双能量的归约方法
Reduction methods for the bienergy
摘要：
本文中，我们开发的想法介绍了\cite{MR}，侧重于\emph{reduction methods}（基本上，群作用，或更一般地说，simmetries）的bienergy。这种类型的技术使我们能够产生的双能临界点的例子，通过减少相关的四阶偏微分方程的系统的ODE的研究。特别地，我们将研究\emph{models}之间的旋转对称双调和共形同态。接下来，我们将采用约简方法来研究$G-$不变浸入到欧几里得空间中的一个丰富的类。目前，在这些背景下的已知实例远未达到其同伴的深度和多样性，这些同伴为调和映射和极小浸入理论中的经典问题提供了基本解。然而，我们认为这些例子代表了一个重要的起点，可以激发进一步的研究双调和性。在这个顺序的想法，我们结束本文讨论了一些开放的问题和可能的方向，进一步的发展。
英文摘要：
This paper, in which we develop ideas introduced in \cite{MR}, focuses on \emph{reduction methods} (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's system to ODE's. In particular, we shall study rotationally symmetric biharmonic conformal diffeomorphisms between \emph{models}. Next, we will adapt the reduction method to study an ample class of $G-$invariant immersions into the Euclidean space. At present, the known instances in these contexts are far from reaching the depth and variety of their companions which have provided fundamental solutions to classical problems in the theories of harmonic maps and minimal immersions. However, we think that these examples represent an important starting point which can inspire further research on biharmonicity. In this order of ideas, we end this paper with a discussion of some open problems and possible directions for further developments.
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