Geometric properties of surfaces with the same mean curvature in and

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• 作者：Alma L. Albujer ^{alma.albujer@uco.es" class="auth_mail" title="E-mail the corresponding author} ; Magdalena Caballero ; ^{magdalena.caballero@uco.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Spacelike surface ; Mean curvature ; Dirichlet problem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：1013-1024
• 全文大小：399 K

文摘

Spacelike surfaces in the Lorentz–Minkowski space 7383639dfc" title="Click to view the MathML source">L³${\mathbb{L}}^3$ can be endowed with two different Riemannian metrics, the metric inherited from 7383639dfc" title="Click to view the MathML source">L³${\mathbb{L}}^3$ and the one induced by the Euclidean metric of R³${\mathbb{R}}^3$. It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. One of our main results states that those surfaces have non-positive Gaussian curvature in R³${\mathbb{R}}^3$. As an application of this result, jointly with a general argument on the existence of elliptic points, we present several geometric consequences for the surfaces we are considering. Finally, as any spacelike surface in 7383639dfc" title="Click to view the MathML source">L³${\mathbb{L}}^3$ is locally a graph, our surfaces are locally determined by the solutions to the H_R=H_L$H_R=H_L$ surface equation. Some uniqueness results for the Dirichlet problem associated to this equation are given.