부산대학교 도서관

For an immersed compact surface $N$ in ${\bf R}^3$, the Willmore functional is defined as $W(N)=\int\sb N\,H\sp 2\,dN$, where $H$ is the mean curvature function. The critical points of $W$ are called Willmore surfaces. Finding Willmore surfaces has proved to be a very fruitful field of research in recent years. The stereographic projections of minimal surfaces in $S\sp 3$ are examples of Willmore surfaces. For surfaces of revolution, R. Bryant and P. Griffiths showed that one can relate $W$ with a functional defined for closed curves [Amer. J. Math. {\bf 108} (1986), no.~3, 525--570; MR0844630 (88a:58044)]. To be precise, let $\gamma$ be a closed curve in the upper half-plane and $N\sb \gamma$ the surface obtained by revolving $\gamma$ about the $x$-axis; then $$ W(N\sb \gamma)=\int_{N\sb\gamma}\,H\sp 2\,dN={\pi\over 2}\int \sb \gamma\,k\sp 2\,ds, $$ where $k$ is the geodesic curvature of $\gamma$ with respect to the hyperbolic metric on the upper half-plane. Thus, if we define $F(\gamma)=\int\sb \gamma\,k\sp 2\,ds$, then the critical points for $F$ yield critical points for (constrained) $W$ (see also the paper by J. Langer\ and D. A. Singer\ [Bull. London Math. Soc. {\bf 16} (1984), no.~5, 531--534; MR0751827 (85k:53006)]). \par The higher-dimensional analogue of the Willmore functional is defined as $$ W(N)=\int\sb N\,(H\sp 2-\tau\sb e)\sp {n/2}\,dN, $$ where $N$ is an $n$-dimensional compact submanifold in a Riemannian or pseudo-Riemannian manifold, $H$ is the mean curvature function and $\tau\sb e$ is the difference between the scalar curvature of $N$ and a part of the scalar curvature of $P$ along $N$. This functional, called the Willmore-Chen functional, was proved by B. Y. Chen to be invariant under conformal changes of the metric in $P$ [Boll. Un. Mat. Ital. (4) {\bf 10} (1974), 380--385; MR0370436 (51 \#6663)]. \par The paper under review considers the Willmore-Chen functional and proves an analogous result to that of Bryant and Griffiths mentioned above. The main result can be described as follows. Let $G$ be an $m$-dimensional compact Lie group endowed with a bi-invariant metric $d\sigma\sp 2$. Let $(\Gamma, \omega)$ be a flat principal connection on a principal fibre bundle $P$ with base space $(M,h)$ and structure group $G$, where $\omega$ is the connection $1$-form. For any positive function $u$ on the Riemannian manifold $M$, define a metric on $P$ by $\overline{h}=\pi\sp\ast(u\sp {-2}h)+\omega\sp\ast(d\sigma\sp 2)$, where $\pi$ is the projection map from $P$ to $M$. Let $\gamma$ be any closed curve in $M$ and denote $\pi\sp{-1}(\gamma)$ by $N\sb\gamma$. The authors show that $$ W(N\sb\gamma)=\frac{{\rm Vol}\,G}{(m+1)^{m+1}}\,\int\sb\gamma\,(k^2)\sp {(m+1)/2}\,ds, $$ where $k$ is the geodesic curvature of $\gamma$ in $(M, u\sp{-2} h)$. The authors then define $F\sp m(\gamma)=\int\sb\gamma\,(k^2)\sp {(m+1)/2}\,ds$ and call the critical points of $F\sp m$ the $m$-generalized elasticae. By applying the principle of symmetric criticality (see the paper by R. S. Palais\ [Comm. Math. Phys. {\bf 69} (1979), no.~1, 19--30; MR0547524 (81c:58026)]), they conclude that $N\sb \gamma$ is a critical point of the Willmore-Chen functional (called the Willmore-Chen manifold) if and only if $\gamma$ is an $m$-generalized elastica. \par The paper also gives examples of generalized elasticae in lens spaces and surfaces of revolution. These in turn give rise to examples of Willmore-Chen manifolds in flat principal fibre bundles over these spaces.