The Gauss-Bonnet theorem and Crofton-type formulas in complex space forms

Abstract

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms.1991 Mathematics Subject Classification. Primary 53C65; Secondary 52A22, 53C55.

Valuations and Curvature Measures on Complex Spaces

“…Roughly speaking, the knowledge of the local kinematic formulas and the knowledge of the different global kinematic formulas is equivalent. Some partial results were known before [20]: Park established local kinematic formulas in small degrees (n ≤ 3), while Abardia-Gallego-Solanes [1] proved Crofton-type formulas, which are special cases of the general kinematic formulas.…”

Aleksandrov-Fenchel inequalities for unitary valuations of degree $$2$$ 2 and $$3$$ 3

“…, K n in R n (n ≥ 2). A whole series of important inequalities between mixed volumes of convex bodies, including the Brunn-Minkowski and isoperimetric inequalities for quermassintegrals, can be deduced from (1) and hence the Aleksandrov-Fenchel inequality can be regarded as the main inequality in the Brunn-Minkowski theory of convex bodies. Special cases of (1) have been extended to non-convex domains, see [20,25,40].…”

“…A whole series of important inequalities between mixed volumes of convex bodies, including the Brunn-Minkowski and isoperimetric inequalities for quermassintegrals, can be deduced from (1) and hence the Aleksandrov-Fenchel inequality can be regarded as the main inequality in the Brunn-Minkowski theory of convex bodies. Special cases of (1) have been extended to non-convex domains, see [20,25,40]. For applications of the Aleksandrov-Fenchel inequality to the geometry of convex bodies and other fields such as combinatorics, geometric analysis and mathematical physics, we refer the reader to [11,19,33,37,38,45,46] and the references therein.…”

“…In the 1930s, Aleksandrov [2,3] gave two different proofs of his inequality, one based on strongly isomorphic polytopes and another, building on ideas of Hilbert [29,Chapter 19], based on elliptic operator theory. Around the same time, also Fenchel [21] sketched a proof of the inequality (1). In the 1970s, Khovanskiȋ [18,Section 27] and Teissier [50] independently discovered that the Aleksandrov-Fenchel inequality can be deduced from the Hodge index theorem from algebraic geometry.…”

Volume inequalities for complex isotropic measures

“…The idea to find analogs of known results from Euclidean geometry in complex vector spaces is not new. In recent years, the study of convex bodies in C n has received considerable attention (see, e.g., [1][2][3][4][5]13,15,19,[22][23][24][25][26]33,34,37,41,42]). …”