Geometry and Analysis of Extremal Mappings of Finite Energy

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The proposed research centers around extremal problems of finite distortion. The relevance of these problems comes out of a confirmation that quasiconformal theory and nonlinear elasticity share compelling mathematical challenges. The theory of mappings of finite distortion arose from the need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting. There are many natural reasons for studying extremal problems in such a general degenerated setting. We eventually hope to lay down the analytical foundations for compactifying the moduli spaces. In such an extremal problem our mappings are not constrained on the boundary; the boundary does not even exist for compact Riemann surfaces. Annuli are where one first observes nontrivial conformal invariants. There is a close relationship between the existence of mappings with smallest mean distortion and the well known conjecture of Nitsche (1962) on harmonic homeomorphisms between circular annuli. The Nitsche conjecture was originally motivated by the study of the non-existence of doubly connected minimal graphs. Very recently, the PI in collaboration with T. Iwaniec and L. V. Kovalev gave an affirmative answer to this question. Our approach opens several future directions. There are many new and unexpected phenomena concerning existence (hammering phenomenon), uniqueness, regularity and failure of radial symmetry of the extremal mappings; some already prepared for answers, while others remain long term goals. Mathematical analysis relies on physical and geometric intuition for its future development. In recent years, this trend has become more pronounced, leading to concerted efforts of pure and applied mathematicians to work together. Problems in the intersection of Geometric Function Theory and Non-linear Elasticity that the PI proposes here contribute to these efforts. The proposed existence theorems of hyperelastic deformations might interest not only mathematicians but also engineers and physicists. The project also involves working with small colleges located near to Syracuse University with the aim of attracting young researchers from underrepresented groups to mathematics and also involves mentoring graduate students.