1 apr 2015 -- 14:30
Aula di Consiglio, Dip. Matematica, Università "La Sapienza", Roma
A celebrated result by Poincaré states that a compact Riemann surface has a conformal metric of constant curvature, unique up to rescaling and biholomorphisms. Clearly, the case of genus 0 is not so exciting, being trivially solved by any Fubini-Study metric on the complex projective line. The problem becomes more interesting if we require such metrics to have conical singularities of prescribed angles at a finite subset of n marked points. The case of negative and zero curvature was settled by McOwen and Troyanov in 1989-1991: they established the existence and uniqueness of such a metric in each conformal class. The case of positive curvature is more delicate: existence and uniqueness results are known for small angles (Troyanov), whereas non-uniqueness results are known in positive genus (Bartolucci-De Marchis-Malchiodi). In a joint work with D.Panov (still in progress), we determine for which angle assignment there exists a surface of genus 0 with a metric of curvature 1 and conical singularities of such prescribed angles (and non-coaxial holonomy).