Harmonic determinants and unique continuation

Abstract

We give partial answers to the following question: if $F$ is an $m$ by $m$ matrix on ${\mathbb{R}}^n$ satisfying a second order linear elliptic equation, does $\det F$ satisfy the strong unique continuation property? We give counterexamples in the case when the operator is a general non-diagonal operator and also for some diagonal operators. Positive results are obtained when $n=1$ and any $m$, when $n=2$ for the Laplace-Beltrami operator and also twisted with a Yang-Mills connection. Reductions to special cases when $n=2$ are obtained. The last section considers an application to the Calderón problem in 2D based on recent techniques.

22 pages, comments are welcome