Gromov-hyperbolicity of strictly pseudoconvex domains and applications

The reason of interest for this thesis is to investigate bounded strictly pseudoconvex domains with C^2−smooth boundary, that is domains Ω ⊂ C^n which admit a strictly plurisubharmonic defining function of class C^2. Central part of the thesis is to show that bounded strictly pseudoconvex domains with C^2−smooth boundary, when endowed with the Kobayashi metric, are Gromov-hyperbolic. Chapter 4 is entirely devoted to the proof of this Theorem. Chapter 1, 2 and 3 introduce the terminology and the preliminary results needed to carry out the proof in Chapter 4. Chapter 1 deals in detail with the geometric structure of strictly pseudoconvex domains and the construction of the Carnot-Carathéodory metric on the boundary. In Chapter 2 we present the notion of Gromov-hyperbolicity. Such notion is a generalization of the ”classical” notion of hyperbolicity to general metric spaces. In Chapter 3 we introduce another notion of hyperbolicity, due to Kobayashi, for complex spaces. It is based on the definition of an intrinsic semi-distance function on any complex space, and such space is said to be Kobayashi-hyperbolic if such semi-distance function is an actual distance function. Notice that a Kobayashi-hyperbolic space is a metric space, hence it makes sense to investigate its Gromov-hyperbolicity. The domains of interest in this Thesis result to be Kobayashi-hyperbolic and Chapter 4, as previously mentioned, deals with the proof of their Gromov-hyperbolicity In the final part of the thesis we present a recent application of the results by Balogh and Bonk to the theory of functions with Bounded Mean Oscillation (BMO spaces, for short). In the setting of strictly pseudoconvex domains there are at least two notions of such spaces: BMO spaces defined via balls in the Kobayashi metric and dyadic BMO spaces. It was recently proved that such notions are equivalent. A key role in the proof of such equivalence is indeed played by the results presented by Balogh and Bonk.