SUPERFICI LAGRANGIANE IN R^4 E TS^2 - Wilhelm Klingenberg (Durham University)

Gromov proved in 1985 that there is no exact
Lagrangian torus in standard R^4. In collaboration with B
Guilfoyle, we proved that there is no closed Lagrangian
(necessarily exact) section of TS^2 with only one complex
point. The symplectic and complex structures of TS^2 give
rise to a neutral metric of index (2,2). We will review
both proofs, which use the Cauchy-Riemann equation for
holomorphic discs with boundary on L and its analytic
index.