Spacelike convex surfaces with prescribed curvature in (2 + 1)-Minkowski space

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• 作者：Francesco Bonsante ; ^{bonfra07@unipv.it" class="auth_mail" title="E-mail the corresponding author} ; Andrea Seppi ^{andrea.seppi01@ateneopv.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Monge&ndash ; Ampè ; re equation ; Constant curvature surfaces ; Minkowski space ; Universal Teichm&uuml ; ller theory
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：434-493
• 全文大小：1090 K

文摘

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence D   in (2+1)$(2+1)$-dimensional Minkowski space, provided D   is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge–Ampère equation det⁡D²u(z)=(1/ψ(z))(1−|z|²)⁻²$\det D^{2}u(z)=(1/\psi (z)){(1-|z{|}^2)}^{-2}$ on the unit disc, with the boundary condition u|_∂D=φ$u{|}_{\partial \mathbb{D}}=\phi$, for ψ a smooth positive function and φ a bounded lower semicontinuous function.

We then prove that a domain of dependence D contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function φ is in the Zygmund class. Moreover in this case the surface of constant curvature K contained in D   has bounded principal curvatures, for every K<0$K<0$. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of ∂D$\partial \mathbb{D}$.

Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature K, as K   varies in (−∞,0)$(-\infty ,0)$.