Scaling of curvature measures and the modified Weyl-Berry conjecture
Research Project funded by the Deutsche Forschungsgemeinschaft (started in Mai 2020)
In the last years a theory of fractal curvature measures and associated scaling exponents has been established as a tool to describe the geometry of fractal sets beyond their fractal dimension. Exact theoretical results are up to now limited to some classes of fractals such as self-similar and self-conformal sets and the practical computation methods are based on a rather complicated direct implementation of the definition. In the proposed research project we aim at a simpler approach to the curvature scaling exponents, an extension of the theory to a larger class of (random) sets and applications of fractal curvatures to spectral problems. More precisely, we want to
- develop a box counting approach for curvature scaling exponents that parallels the well established box counting methods for the fractal dimension; explore the exact relations (expectedly the equivalence) of this new approach to the existing definition based on parallel sets; implement a box counting algorithm to estimate curvature scaling exponents from digital images;
- extend the theory of fractal curvatures to some further classes of sets including V-variable fractals and fractal sprays; explore the geometric meaning of curvature scaling exponents and fractal curvatures further and the interrelations between them;
- investigate the role of curvature scaling exponents and fractal curvatures for the spectral asymptotics of domains with fractal boundary; come up with a suitable modification of the modified Weyl-Berry conjecture.
Part 1 will increase the practical applicability of curvature scaling exponents, while both parts 1 and 2 will broaden and consolidate the theoretical foundations. Progress on part 3 will hopefully stimulate the interest of physicists in this theory, as it would underline the physical relevance of these geometric concepts.