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

Abstract

Abstract In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold ( R n , g ) where the metric is of the form g ( x ) = c ( x ) ( g ^ ⊕ e ) . Here g ^ is a simple Riemannian metric on R n − 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 x n = 0 is equal to a positive constant. We also show a partial data result when n = 3.