Total curvature of graphs in Euclidean spaces

Embedding Products of Graphs into Euclidean Spaces

For any collection of graphs G1, . . . , GN we find the minimal dimension d such that the product G1 × · · · ×GN is embeddable into R . In particular, we prove that (K5) and (K3,3) are not embeddable into R, where K5 and K3,3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we s...

Total Curvature of Graphs in Space

The Fáry-Milnor Theorem says that any embedding of the circle S1 into R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs Γ, what limitations on the isotopy class of Γ are implied by a bound on total curvature? Especially: what does “total curvature” mean for a graph? I shall disc...

Constant Mean Curvature Surfaces in Euclidean and Minkowski 3-spaces

Spacelike constant mean curvature surfaces in Minkowski 3-space L have an infinite dimensional generalized Weierstrass representation. This is analogous to that given by Dorfmeister, Pedit and Wu for constant mean curvature surfaces in Euclidean space, replacing the group SU(2) with SU(1, 1). The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop grou...

On the Total Curvature of Semialgebraic Graphs

We define the total curvature of a semialgebraic graph Γ ⊂ R as an integral K(Γ) = R Γ dμ, where μ is a certain Borel measure completely determined by the local extrinsic geometry of Γ. We prove that it satisfies the Chern-Lashof inequality K(Γ) ≥ b(Γ), where b(Γ) = b0(Γ) + b1(Γ), and we completely characterize those graphs for which we have equality. We also prove the following unknottedness r...

On the Embeddability of Weighted Graphs in Euclidean Spaces