Deformations of finite conformal energy: Existence and removability of singularities

This paper features a class of mappings h = (h¹,...,h ⁿ) :X→^ontoY between bounded domains X,Y ⊂ ℝⁿ, having finite n-harmonic energy, such that we have ℰ[h]=∫_x ∥Dh(x)∥ⁿdx, ∥Dh∥ ²=Tr(D*h Dh). The fundamental question is whether or not the domains X, Y ⊂ ℝⁿ of the same topological type admit a homeomorphism h: X → ^onto Y in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets x ⊂ X and γ ⊂ Y, referred to as cracks or fractures. We are looking for homeomorphisms h: X\x → ^ontoY\γ of finite energy for which γis the cluster set of h over x. In general, infinite energy is required in order to increase the dimension of a crack x ⊂ X;that is, when dim x < dim γ ≤ n-1. Suppose now that a bounded deformation h: X\x → ℝⁿ of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X. We give affirmative answers to these questions.