---
abstract: 'We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain’s $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate.'
address:
- |
    Department of Mathematics, Hylan Building\
    University of Rochester, Rochester, NY 14627
- |
    Department of Mathematics, Evans Hall\
    University of California at Berkeley, Berkeley, CA 94720-3840
author:
- 'Dan-Andrei Geba and Daniel Tataru'
bibliography:
- 'tril.bib'
title: Gradient NLW on curved background in $4+1$ dimensions
---

Introduction
============

In this article we are investigating the issue of local well-posedness for a variable coefficient semilinear wave equation in $4+1$ dimensions. To describe the context and motivate the interest in our problem we introduce three related equations. We begin with a generic gradient NLW equation in ${\mathbb R}^{n+1}$, $$\Box u\,=\,\Gamma(u)(\nabla u)^2
\label{sem}$$ with the nonlinearity $$\Gamma(u) (\nabla u)^2\,=\,q^{ij}(u) {\partial}_i u\, {\partial}_j u$$ where $q^{ij}$ are smooth functions and the standard summation convention is used.

Then we move on to a similar equation but on a curved background, $$\Box_g u\,=\,\Gamma(u)(\nabla u)^2
\label{eq}$$ with $\Box_{g}\,=\,g^{ij}\,\partial_i\partial_j$, where the summation occurs from $0$ to $n$ 