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The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently.

We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2, 2n − 1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank. 1.

The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

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