Hessian energy and optimal parametrisations of surfaces
type: Item
instance of: scholarly article
title: Hessian energy and optimal parametrisations of surfaces
zbMATH Open document ID: 0879.49029
DOI: 10.1007/PL00004316
author: Yi Fang
author: John E. Hutchinson
published in: Mathematische Zeitschrift
publication date: 1998-01-25T00:00:00Z
review text: 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 Lagra…
Mathematics Subject Classification ID: 49Q99
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J50
zbMATH DE Number: 1057303
zbMATH Keywords: Hessian energy
zbMATH Keywords: existence
zbMATH Keywords: optimal parametrization
zbMATH Keywords: Hessian functional
zbMATH Keywords: conformal parametrization
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: branched immersions
zbMATH Keywords: univalent maps
zbMATH Keywords: Enneper surface
zbMATH Keywords: energy functional
MaRDI profile type: Publication
full work available at URL: pl00004316
OpenAlex ID: W2088393960
description: scientific article; zbMATH DE number 1057303