A PhD course (6 lectures  2
hours each)
Dipartimento di Matematica Università di Roma "Tor Vergata" First Lecture:
Monday February 3 at 14:00 in Aula D’Antoni Abstract In this minicourse we are interested in the relationships between the geometric curvature of a domain and its metric curvature. More precisely, the picture is as follows. On one hand, a domain D in the complex space comes with a natural boundary, on which one can put geometric conditions (examples: convexity, pseudoconvexity, finite type). On the other hand, D can be viewed as a complex manifold that possesses under mild conditions metrics (the Kobayashi metric, the Bergman metric, KählerEinstein metrics to name a few) , that is, objects giving an intrinsic way of measuring speed of vectors, distances between points. Moreover if the metric we work with is smooth enough one can define several notions of curvatures (such as Riemannian sectional curvatures or holomorphic sectional/bisectional curvatures). The game is now to relate the geometric boundary conditions with the behaviour of certain metrics. Naturally, one expects the problem to be local. In other words, the geometric properties of the boundary of D at/near a boundary point p might impact the behaviour of metrics only near p, and viceversa. The program of this minicourse is the following. After a more detailed introduction (chapter 0), we will define the notions that are in parenthesis in the above description, and study it on several "toy" examples. This study will lead us to formulate a conjecture regarding the possible relationships between the notions of curvatures described above. In chapter 2 we will study this conjecture in an easy case, that is when a domain "looks like a ball near its boundary". Finally we will touch on the state of this conjecture in the general case in chapter 3. In chapter 2 and 3 we will pay particular attention to the local nature of our problem. To that effect we will discuss about a scaling technique and also about the problem of localisation of metrics. Key Words: Holomorphicity, Pseudoconvexity, Finite Type, Invariant Metrics, Localisation of Metrics, Scaling. 
