Curvature Measures in Convex and Integral Geometry

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This project builds on decisive progress in Convex and Integral Geometry arising from the work of S. Alesker on valuations, i.e., finitely additive functions on the space of convex bodies. The space of continuous and translation-invariant valuations admits a natural finite grading. By Alesker's Irreducibility Theorem, each graded component is irreducible under the action of the general linear group. This has strong implications for the structure of the space of valuations and led S. Alesker to discover a range of natural algebraic operations on valuations, in particular a commutative product. Furthermore the restriction to the convex setting turned out to be unnatural, and S. Alesker has introduced a theory of smooth valuations on general smooth manifolds.Smooth valuations may be localized, albeit not uniquely, to smooth curvature measures, a crucial new concept introduced by A. Bernig and J.H.G. Fu. In more geometric terms, a smooth curvature measure is an integral of invariants of the second fundamental form that persists under certain singular degenerations. The prime classical examples are Federer's curvature measures, which localize the intrinsic volumes. By Alesker's Irreducibility Theorem, the closure of smooth valuations in the topology of uniform convergence on compact sets is precisely the space of continuous valuations. A fundamental issue and main objective in this project is to similarly complete the space of smooth curvature measures and to characterize the elements of the completion by a short list of inevitable properties. The project will persue the idea that many theorems on translation-invariant valuations should have a counterpart for curvature measures in a few different contexts. In particular, the action of the general linear group will be investigated. Smooth curvature measures also open up a new perspective on the intrinsic volumes. Alesker has shown that the notion of intrinsic volumes from Convex Geometry associates to each Riemannian manifold a finite-dimensional algebra of valuations, the Lipschitz-Killing algebra. The fact that the space of smooth curvature measures is naturally a module over smooth valuations opens up a new and evidently very natural approach: By a conjecture of A. Bernig, J.H.G. Fu, and S. Solanes, a smooth valuation leaves invariant (with respect to the Alesker product) the subspace angular curvature measures, which is distinguished by a simple geometric property, if and only if it is an element of the Lipschitz-Killing algebra. Recently we have established the ``if part'' of this conjecture. One of our primary goals in this project is to explore the ``only if part'' of this conjecture.