On well-posedness for the timelike minimal surface equation

Abstract

We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation of this type. The first result of this kind, going substantially beyond a general Strichartz threshold for quasilinear hyperbolic equations, was shown for minimal surfaces of co-dimension one with a gain of $1/4$ regularity by Ai-Ifrim-Tataru. We use a geometric formulation of the problem, relying on its parametric representation. The natural dynamic variables in this formulation are the parametrizing map, the induced metric and the second fundamental form of the immersion. The main geometric observation used in this paper is the Gauss (and Ricci) equation, dictating that the Riemann curvature of the induced metric (and the curvature of the normal bundle) can be expressed as the wedge product of the second fundamental form with itself. The second fundamental form, in turn, satisfies a wave equation with respect to the induced metric. Exploiting the problem's diffeomorphism freedom, stemming from the non-uniqueness of the parametrization, we introduce a new gauge - a choice of a coordinate system on the parametrizing manifold - in which the metric recovers the full regularity of its Riemann curvature, including the crucial $L^1L^\infty$ estimate for the first derivatives of the metric. Analysis of minimal surfaces with co-dimension bigger than one requires that we impose and take advantage of an additional special gauge on the normal bundle of the surface. The proof also uses both the additional structure contained in the wedge product and the Strichartz estimates with losses developed earlier in the context of a well-posedness theory for quasilinear hyperbolic equations.