take $t_i \nearrow T$ and dilate the metric<br> $g(t_i)$ so that the rescaled sequence of manifolds have uniformly<br> bounded curvatures; then, we prove that they stay within a<br> precompact class and take a limit of such sequence. At this point,<br> one has to study the properties of such possible limit manifold<br> (this may require a full classification result) in order to<br> proceed in one of the ways described above.<br><br> 4. In our case, we actually want to find a contradiction in this<br> procedure by studying the limit manifold. This would imply that<br> the flow cannot actually be singular in finite time and the<br> maximal time of smooth existence has to be $+\infty$.<br><br> 5. Then, once we know that the flow is defined for all times, we<br> prove again that there is a limit manifold as $t \to+ \infty$ and<br> we study its properties. For example, if the limit manifold turns<br> out to have constant positive sectional curvature, it must be the<br> quotient of the standard sphere. Hence, the initial manifold too<br> is topologically a quotient of the sphere, concluding the<br> geometric program.<br><br>Among the geometric flows, a special class is given by the ones<br>arising as gradients of geometric functionals of the metric and the<br>curvature. In such cases, because of the variational structure of the<br>flow, the natural energy (the value of the functional) is decreasing<br>in time and one can take advantage of this fact to carry out some of<br>the arguments mentioned above.<br><br>Our work, which fits in this context, is based upon the PhD thesis of<br>Vincent Bour, who studied a class of geometric gradient flows of the<br>fourth order. To briefly describe it, we recall that the Riemann<br>curvature tensor $Riem_g$ of a Riemannian manifold $(M^n,g)$ can be<br>orthogonally decomposed as<br><br> Riem_g = W_g + Z_g + S_g<br><br>with<br><br> S_g = R_g / (2n(n-1)) g . g<br> Z_g = 1 / (n-2) ( Ric_g - R_g / n * g ) . g<br><br>where $Ric_g$ is the Ricci tensor, $R_g$ the scalar curvature and the<br>remaining Weyl curvature $W_g$ is a fully traceless tensor (the<br>operation $.$ indicates the Kulkarni--Nomizu product, see the next<br>chapter for all the definitions).<br><br>Then, we define for $0 &lt; \lambda &lt; 1$<br><br> F^\lambda (g) = (1-\lambda) \int_{M^n} \abs{W_g}^2 \, dv_g + \lambda \int_{M^n} \abs{Z_g}^2 \, dv_g \]<br><br>and consider the gradient flow<br><br> \partial_t g(t) = -2 \nabla \F^\lambda(g(t)) . (1)<br><br>We will follow the steps outlined above in order to prove that if we<br>consider a compact manifold $M^4$ with an initial smooth metric $g_0$<br>such that $(M^4, g_0)$ has positive scalar curvature and initial<br>energy $\F^\lambda(g_0)$ sufficiently low, the flow (1) exists for all<br>times and converges in the $C^\infty$ topology to a smooth metric<br>$g_\infty$ on $M$ of positive constant sectional curvature. Thanks to<br>the Uniformization Theorem, we have that $M$ is diffeomorphic to a<br>quotient of the $4$--sphere, thus, it can only be either the<br>$4$--sphere or the $4$ dimensional real projective space.