The Transfer Principle of Integral Geometry and Isoperimetric Inequalities

• Applicant: Professor Dr. Thomas Wannerer
• Subject Area: Mathematics
• Term: from 2016 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 289866435

• Final Report Year: 2020

Final Report Abstract

About a decade ago it was realized that the fundamental operations discovered by pioneering work of S. Alesker on valuations, i.e., finitely additive measures, are intimately connected with the integral geometry of isotropic spaces. This radically new perspective opened the path to solve long-standing open problems such as the description of explicit kinematic formulas in complex space forms, a problem first taken up by Blaschke and his school in the 1930s, but also gave rise to interesting new questions and phenomena. Particularly striking among those is a mysterious transfer between certain isotropic spaces leading to isomorphic algebras of invariant valuations. For the real space forms—flat Euclidean space, the sphere, and hyperbolic space—this transfer is a now well-understood classical discovery due to R. Howard. But it was not expected for the complex space form when discovered by A. Bernig, J.H.G. Fu, and G. Solanes in 2014 and has so far resisted explanation. In collaboration with G. Solanes, I have discovered that the spheres of dimensions 6 and 7, that have an exceptional status owing to the octonions, provide two more instances where such a transfer happens. While this provided further evidence for a conjectural general transfer principle, the underlying mechanism responsible for the transfer remained mysterious. Key to understanding this mechanism—for the complex space forms and potentially in general—was a new perspective put forward in a joint collaboration with J.H.G. Fu that can loosely be described as valuationtheoretic Riemannian geometry or, more paradoxically, as integral geometry without a group. A surprising discovery was made by J. Kotrbatý, a PhD student in this project, who found that a certain pattern discovered in his investigation of the integral geometry of the octonionic plane perfectly matched with a pattern from a completely a different part of mathematics, the Hodge-Riemann bilinear relations of Kähler geometry. Maybe even more remarkable and vindicating an old dream of Blaschke and his school of integral geometry is the observation that the Hodge-Riemann bilinear relations for valuations imply isoperimetric inequalities. This discovery opens up a completely new perspective on isoperimetric inequalities and confirms on an even larger scale the fundamental importance of the Alesker product.

Publications

• Integral Geometry on the Octonionic Plane, PhD thesis, Jena, 2020
  J. Kotrbaty
  (See online at https://dx.doi.org/10.22032/dbt.45670)
• Classification of angular curvature measures and a proof of the angularity conjecture
  T. Wannerer
  
• Riemannian curvature measures. Geometric and Functional Analysis, 29 (2019), 343–381
  J.H.G. Fu and T. Wannerer
  (See online at https://doi.org/10.1007/s00039-019-00484-6)
• Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a Spin(9)-structure. Journal of Geometric Analysis
  J. Kotrbaty
  (See online at https://doi.org/10.1007/s12220-019-00209-z)
• On the extendability by continuity of angular valuations on polytopes, Journal of Functional Analysis
  T. Wannerer
  (See online at https://doi.org/10.1016/j.jfa.2020.108665)
• Integral geometry of exceptional spheres. Journal of Differential Geometry
  G. Solanes and T. Wannerer
  (See online at https://doi.org/10.4310/jdg/1609902019)