Geometric curvature functionals: energy landscape and discrete methods

• Applicants: Professor Dr. Heiko von der Mosel; Professor Dr. Max Wardetzky
• Subject Area: Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 282535003

• Final Report Year: 2021

Final Report Abstract

We report on our investigation of various geometric curvature functionals. In our investigation we covered classical curvature energies, such as Euler elasticae and bending energies, as well as geometrically defined self-avoidance energies for curves, surfaces, or more general m-dimensional sets in Rn . As planned, our research was driven by a combination of analytic techniques for studying knot energies and numerical algorithms and a convergence analysis thereof. We contributed to a deeper understanding of the energy landscape of highly nonlinear and partially singular and non-local geometric curvature energies. We investigated the impact of these energies on geometric knot theory alongside suitable structure-preserving discrete versions that we developed and analyzed. Our tools ranged from non-local fractional differential operators over measure theory, geometric topology, smooth and discrete variational calculus, discrete differential geometry to numerical algorithms and optimisation, using tools from hierarchical clustering and multigrid methods. We embarked on the investigation of the main questions outlined in the proposal of this project and made significant progress in terms of the exploration of the energy landscape of various curvature functionals, both from a theoretical and from a numerical perspective. Keywords: geometric curvature energies, singular integrals, geometric knot theory, discrete methods, variational calculus

Publications

• “Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies”. In: Comm. Anal. Geom. 26.6 (2018), pp. 1251–1316
  Sławomir Kolasiński and Heiko Strzelecki Pawełand von der Mosel
  (See online at https://doi.org/10.4310/CAG.2018.v26.n6.a2)
• “Natural Boundary Conditions for Smoothing in Geometry Processing”. In: ACM Trans. Graph. 37.2 (2018)
  Oded Stein, Eitan Grinspun, Max Wardetzky, and Alec Jacobson
  (See online at https://doi.org/10.1145/3186564)
• “A Reifenberg type characterization for m-dimensional C1-submanifolds of Rn ”. In: Ann. Acad. Sci. Fenn. Math. 44.2 (2019), pp. 693–721
  Bastian Käfer
  (See online at https://doi.org/10.5186/aasfm.2019.4443)
• Sobolev Gradients for the Möbius Energy. Archive of Rational Mechanics and Analysis. May 2020
  Philipp Reiter and Henrik Schumacher
  (See online at https://doi.org/10.1007/s00205-021-01680-1)
• “A Simple Discretization of the Vector Dirichlet Energy”. In: Computer Graphics Forum 39.5 (2020), pp. 81–92
  Oded Stein, Max Wardetzky, Alec Jacobson, and Eitan Grinspun
  (See online at https://doi.org/10.1111/cgf.14070)
• “A Smoothness Energy without Boundary Distortion for Curved Surfaces”. In: ACM Trans. Graph. 39.3 (2020)
  Oded Stein, Alec Jacobson, Max Wardetzky, and Eitan Grinspun
  (See online at https://doi.org/10.1145/3377406)
• “Variational convergence of discrete elasticae”. In: IMA Journal of Numerical Analysis (Dec. 2020)
  Sebastian Scholtes, Henrik Schumacher, and Max Wardetzky
  (See online at https://doi.org/10.1093/imanum/draa084)
• “Variational Methods for Discrete Geometric Functionals”. In: Handbook of Variational Methods for Nonlinear Geometric Data. Ed. by Philipp Grohs, Martin Holler, and Andreas Weinmann. Cham: Springer International Publishing, Apr. 2020, pp. 153–172
  Henrik Schumacher and Max Wardetzky
  (See online at https://doi.org/10.1007/978-3-030-31351-7_5)
• “Repulsive Curves”. In: ACM Trans. Graph. 40.2 (May 2021)
  Chris Yu, Henrik Schumacher, and Keenan Crane
  (See online at https://doi.org/10.1145/3439429)