Crofton type formulas for complex geodesic planes:

In the real Euclidean space, the Euler characteristic of a domain is a multiple of the Gauss curvature integral of its boundary. This is no longer true in the hyperbolic space or in the real projective space. A possible way to express the Euler characteristic in these spaces is using some other mean curvature integrals and the volume of the domain or its boundary.

Euler characteristic, mean curvature integrals and volume are examples of the so-called valuations. In a real vector space, a valuation is a scalar valued functional of the space of non-empty compact convex subsets that satisfies an additive property. Hadwiger’s theorem states that the above valuations constitute a basis for the continuous invariant valuations under the full isometry group of R^n. In the standard Hermitian space, Euler characteristic, mean curvature integrals and volume are also valuations but Alesker recently proved that they do not constitute a basis for the continuous invariant valuations under the full isometry group of C^n.

In this talk, first, I will introduce the concept of valuation and the Hadwiger theorem. Then, I will give a basis for the continuous invariant valuations under the full isometry group of C^n. Finally, I will use it to give an expression for the Euler characteristic of a domain in the complex projective space and in the complex hyperbolic space.