Physical Sciences and Mathematics Commons^™

Criticality For The Gehring Link Problem, Jason Cantarella, Joseph H.G. Fu, Robert Kusner, John M. Sullivan, Nancy C. Wrinkle

Robert Kusner

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring’s problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality.

Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn–Tucker theorem. We use this to prove that every critical link …

On The Nondegeneracy Of Constant Mean Curvature Surfaces, Nick Korevaar, Robert Kusner, Jesse Ratzkin

Robert Kusner

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\Sigma \to \R^3$ are nondegenerate; that is, the Jacobi operator Δf+|Af|2 has no L2 kernel. In fact, if Σ has genus zero and f(Σ) is contained in a half-space, then we find an explicit upper bound for the dimension of the L2 kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising …