The Willmore energy and curvature concentration
Raz Kupferman, Cy Maor, David Padilla-Garza
TL;DR
This work derives two complementary lower bounds for the bending energy $E_B$ of isometric immersions of a surface into $\mathbb{R}^3$, addressing curvature concentration and curvature-dipoles relevant to non-Euclidean elasticity. The first bound uses a curvature-based approach via a framed-loop isoperimetric inequality and a dual electrostatic interpretation of the Gauss curvature, yielding $\|d\mathcal{N}\|^2_{L^2} \ge \frac{4\pi - \|K\|_{L^1_{MC}}}{\|K\|_{L^1_{MC}}} \|K\|^2_{H^{-1}_{MC}}$ for extendable immersions on finitely-connected domains. The second bound detects curvature dipoles through a Burgers-vector construction for framed loops, establishing $\int_{\mathbb{S}^1} |D_s \mathcal{N}|\,d\ell \ge \frac{|B_{\gamma,\mathcal{N},\varphi_0}|}{\operatorname{Length}(\gamma(\mathbb{S}^1))}$, and together with a foliation-based coarea argument yields robust lower bounds on $E_B$ in dipole-rich geometries. The results yield optimal blow-up rates for cone singularities and provide lower bounds for the elastic energy of thin sheets, with potential impact on material science models of dislocations and curvature defects; an isoperimetric inequality for framed loops is also established as a potentially independently useful tool.
Abstract
We study isometric immersions of a Riemannian surface $(Ω,\frak{g})$, where $Ω\subset \mathbb{R}^2$, into $\mathbb{R}^3$. We consider their bending energy, i.e., the square of the $L^2$-norm of their second fundamental form, which is equivalent to the Willmore functional. We obtain two new lower bounds for this energy, one in terms of the Gaussian curvature of the surface, and the other in terms of a Burgers vector -- a measure of non-flatness connected to torsion. These new estimates provide optimal blowup rates of the energy when the curvature is concentrated (e.g., in a conical geometry). In the more subtle case of dipoles of concentrated curvature, we use the Burgers vector estimates to obtain an optimal blowup rate in terms of the size of the system. Our motivation comes from non-Euclidean elasticity, in which cones and curvature-dipoles play a central role. The lower bounds derived in this work directly yield lower bounds for the elastic energy of thin elastic sheets. The derivation of the curvature-based lower bound involves an isoperimetric inequality for framed loops, which we believe to be of independent interest.