Geometric curvature energies

本研究计划的中心议题是研究曲线、曲面和低规则性高维物体上的几何曲率能量。这些能量表现出自我回避和正则化效应：有限的能级保证和量化几何物体的嵌入性，并导致更高的规律性。几何曲率能量种类繁多，从硬空间约束到软排斥势，如全局曲率、门格尔曲率的积分版本或自排斥结能。我们研究了这些能量的解析性质，并研究了它们在曲线或曲面族上的行为。研究结果将应用于嵌入类中的拓扑约束变分问题，并研究相应的随时间变化的几何流动问题。

The central issue of this research proposal is the study of geometric curvature energies on curves, surfaces, and higher-dimensional objects of low regularity. These energies exhibit self-avoidance and regularizing effects: a finite energy level guarantees and quantifies embeddedness of the geometric object, and leads to higher regularity. There is a variety of geometric curvature energies ranging from hard steric constraints to soft repulsive potentials, such as global curvature, integral versions of Menger curvature, or self-repulsive knot energies. We investigate the analytical properties of these energies and study their behaviour on families of curves or surfaces. The results will be applied to topologically constrained variational problems in the class of embeddings, and we examine the corresponding timedependent geometric flow problems.