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

详细信息    查看全文

• 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 Teichmü ; 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 870815302577&_mathId=si1.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=ae01a48cfe40dcfe0d5fef57eb168fff" title="Click to view the MathML source">(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 870815302577&_mathId=si2.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=e84cb53d4ffec5b59300563e9d36b648" title="Click to view the MathML source">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 870815302577&_mathId=si3.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=55c7dff8cda1880255fb7173afd9dd1b" title="Click to view the MathML source">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 870815302577&_mathId=si4.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=b4fca8723f12c207366643b9f77f38e0" title="Click to view the MathML source">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 870815302577&_mathId=si5.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=90dac9de45eec8af3c672e0dd0f221a4" title="Click to view the MathML source">∂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 870815302577&_mathId=si6.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=e3b9a5a8544410f1045889e50e0913ee" title="Click to view the MathML source">(−∞,0)$(-\infty ,0)$.