#### Vol. 14, No. 4, 2021

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

### Cristian Gavrus, Casey Jao and Daniel Tataru

Vol. 14 (2021), No. 4, 985–1084
##### 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.

##### Keywords
wave maps, curved spacetimes, low regularity, wave packets
##### Mathematical Subject Classification 2010
Primary: 35L05, 35L15, 35L70