Hessian energy and optimal parametrisations of surfaces (Q1365454)

Suppose \(S\subset\mathbb{R}^n\) is a disc-like smooth surface. The existence of ``optimal'' prametrizations \(Y:\Omega\) \((\subset\mathbb{R}^2)\to S\) is established, where \textit{both} the domain \(\Omega\) and the parametrization \(Y\) are allowed to vary. Some examples and model problems are analyzed. For simplicity, first suppose \(Y\) is conformal. Minimizing the Dirichlet energy is not appropriate because of its invariance under conformal change of domain. Instead, motivated by the idea that an optimal parametrization should be as close to ``flat'' as possible, one attempts to minimize the Hessian functional \(\int_\Omega|D^2Y|^2\) over arbitrary disc-like \(\Omega\subset\mathbb{R}^2\) and arbitrary conformal parametrizations \(Y:\Omega\to S\). This is essentially equivalent to considering a \textit{fixed} conformal parametrization \(X:D\to S\), where \(D\) is the unit disc, an arbitrary \textit{flat} metric \(g\) on \(D\), and minimizing the covariant Hessian functional \(\int_D|\nabla^2_g X|^2\). The metric \(g\) is normalized by fixing the corresponding area of \(D\). Without loss of generality, let \(g= e^{2h}\delta_{ij}\), where \(\Delta h=0\) and \(\int_D e^{2h}=\pi\). A computation shows the energy functional \({\mathcal E}(h)\) equals \(\int_D\sum^n_{\alpha=1}|e^{-h}D^2X^\alpha+ D(e^{-h})\odot DX^\alpha|^2\). The quantity \(D(e^{-h})\odot DX^\alpha\) denotes the symmetric product of the vectors \(D(e^{-h})\) and \(DX^\alpha\), and is a \(2\times 2\) symmetric matrix. It is instructive to first analyze a simple model linear problem, where one minimizes the \(L^2\) (or \(W^{1,2}\)) distance from a fixed \(\overline h\in L^2(D)\) (or \(W^{1,2}(D)\)) to the corresponding subset of harmonic functions. In the current setting, it is natural to take \(e^{2\overline h}\delta_{ij}\) to be the pullback metric on \(D\) induced by the fixed map \(X\). Existence, uniqueness and regularity up to the boundary of a minimizer \(h\), are readily established. The Lagrange multiplier for the problem is a member of \(W^{2,2}_0(D)\) (or \(W^{1,2}_0(D)\)). For the original nonlinear problem, coercivity is not obvious, but computations using the integrand's precise algebraic structure show that the integrand is bounded from below by \(|D(e^{-h}|DX|)|^2\) (even if \(X\) is not conformal). If \(S\) is \textit{not} flat, again by analyzing the algebraic structure of the integrand, one can show \(|e^{-h}|^2_{W^{1,2}(D)}\leq c{\mathcal E}(h)\), where \(c\) depends in particular on how far the surface \(S\) is from being flat. The existence of a minimizer is established by considering separately the cases where \(S\) is flat and not flat. The arguments are valid even if \(X\) is not conformal, and in this case establish the existence of an optimal parametrization in the conformal class determined by \(X\). Analogous results for branched immersions and univalent maps are also obtained. Interior regularity is immediate and boundary regularity will be considered elsewhere. In the conformal case, alternative expressions for the energy integrand are established. From these one shows that the standard Weierstrass representation of Enneper's surface is optimal. Standard stereographic projection onto any spherical cap is also optimal, but the argument is somewhat more subtle. One first finds another form of the energy functional and then, with Mathematica\(^{\text{TM}}\) as a guide, shows the positivity of various integrals.