Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral. (English) Zbl 0874.49038
Chow, Ben (ed.) et al., Elliptic and parabolic methods in geometry. Proceedings of a workshop, Minneapolis, MN, USA, May 23–27, 1994. Wellesley, MA: A K Peters. 67-72 (1996).
The Willmore problem seeks the compact surface \(M\) embedded or immersed in \(\mathbb{R}^3\) that minimizes the squared-mean-curvature integral
\[
W(M)= \int_M H^2dA
\]
among surfaces of fixed topological type, such as prescribed genus or regular homotopy class. The main result is the following proposition.
Suppose that \(M_i\subset\mathbb{R}^3\) is a sequence of embedded surfaces of genus \(g\), with \(W(M_i)\) converging to the infimum \(W_g\) of \(W\) among surfaces of genus \(g\). Then there is a subsequence \(M_j\) and a sequence of Möbius transformations \(G_j\), such that \(G_j(M_j)\) converges smoothly to an embedded surface \(M\subset\mathbb{R}^3\) of genus \(g\), with \(W(M)= W_g\).
For the entire collection see [Zbl 0853.00042].
Suppose that \(M_i\subset\mathbb{R}^3\) is a sequence of embedded surfaces of genus \(g\), with \(W(M_i)\) converging to the infimum \(W_g\) of \(W\) among surfaces of genus \(g\). Then there is a subsequence \(M_j\) and a sequence of Möbius transformations \(G_j\), such that \(G_j(M_j)\) converges smoothly to an embedded surface \(M\subset\mathbb{R}^3\) of genus \(g\), with \(W(M)= W_g\).
For the entire collection see [Zbl 0853.00042].
Reviewer: V.Pavlenko (Chelyabinsk)
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 58E30 | Variational principles in infinite-dimensional spaces |
| 31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |
