Willmore functional

The Willmore functional of an immersed surface $\Sigma$ into the Euclidean space $\mathbf{R} ^ { 3 }$ is defined by

\begin{equation*} W = \int _ { \Sigma } H ^ { 2 } d A, \end{equation*}

where $H = ( \kappa _ { 1 } + \kappa _ { 2 } ) / 2$ is the mean curvature of the surface. Here $\kappa_1$, $\kappa_2$ are the two classical principal curvatures of the surface (cf. also Principal curvature) and $d A$ is the area element of the induced metric on $\Sigma$. Moreover, it is assumed that the integral $W$ converges, which is guaranteed if $\Sigma$ is compact, as is usually assumed. Critical points of the functional $W$ are called Willmore surfaces and are characterized by the Euler equation $\Delta H + 2 H ( H ^ { 2 } - K ) = 0$ corresponding to the variational problem $\delta W = 0$ (cf. also Variation of a functional; Variational calculus; Variational problem). Here, $K = \kappa _ { 1 } \quad \kappa _ { 2 }$ is the Gaussian curvature of the surface and $\Delta$ is its Laplace–Beltrami operator (cf. Laplace–Beltrami equation). An alternative functional to $W$ is the functional given by

\begin{equation*} \tilde { W } = \int _ { \Sigma } ( H ^ { 2 } - K ) d A. \end{equation*}

Because of the Gauss–Bonnet theorem, if $\Sigma$ is assumed to be compact and without boundary, then $\tilde { W } = W - 2 \pi \chi ( \Sigma )$, where $\chi ( \Sigma )$ denotes the Euler characteristic of the surface, so that $W$ and $\tilde { W }$ have the same critical points.

The functional $W$ was first studied by W. Blaschke (1929) and G. Thomsen (1923), who established the most important property of $W$: The functional $W$ is invariant under conformal changes of metric of the ambient space $\mathbf{R} ^ { 3 }$. They considered it as a substitute for the area of surfaces in conformal geometry. For that reason, Willmore surfaces were called Konformminimalflächen (conformally minimal surfaces; cf. also Minimal surface).

These results were forgotten for some time and were rediscovered by T.J. Willmore in 1965, reopening interest in the subject. He proved that for any compact orientable surface $\Sigma$ immersed in $\mathbf{R} ^ { 3 }$ one has $W \geq 4 \pi$, equality holding if and only if $\Sigma$ is embedded as a round sphere. In an attempt to improve this inequality for surfaces of higher genus, Willmore also showed that for anchor rings, obtained by rotating a circle of radius $r$ about an axis in its plane at distance $R > r$ from its centre, it holds that $W \geq 2 \pi ^ { 2 }$, equality holding when $R / r = \sqrt { 2 }$. For various reasons, Willmore also conjectured that any torus immersed in $\mathbf{R} ^ { 3 }$ satisfies the inequality $W \geq 2 \pi ^ { 2 }$. This inequality is known as the Willmore conjecture.

Although the general case remains an open problem (as of 2000), the Willmore conjecture has been proved for various special classes of tori. For instance, it is known to be true for a torus embedded in $\mathbf{R} ^ { 3 }$ as a tube of constant circular cross-section (K. Shiohama and R. Takagi, 1970, and Willmore, 1971) as well as for tori of revolution (J. Langer and D. Singer, 1984). In 1982, P. Li and S.T. Yau showed that the Willmore conjecture is true for conformal structures near that of the special $\sqrt { 2 }$-torus. The set of conformal structures for which the Willmore conjecture is true was enlarged by S. Montiel and A. Ros (1985). Recently (2000), B. Ammann proved it under the condition that the $L ^ { p }$-norm of the Gaussian curvature is sufficiently small.

On the other hand, in 1978 J.L. Weiner generalized the Willmore functional by considering immersions of an orientable surface $\Sigma$, with or without boundary, into a Riemannian manifold of constant sectional curvature $c$. Instead of $W$ he considered the integral

\begin{equation*} \int _ { \Sigma } ( | \mathcal{H} | ^ { 2 } + c ) d A, \end{equation*}

where $\mathcal{H}$ denotes the mean curvature vector field of the surface, and obtained the corresponding Euler equation. In the particular case when the ambient space is the unit sphere $S ^ { 3 } \subset \mathbf{R} ^ { 4 }$, the Euler equation becomes $\Delta H + 2 H ( H ^ { 2 } - K + 1 ) = 0$, so that every minimal surface in $S ^ { 3 }$ is a Willmore surface. An interesting consequence of Weiner's result is the proof that stereographic projections of compact minimal surfaces in $S ^ { 3 }$ produce Willmore surfaces in $\mathbf{R} ^ { 3 }$. For instance, the special $\sqrt { 2 }$-torus in $\mathbf{R} ^ { 3 }$ for which $W = 2 \pi ^ { 2 }$ corresponds to the stereographic projection of the Clifford torus, embedded as a minimal surface in $S ^ { 3 }$. Moreover, H.B. Lawson proved in 1970 that every compact, orientable surface can be minimally embedded in $S ^ { 3 }$; it follows from this that there are embedded Willmore surfaces in $\mathbf{R} ^ { 3 }$ of arbitrary genus (cf. also Genus of a surface). On the other hand, Weiner also considered questions of stability of Willmore surfaces by considering the second variation of the Willmore functional. In particular, he showed that the special minimizing $\sqrt { 2 }$-torus is stable.

Most of the known examples of embedded Willmore surfaces in $\mathbf{R} ^ { 3 }$ come from compact minimal surfaces in the unit sphere $S ^ { 3 } \subset \mathbf{R} ^ { 4 }$. In 1985, U. Pinkall found the first examples of compact embedded Willmore surfaces that are not stereographic projections of compact embedded minimal surfaces in $S ^ { 3 }$. Using results of Langer and Singer on elastic curves on $S ^ { 2 }$, he exhibited an infinite series of embedded Willmore tori in $\mathbf{R} ^ { 3 }$ that cannot be obtained by stereographic projection of minimal surfaces in $S ^ { 3 }$. All of these tori are, however, unstable critical points of $W$ and hence are not candidates for absolute minima.

Important contributions to the study of Willmore surfaces are made by R.L. Bryant, who classified all Willmore immersions of a topological sphere into $S ^ { 3 }$ and determined the critical values of the Willmore functional; Li and Yau, who introduced the concept of conformal volume; M. Gromov, who introduced the concept of visual volume, closely related to the conformal volume; and others.

Finally, it is worth pointing out the relation between the Willmore functional and quantum physics, including the theory of liquid membranes, two-dimensional gravity and string theory. For instance, in string theory the functional $W = \int H ^ { 2 } d A$ is known as the Polyakov extrinsic action and in membrane theory it is the Helfrich free energy.

References

[a1]	U. Pinkall, I. Sterling, "Willmore surfaces" Math. Intelligencer , 9 : 2 (1987) pp. 38–43
[a2]	T.J. Willmore, "A survey on Willmore immersions" , Geometry and Topology of Submanifolds, IV (Leuven, 1991) , World Sci. (1992) pp. 11–16
[a3]	T.J. Willmore, "Riemannian geometry" , Oxford Univ. Press (1993) pp. Chap. 7
[a4]	T.J. Willmore, "Total mean curvature squared of surfaces" , Geometry and Topology of Submanifolds, VIII (Brussels/Nordfjordeid, 1995) , World Sci. (1996) pp. 383–391