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I am looking some references on the existence and non-existence of (space-like) harmonic maps solving the Dirichlet into the de-Sitter space.

More precisely: Let, for $n\geq 3$, $$dS^n=\{ u\in \mathbb{R}^{n,1} \, \vert \, \langle u,u\rangle =1 \}$$ where

$$\langle .,.\rangle= -dx_0^2 +\sum_{i=1}^n dx_i^2 .$$

Given $\phi : S^1 \rightarrow dS^n$ (smooth), we look for an harmonic map $$u:\mathbb{D} \rightarrow dS^n$$ such that
$$u_{\partial \mathbb{D}} =\phi.$$

As in the Euclidean(and Riemannian) case, one idea is to proceed by minimization, but since the metric is not positive, there is no chance for this method to work.

One idea is to assume that $\phi$ can be bounded by a disc in de Sitter whose tangent planes are space-like and then to minimize only among the space-like map: $$\{ u:\mathbb{D} \rightarrow dS^n \, \vert \, \vert \nabla u\vert^2\geq 0 \}.$$

This will provide an inferior bound on the functional and give some some chance to a minimization process to work. Unfortunately I found nothing in this spirit in the Literature. I will be grateful for any result reference on existence, uniqueness, etc...

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    $\begingroup$ The analogous problem for the minimal surface equation (which in this case would be the Dirichlet problem for maximal hypersurfaces) is fairly well studied. Maybe some of the theory developed there could help? In particular, when $n = 3$, your disc would be a hypersurface and I think you can look for maximizers of the energy functional, assuming the boundary is acausal. (Not 100% sure.) (When $n > 3$ then I don't think you can use maximizers anymore, so things becomes more tricky.) $\endgroup$ Apr 2, 2019 at 17:16
  • $\begingroup$ Have you any reference. I am looking for some parametric approach, no-graph if possible. $\endgroup$
    – Paul
    Apr 5, 2019 at 17:57
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    $\begingroup$ Why "no-graph"? If you want a space-like solution, using that dS is globally hyperbolic, the solution is guaranteed to be a graph over any Cauchy hypersurface. Anyway, here are some links: arxiv.org/abs/1112.4219 mathscinet.ams.org/mathscinet-getitem?mr=545611 mathscinet.ams.org/mathscinet-getitem?mr=3194356 $\endgroup$ Apr 8, 2019 at 12:46
  • $\begingroup$ Thanks for the link. Yes we are necessary graph but looking at $(x,y)\mapsto (x,y,u(x,y))\in dS^3$ seems very rigid to me, when dealing with harmonic maps I prefer to let the parametrization more free, up to fix it later, so usually I look to $(x,y)\mapsto u(x,y)$ where $u$ is vector. $\endgroup$
    – Paul
    Apr 9, 2019 at 13:16

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