An inverse problem for the minimal surface equation in the presence of a riemannian metric

Abstract

In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold (Rn,g) where the metric is of the form g(x)=c(x)(g^⊕e) . Here g^ is a simple Riemannian metric on Rn−1 , e is the Euclidean metric on R and c a smooth positive function. We show that if the associated Dirichlet-to-Neumann maps corresponding to metrics g and c~g agree, then the Taylor series of the conformal factor c~ at xn=0 is equal to a positive constant. We also show a partial data result when n = 3.