Geometric curvature functionals: energy landscape and discrete methods

几何曲率函数：能量景观和离散方法

• 批准号: 282535003
• 负责人: Professor Dr. Heiko von der Mosel
• 金额: --
• 依托单位: Institut für Numerische und Angewandte Mathematik (NAM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/282535003?language=en
• 关键词: Geometric; curvature; functionals; energy; landscape

This research proposal focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces, or more general k-dimensional sets in Euclidean d-space. Previous investigations of the principal investigators concentrated on the regularizing effects of such energies, with a priori estimates that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. The present project aims at a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Their impact on geometric knot theory is going to be investigated, and suitable structure-preserving discrete versions need to be developed and analysed. The tools range from non-local fractional differential operators, measure theory and integral geometry, to variational calculus and geometric topology. This project touches related mathematical disciplines such as knot theory, harmonic analysis, discrete differential geometry, and Riemannian geometry, and there are interesting connections to the Sciences and Engineering.

本研究的重点是几何曲率泛函，即欧氏d-空间中曲线、曲面或更一般的k维集的几何定义的自回避能量。以前主要研究者的研究集中在这些能量的正则化效应上，先验估计允许在拓扑约束下的纽结曲线和曲面的紧致性和变分应用。本项目的目的是更深入地了解这些高度奇异和非线性的非局域相互作用能量的能量格局。它们对几何纽结理论的影响将被研究，并需要开发和分析合适的保持结构的离散版本。这些工具的范围从非局部分数次微分算子、测度论和积分几何，到变分和几何拓扑。这个项目涉及到相关的数学学科，如纽结理论、调和分析、离散微分几何和黎曼几何，并且与科学和工程有有趣的联系。

项目成果

A Smoothness Energy without Boundary Distortion for Curved Surfaces

• DOI: 10.1145/3377406
• 发表时间: 2019-05
• 期刊: ACM Transactions on Graphics (TOG)
• 作者: Oded Stein;Alec Jacobson;M. Wardetzky;E. Grinspun

Sobolev Gradients for the Möbius Energy

莫比乌斯能量的索博列夫梯度

• DOI: 10.1007/s00205-021-01680-1
• 发表时间: 2020
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Philipp Reiter;Henrik Schumacher
• 通讯作者: Henrik Schumacher

Variational Methods for Discrete Geometric Functionals

离散几何函数的变分法

• DOI: 10.1007/978-3-030-31351-7_5
• 发表时间: 2020
• 作者: Henrik Schumacher;Max Wardetzky
• 通讯作者: Max Wardetzky

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

具有均匀有界几何曲率能的子流形的紧致性和同位素有限性

• DOI: 10.4310/cag.2018.v26.n6.a2
• 发表时间: 2018
• 期刊: arXiv: Differential Geometry
• 作者: Sławomir Kolasiński;Heiko Strzelecki Pawełand von der Mosel
• 通讯作者: Heiko Strzelecki Pawełand von der Mosel

Natural Boundary Conditions for Smoothing in Geometry Processing

• DOI: 10.1145/3186564
• 发表时间: 2017-07
• 期刊: ACM Transactions on Graphics (TOG)
• 作者: Oded Stein;E. Grinspun;M. Wardetzky;Alec Jacobson