On the existence of spherical metrics with conical singularities on a 2-sphere – Gabriele Mondello (Università di Roma “La Sapienza”)

Venue

Sala Seminari (Dip. Matematica).

Abstract

A celebrated result by Poincaré states that a compact Riemann surface ofpositive genus has a conformal metric of constant curvature, unique up torescaling. Clearly, the case of genus 0 is not so exciting: there is aunique complex structure and a unique metric of curvature 1 up to Möbiustransformations. The problem becomes more interesting if we require such metrics tohaveconical singularities of prescribed angles at a finite subset ofmarkedpoints. The case of negative and zero curvature was settled by McOwenandTroyanov in 1989-1991: they established the existence and uniqueness ofsucha metric in each conformal class. The case of positive curvature is more delicate: existence and uniquenessresults are known for small angles (Troyanov), whereas non-uniquenessresults are known in positive genus (Bartolucci-De Marchis-Malchiodi).In a joint work with D.Panov (still in progress), wedetermine for whichangle assignment there exists a surface of genus 0 witha metric ofcurvature 1 and conical singularities of such prescribed angles(andnon-coaxial holonomy).

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