Wave maps on (1+2)-dimensional curved spacetimes

Abstract

We initiate the study of $\left(1+2\right)$-dimensional wave maps on a curved space time in the low-regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical regularity.

As a key part of the proof of this result, we generalize the classical optimal bilinear $L^2$ estimates for the wave equation to variable coefficients by means of wave packet decompositions and characteristic energy estimates. This allows us to iterate in a curved $X^{s,b}$ space.