The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r (\varphi)|^2\,dV$, where $ \varphi:M \to N$ is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an $ES-r$-harmonic map, i.e, a critical point of $ E_r^{ES}(\varphi)$. That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of $E_r^{ES}(\varphi)$ when $N={\mathbb S}^m$ $(r \geq4,\, m\geq3)$, and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for $E_r^{ES}(\varphi)$ for $r=4$. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if $2 r > \dim M$, the functionals $ E_r^{ES}(\varphi)$ may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.

中文翻译：

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高阶能量泛函

高阶能量泛函的研究首先由 Eells 和 Sampson 在 1965 年提出，后来由 Eells 和 Lemaire 在 1983 年提出。这些泛函提供了经典能量泛函的自然推广。更准确地说，Eells 和 Sampson 建议研究所谓的 $ES-r$-energy 泛函 $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^* +d)^r (\varphi)|^2\,dV$，其中 $ \varphi:M \to N$ 是两个黎曼流形之间的映射。在本文的开头部分，我们将澄清有关$ES-r$-谐波映射定义的一些相关问题，即$E_r^{ES}(\varphi)$ 的临界点。这对我们来说似乎很重要，因为在文献中，几位作者已经研究了其他高阶能量泛函，因此需要讨论和扩展一些最近的例子：这将在这项工作的前两部分完成，在那里我们获得了 $E_r^{ES}(\varphi)$ 当 $N={\mathbb S}^m$ $(r \geq4,\, m\geq3)$，我们还证明了一些对本主题未来发展有用的一般事实。接下来，我们将计算 $E_r^{ES}(\varphi)$ 的 Euler-Lagrange 方程组，其中 $r=4$。我们将把这个结果应用到空间形式地图和旋转对称地图的研究中：特别是，我们将专注于各种等角地图系列的研究。在第 4 节中，我们还将表明，即使 $2 r > \dim M$，泛函 $ E_r^{ES}(\varphi)$ 也可能不满足经典的 Palais-Smale 条件 (C)。在论文的最后部分，我们将研究第二个变化并计算一些重要例子的指数和无效性。