Identification of a connection from Cauchy data on a Riemann surface with boundary

Colin Guillarmou (1) , Leo Tzou (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 Department of Mathematics

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the connection up to gauge isomorphism, and the potential $q$.

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Colin Guillarmou, Leo Tzou. Identification of a connection from Cauchy data on a Riemann surface with boundary. Geometric And Functional Analysis, 2011, 21 (2), pp.393-418. ⟨hal-00497657v2⟩