We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.
Si intende presentare, almeno in parte, l'imponente intreccio di idee, tecniche e acquisizioni concettuali che si è sviluppato intorno alla congettura di Poincaré, dalla sua formulazione agli inizi del secolo scorso fino alla soluzione data da Grisha Perelman agli inizi del nuovo millennio, portando a compimento il programma basato sullo studio del flusso di Ricci di metriche riemanniane su una data 3-varietà, delineato e sviluppato da Richard Hamilton dagli anni '80. Pur nei limiti e nelle possibilità...
We give an overview of the classification of networks in the plane with at most two triple junctions with the property that under the motion by curvature they are self-similarly shrinking. After the contributions in [7, 9, 20], such a classification was completed in the recent work in [4] (see also [3]), proving that there are no self-shrinking networks homeomorphic to the Greek “theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees. We present...
We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.
We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...
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