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

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. Contents1. Introduction 1 2. Curved X s,b spaces 13 3. Null frames 22 4. Wave packets analysis 26 5. Microlocalized characteristic energy estimates 38 6.The algebra property (1.15) 56 7. The product estimate (1.16) 75 8.The Moser estimate (1.18) 77 Appendix A. Smith's wave packets 88 Appendix B. Microlocal analysis tools 92 Appendix C.An angular partition of unity 96 References 98

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