In a 2004 paper, Lindblad demonstrated that the minimal surface equation on describing graphical timelike minimal surfaces embedded in enjoy small data global existence for compactly supported initial data, using Christodoulou’s conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity.
The equation describing graphical timelike minimal surfaces in can be written as the quasilinear wave equation
for a real-valued function where , denotes the Minkowski metric on In dimensions . the small data global existence follows largely from the linear decay of solutions to the wave equation which has the rate and is by now standard ,Reference Sog08. In dimensions the linear decay of the wave equation has generic rates , respectively, which, not being integrable in time, can lead to finite-time singularity formation for even small data for generic quasilinear wave equations; see the survey article Reference HKSW16. Equation Equation 1 however exhibits the null condition, which is a structural condition on the nonlinearities identified first by Klainerman Reference Kla86 and Christodoulou Reference Chr86 in and later generalized by Alinhac in Reference Ali01aReference Ali01b. Using this fact Brendle Reference Bre02 and Lindblad Reference Lin04 established the small data global existence for Equation 1 in dimensions and respectively. Brendle’s proof followed the commuting vector field method of Klainerman Reference Kla86. Lindblad, however, gave two different proofs in his paper using respectively the commuting vector field method as well as Christodoulou’s compactification method.
In this paper we focus on the case The linear wave equation exhibits no decay in . and hence the classical null condition cannot be used to assert that the perturbation is effectively short range, as in the case for , As a side effect this means that a direct proof of global existence for .Equation 1 in modeled after the commuting vector field method is not possible. A striking aspect of Reference Lin04 is that via the conformal compactification method, Lindblad was also able to prove the global existence of solutions to Equation 1 in for small initial data. As Lindblad observed, the main trade-off is that for the conformal compactification method, the data must be of compact support, while in the vector field method the data is merely required to have “sufficiently fast” decay at infinity. The purpose of this paper is to produce an alternative proof of the case allowing initial data that is not necessarily compactly supported. In the course of the discussion we will also extract some more detailed geometric information concerning the solution when the data is of compact support.
2. General geometric formulation
The result of Brendle and Lindblad are based on comparing the timelike minimal surface to a flat hyperplane by normal projection, and the scalar function in Equation 1 describes the deviation, or height, of the minimal surface as a graph over the hyperplane. The minimal surface equation can however be written intrinsically by way of the Gauss and Codazzi equations. To quickly recall: let be a timelike hypersurface, then the Gauss equation requires that the Riemann curvature tensorFootnote1 for the induced Lorentzian metric obeyFootnote2
We use the convention which implies .
We will, throughout this paper, freely raise and lower indices using the induced metric.
where is the second fundamental form of the embedding of The Codazzi equation on the other hand requires .
where is the Levi-Civita connection associated to It is well known that equations .Equation Gauss and Equation Codazzi are the only obstructions to the existence of an isometric embedding. More precisely, we have the following theorem.
The upshot of this is that, the initial value problem for Equation 1 can be reformulated as finding the metric and the second fundamental form from the initial data, requiring that they obey Equation Gauss and Equation Codazzi.
Equations Equation Gauss and Equation Codazzi do not yet describe an evolution: they are underdetermined. We get a well-posed initial value problem if we combine Equation Codazzi with the minimal surface equation which gives ,
Using that the Riemann curvature tensor is at the level of two derivatives of the metric, the initial data for the system Equation Gauss, Equation Codazzi, and Equation Minimality consists of the metric the connection coefficients , and the second fundamental form , restricted to an initial slice. The initial value problem for Equation 1 requires the data for and at time these determine fully the first jet of ; restricted to the initial slice. As the equation Equation 1 is second order hyperbolic, this formally provides us the of -jet for any restricted to the initial slice, and a simple computation shows that the of -jet fully determines , and , on the initial slice. And thus we can indeed approach the vanishing-mean-curvature evolution through the initial value problem written in terms of the Gauss-Codazzi equations.
3. Double null formulation when
For geometric partial differential equations, it is necessary to fix a gauge (i.e. make a coordinate choice). For hyperbolic equations it is convenient to use a double-null formulation. Since is a two-dimensional Lorentzian manifold, it is foliated by two transverse families of null curves. Equivalently, there exists two independent functions satisfying
and Note that replacing . by for any with gives an equivalent reparametrization; we fix by prescribing their initial values on the initial slice, satisfying in particular that
Since and are solutions to the eikonal equation Equation 2, as is well known and are geodesic null vector fields, and at any point form a null frame of the tangent space .
The metric takes the form
where is some real-valued function. This implies that
so that the coordinate derivatives are
and that the inverse metric is
We can also compute
The assumption that is trace-free and symmetric requires that there exist scalar functions such that
Then Equation Codazzi implies
and Equation Minimality becomes
Taking linear combinations we get finally the null propagation equations for the second fundamental form
An immediate consequence of Equation -NP is the following proposition.
The evolution of the metric is now a scalar equation for the conformal factor The equation .Equation Gauss implies
Combined with the definition that
we arrive at
which we can rewrite as
The equations Equation -NP and Equation -wave can be regarded as evolution equations on a Minkowski space (not the same one as the Minkowski space over which is a graph in the original setup; it is a manifold conformally equivalent to the graph itself with its induced Lorentzian metric) with the canonical null coordinates with initial data prescribed at , In the remainder of this paper we discuss the easy consequences from the formulation above. .
4. Spatially compact initial data
Since Equation -wave is now a semilinear wave equation, with the nonlinearity only supported in a fixed space-time compact region, immediately by Cauchy stability we have
5. Noncompact initial data
We can also handle the case where are not compactly supported initially, but with sufficiently strong decay. This is the main theorem of this paper. Below we let be the coordinate function on the initial slice, and the time-function on ; and refer to the coordinate derivatives in the coordinate system.