Geometry & Topology

In this talk, given a closed oriented surface of genus at least two and a unitary representation of its fundamental group, we introduce the variational problem for the pullback area associated with that surface and representation. In the case that the representation is weakly equivalent to the regular representation, we present a lower bound of the pullback area and give a complete classification of the extrinsic geometry when the equality is achieved. As a notable corollary, we solve the existence and rigidity aspect of the variational problem in the above case. Time permitting, the (extrinsic) area rigidity conjecture in higher dimensions will be discussed. This talk is based on joint work with Riccardo Caniato and Antoine Song.