On the Distributional Hessian of the Distance Function

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian

Fourth-order geometric flows and integral piching of the curvature

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

CMS: a Growing Grading System

We give an update on CMS, the free and open source grading system used in IOI 2012, 2013 and 2014. In particular, we focus on the new features and development practices; on what we learned by running dozens of contests with CMS; on the community of users and developers that has started to grow around it

Some sphere theorems in linear potential theory

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain Ω ⊂ Rn, n ≥ 3, we prove that if the mean curvature H of the boundary obeys the condition (Formula Presented) then Ω is a round ball.SCOPUS: ar.jDecretOANoAutActifinfo:eu-repo/semantics/publishe