arXiv subjects
Publications and source records attributed to Benjamin B. McMillan.
We demonstrate the existence of quasiconformal mappings on closed manifolds (dimension at least 5) that cannot be decomposed as a composition of mappings with arbitrarily small conformal distortion. This answers in the negative the global version of a classical question of Gehring.
We define a general procedure extending surgery to manifolds with foliation or Haefliger structure. We find a single obstruction to foliation surgery along an attaching sphere. When unobstructed, the surgery can be chosen to preserve characteristic numbers. Studying these obstructions, we obtain two results. First, on every stably trivial manifold of dimension \( n \le 2q+2 \), a transversely framed codimension-\( q \) Haefliger structure can be surgered to a Haefliger structure on the sphere \( S^{n} \), characteristic numbers unchanged. As an application, we modify a construction of Thurston's to give explicit Haefliger structures on \( S^{2q+1} \) whose Godbillon-Vey numbers surject to the real line. Second, the foliation connected sum gives, for each dimension \( n \) and codimension \( q \), a group structure on concordance classes of transversely oriented Haefliger structures on \( S^{n} \). These groups may be identified with the homotopy groups of the Haefliger classifying spaces \( BΓ^{+}_{q} \).
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.
I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants gives a geometric characterization for parabolic equation of Monge-Ampère type. A second family of invariants determines when a parabolic equation has a local choice of coordinates putting it in evolutionary form. In addition to their intrinsic interest, these results are applied in a follow up paper on the conservation laws of parabolic equations. It is shown there that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.