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

TL;DR

This paper establishes local well-posedness for wave maps on curved (1+2)-dimensional spacetimes at near-critical regularity, extending bilinear estimates to variable coefficient settings using wave packet techniques.

Contribution

It introduces a novel approach to analyze wave maps on curved spacetimes at low regularity, generalizing bilinear estimates for variable coefficients.

Findings

Proves local well-posedness at almost critical regularity.

Extends bilinear L^2 estimates to variable coefficient wave equations.

Develops wave packet and characteristic energy methods for curved spacetime analysis.

Abstract

In this article we initiate the study of 1+ 2 dimensional wave maps on a curved spacetime 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.