View source for Willmore functional

The Willmore functional of an immersed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301301.png" /> into the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301302.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301303.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301304.png" /> is the [[Mean curvature|mean curvature]] of the surface. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301306.png" /> are the two classical principal curvatures of the surface (cf. also [[Principal curvature|Principal curvature]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301307.png" /> is the area element of the induced metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301308.png" />. Moreover, it is assumed that the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w1301309.png" /> converges, which is guaranteed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013010.png" /> is compact, as is usually assumed. Critical points of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013011.png" /> are called Willmore surfaces and are characterized by the Euler equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013012.png" /> corresponding to the variational problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013013.png" /> (cf. also [[Variation of a functional|Variation of a functional]]; [[Variational calculus|Variational calculus]]; [[Variational problem|Variational problem]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013014.png" /> is the [[Gaussian curvature|Gaussian curvature]] of the surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013015.png" /> is its Laplace–Beltrami operator (cf. [[Laplace–Beltrami equation|Laplace–Beltrami equation]]). An alternative functional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013016.png" /> is the functional given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013017.png" /></td> </tr></table>

Because of the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013018.png" /> is assumed to be compact and without boundary, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013020.png" /> denotes the [[Euler characteristic|Euler characteristic]] of the surface, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013022.png" /> have the same critical points.

The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013023.png" /> was first studied by W. Blaschke (1929) and G. Thomsen (1923), who established the most important property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013024.png" />: The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013025.png" /> is invariant under conformal changes of metric of the ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013026.png" />. 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013027.png" /> immersed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013028.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013029.png" />, equality holding if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013031.png" /> about an axis in its plane at distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013032.png" /> from its centre, it holds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013033.png" />, equality holding when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013034.png" />. For various reasons, Willmore also conjectured that any torus immersed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013035.png" /> satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013036.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013039.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013040.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013041.png" />, with or without boundary, into a [[Riemannian manifold|Riemannian manifold]] of constant sectional curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013042.png" />. Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013043.png" /> he considered the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013045.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013046.png" />, the Euler equation becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013047.png" />, so that every minimal surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013048.png" /> is a Willmore surface. An interesting consequence of Weiner's result is the proof that stereographic projections of compact minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013049.png" /> produce Willmore surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013050.png" />. For instance, the special <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013051.png" />-torus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013052.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013053.png" /> corresponds to the stereographic projection of the Clifford torus, embedded as a minimal surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013054.png" />. Moreover, H.B. Lawson proved in 1970 that every compact, orientable surface can be minimally embedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013055.png" />; it follows from this that there are embedded Willmore surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013056.png" /> of arbitrary genus (cf. also [[Genus of a surface|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013057.png" />-torus is stable.

Most of the known examples of embedded Willmore surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013058.png" /> come from compact minimal surfaces in the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013059.png" />. In 1985, U. Pinkall found the first examples of compact embedded Willmore surfaces that are not stereographic projections of compact embedded minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013060.png" />. Using results of Langer and Singer on elastic curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013061.png" />, he exhibited an infinite series of embedded Willmore tori in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013062.png" /> that cannot be obtained by stereographic projection of minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013063.png" />. All of these tori are, however, unstable critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013064.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013065.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130130/w13013066.png" /> is known as the Polyakov extrinsic action and in membrane theory it is the Helfrich free energy.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Pinkall,   I. Sterling,   "Willmore surfaces"  ''Math. Intelligencer'' , '''9''' :  2  (1987)  pp. 38–43</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Willmore,   "A survey on Willmore immersions" , ''Geometry and Topology of Submanifolds, IV (Leuven, 1991)'' , World Sci.  (1992)  pp. 11–16</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.J. Willmore,   "Riemannian geometry" , Oxford Univ. Press  (1993)  pp. Chap. 7</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.J. Willmore,   "Total mean curvature squared of surfaces" , ''Geometry and Topology of Submanifolds, VIII (Brussels/Nordfjordeid, 1995)'' , World Sci.  (1996)  pp. 383–391</TD></TR></table>