Mathematical Sciences: Mean Curvature Problems in Euclidean Space and Riemannian Analogs

尼古拉斯·科尔瓦将继续他在三维空间中具有恒定平均曲率的曲面的工作。在历史上，大多数研究都是针对平均曲率为零的曲面，这些曲面被称为极小曲面。典型的例子是肥皂片。在过去的五年里，在非零平均曲率领域有大量的活动；肥皂泡就是典型的例子。这项工作从温特对霍普夫猜想的反例中得到了推动。在过去的一年里，卡普利亚斯展示了如何构建更多不同拓扑类型的这样的例子。科尔瓦的工作在许多方面都是对这一点的补充，也是一种自然的发展，因为他正在研究这些表面必须具有的结构，而不是试图构建更多的例子。科尔瓦早期与他的合作者的工作使人们对这些近乎无穷大的表面的行为有了很好的理解。他的注意力现在将转向对空间有限区域中的结构的调查。这将涉及几何学和偏微分方程式理论的相互作用。他还将对毛细问题和椭圆方程解的凸性定理进行相关研究。

Nicholas Korevaar will continue his work on surfaces of constant mean curvature in three dimensional space. Historically most research was directed at surfaces of zero mean curvature, these are known as minimal surfaces. Prototypical examples being soap films. During the last five years there has been a great deal of activity in the realm of non-zero mean curvature; here soap bubbles are the typical examples. This work received its impetus from Wente's counterexample to the Hopf conjecture. During the last year Kapouleas has shown how to construct many more such examples of different topological types. Korevaar's work is in many ways complementary to this and a natural development in the sense that he is looking at the structure that such surfaces must have rather than attempting to construct more examples. Korevaar's earlier work with his collaborators has led to a good understanding of the behaviour of these surfaces near to infinity. His attention will now turn to an investigation of the structure in bounded regions of space. This will involve an interaction of ideas from geometry and the theory of partial differential equations. He will also carry out related research into capillarity questions and into convexity theorems for solutions of elliptic equations.