Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds


We study maps $\varphi$ from domains $\Omega$ in $\mathbf{R}^3$ to $\mathbf{S}^2$ which have prescribed boundary value functions $\psi$ and which minimize Dirichlet’s integral energy. Such $\varphi$’s can have isolated singular pints, and our principal concern is the number and arrangement of such singularities. Our main result is that the number of singular points is bounded above by a constant times Dirichlet’s integral of $\psi$ on $\partial\Omega$, and we show that this linear law is the best possible in two different senses. Furthermore, singularities of minimizing $\varphi$’s sometimes have peculiar and counterintuitive properties: (i) We illustrate by example that the mapping area (Jacobian integral) of $\psi$ can be zero although $\varphi$ has many singular points. (ii) We also show how to construct uniquely minimizing $\varphi$’s having many singular points stacked up vertically near $\partial\Omega$ (like bubbles in a pan of water about to boil. (iii) Finally, we show by example that symmetries of $\Omega$ and $\psi$ do not insure corresponding symetries of $\varphi$ and it singularities; in particular, singularities of $\varphi$ can be unstable under small perturbations of $\psi$.


Frederick J. Almgren, Jr.

Elliott H. Lieb