According to Pierre Ossian Bonnet's theorem claiming that a surface is known up to a rigid body displacement from the data of its first and second fundamental forms, we give up to exhibit a minimal surface by finding directly its parametric equations. We prefer to discover the two fundamental forms in a first step, and to rebuild the surface in a second step. In this way we develop the following variational plan: "minimize the area regarded as a functional of the two fundamental forms taking into account the Gauss-Codazzi-Mainardi compatibility conditions by Lagrange multipliers". We show that the introduced multipliers satisfy an adjoint partial differential equation, as in optimal control when one applies the Pontriagin principle. The well-known pioneering discovery of Jean-Baptiste Meusnier de la Place (1785) asserting that "the mean curvature should vanish" is revealed as a compatibility condition for this adjoint equation.
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机译:根据Pierre Ossian Bonnet的定理,声称从其第一和第二基本形式的数据中已知一个刚性身体位移,我们通过直接找到其参数方程来放弃最小的表面。我们更愿意在第一步中发现这两个基本形式,并在第二步中重建表面。通过这种方式,我们开发了以下变分计划:“最小化被视为Gauss-Codazzi-Mainardi的兼容性条件通过拉格朗日乘法器的兼容性兼容的区域。我们表明,引入的乘法器满足伴随部分微分方程,如在应用Pontrigagin原理的最佳控制中。众所周知的Jean-Baptiste Meusnier de la Place(1785)断言“平均曲率应该消失”作为这种伴随方程的兼容性条件。
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