Search

Associated to an embedded surface in the three-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, whereform we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs. The main ingredients in the proof of the completeness include a generalization of the Kneser conjecture for three-manifolds with boundary proved here, and extensions of Waldhausen's theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated. [ABSTRACT FROM AUTHOR]

We construct a complete invariant of oriented connected closed surfaces in 𝕊 3 , which generalizes the notion of peripheral system of a knot group. As an application, we define two computable invariants to investigate handlebody knots and bi-knotted surfaces with homeomorphic complements. In particular, we obtain an alternative proof of inequivalence of Ishii, Kishimoto, Moriuchi and Suzuki's handlebody knots 5 1 and 6 4 . [ABSTRACT FROM AUTHOR]

We discuss some examples of nonuniqueness for the crystalline curvature flow, when the Wulff shape is a square not centered at the origin. [ABSTRACT FROM AUTHOR]

In this paper we consider the so-called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend Ref. 10 with the introduction of the same nonregular potential used for the evolution problem in Ref. 9 and prove an optimal O(ε[sup 2]) error estimate for nondegenerate minimizers (where ε represents the perturbation parameter). Following Ref. 10 the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error estimate for discrete minima using the Rannacher–Scott L[sup ∞]-estimates and under appropriate restrictions on the mesh size (h[sup 2]=O(ε[sup σ]) with σ>5/2). [ABSTRACT FROM AUTHOR]