Willmore surfaces in Riemannian manifolds

几何变分问题自然而然地出现在数学、物理、生物学和计算机科学的各个分支中。最突出的例子是等周问题和极小曲面。在这个项目中，我们研究了高阶变分问题，如Willmore泛函及其变种。这种特定的泛函出现在例如广义相对论、生物学和图像恢复理论中。在过去的几年里，Willmore泛函及其变种在欧氏空间中得到了广泛的研究，我们的目标是将该理论发展并部分推广到任意目标流形，因为这与上述应用有关。更准确地说，我们想要研究环境曲率对泛函的几何和解析性质的影响。我们的主要动机来自于广义相对论的应用，我们想要更好地理解威尔莫尔泛函和这一理论中相关物理量之间的关系。为了达到我们的目标，我们必须推广所考虑泛函的现有正则性、紧性和存在性结果，而且由于我们处理的是关键问题，这就需要对基本的偏微分方程组进行仔细而细致的研究。

Geometric variational problems arise naturally in various branches of mathematics, physics, biology and computer science. The most prominent examples are the isoperimetric problem and minimal surfaces. In this project we study higher order variational problems such as the Willmore functional and variants thereof. This specific functional arises for example in general relativity, biology and in the theory of image restoration. In the last few years the Willmore functional and its variants were extensively studied in Euclidean space and our goal here is to develop and partially extend the theory to arbitrary target manifolds since this is relevant in the above mentioned applications. More precisely, we want to study the effect of the ambient curvature on the geometric and analytical properties of the functionals. Our main motivation comes from applications to general relativity and we want to better understand the relation between the Willmore functional and the relevant physical quantities in this theory. In order to achieve our goals we have to extend the existing regularity, compactness and existence results for the functionals under consideration and since we are dealing with critical problems this requires a careful and delicate study of the underlying partial differential equations.