MATHEMATICS

D. V. BEKLEMISHEV

ON STRONGLY MINIMAL SURFACES OF A RIEMANNIAN SPACE

(Presented by Academician P. S. Aleksandrov, 30 XII 1956)

1°. Various connections of two-dimensional minimal surfaces with complex-analytic surfaces were noted by Schwarz ((^1)), Eisenhart ((^2)), Borůvka ((^3)), and others. However, their results cannot be transferred to arbitrary minimal surfaces with a number of dimensions greater than two. In the present note a class of minimal surfaces is introduced which are as closely connected with complex-analytic surfaces of the corresponding number of dimensions as two-dimensional minimal surfaces are with complex-analytic surfaces of two real dimensions.

We shall call a surface of (2n) dimensions in an (N)-dimensional Riemannian space a strongly minimal surface if there exists for it a frame of the first order in which its second fundamental object (\Lambda^\xi_{pq}) ((^4)) and metric tensor (g_{pq}) satisfy the equalities

[
\begin{gathered}
\Lambda^\xi_{ij}=-\Lambda^\xi_{n+i\,n+j},\qquad
\Lambda^\xi_{n+i\,j}=\Lambda^\xi_{i\,n+j},\
g_{ij}=g_{n+i\,n+j},\qquad
g_{i\,n+j}=-g_{n+i\,j},
\end{gathered}
\tag{1}
]

[
i=1,\ldots,n,\qquad \xi=2n+1,\ldots,N,\qquad p=1,\ldots,2n.
]

2°. Theorem 1. In order that a (2n)-dimensional surface of an (N)-dimensional Riemannian space be strongly minimal, it is necessary that the equalities

[
A^{\xi_1\ldots \xi_{2k-1}}
\equiv
\Lambda^{p_1\xi_1}{[p_1}
\Lambda^{p_2\xi_2}
\cdots
\Lambda^{p_{2k-1}\xi_{2k-1}}{p}]
=0
\qquad (k=1,\ldots,n).
\tag{A}
]

hold.

For the proof we fix the normal vectors (e_\xi) of the frame and perform the following transformation of the tangent vectors of the frame:

[
E_j=\frac12 e_j+\frac12 e_{n+j},\qquad
\bar E_j=\frac{1}{2i}e_j-\frac{1}{2i}e_{n+j}
\qquad (j=1,\ldots,n).
]

In the transformed frame ((E_i,\bar E_i,e_\xi)), the equalities (1) take the form

[
\Lambda^\xi_{ij}=0,\qquad
\Lambda^\xi_{\bar i\bar j}=0,\qquad
g_{ij}=g_{\bar i\bar j}=0.
\tag{1'}
]

From ((1')) the fulfillment of condition (A) follows immediately.

Remarks. 1. Condition (A) contains the equality (\Lambda^{p\xi}_{p}=0). This means that strongly minimal surfaces are minimal.

1. The tensors (A^{\xi_1\cdots \xi_{2k-1}}) are tensors of mean curvatures in odd-dimensional directions.

2. Condition (A) is sufficient in the cases (n=1,\ N-2n=1). All two-dimensional minimal surfaces are strongly minimal.

3°. We pass to the clarification of the connections of strongly minimal surfaces with complex-analytic surfaces.

A Kähler manifold is a complex-analytic manifold on which a metric tensor is given satisfying the conditions
(g_{J\bar K}=\bar g_{KJ}), (Dg_{J\bar K}[dz^J d\bar z^K]=0) ((J=1,\ldots,N)), where (z^J) are local coordinates on the manifold.

We attach the Kähler manifold to a moving frame, the components of whose infinitesimal displacement are complex linear differential forms (\omega^J,\omega^J_K). The structural equations hold:
[
D\omega^J=[\omega^K\omega^J_K],\qquad
D\omega^J_K=[\omega^L_K\omega^J_L]+R^J_{KL\bar M}[\omega^L\omega^{\bar M}],
]
[
dg_{J\bar K}=g_{J\bar L}\omega^{\bar L}{\bar K}+g\omega^L_J.
]

We shall denote (\omega^{\bar J}=\bar\omega^J,\ \omega^{\bar J}_{\bar K}=\bar\omega^J_K).

If one writes also the complex-conjugate equations, then the system obtained may be regarded as the structural equations of a (2N)-dimensional Riemannian manifold written in complex-conjugate coordinates. In this case the motions of the frame determined by the forms (\omega^P,\omega^P_Q) ((P=1,\ldots,N,\bar1,\ldots,\bar N)) are restricted by the equalities
[
\omega^{\bar J}K=0,\qquad \omega^J=0.
]

If we free ourselves from this restriction, then we obtain a Riemannian manifold which we shall call a Kähler manifold deprived of complex structure.

A surface of a Kähler manifold, generally speaking not complex-analytic, may locally be given by the differential equations
[
\omega^J=\Lambda^J_k\pi^k+\Lambda^J_{\bar k}\pi^{\bar k}\quad
(k=1,\ldots,n;\ \bar k=\bar1,\ldots,\bar n),
]
where (\pi^k,\ \pi^{\bar k}=\overline{\pi^k}) are invariant forms of the complex-analytic transformation group of the parameters.

The quantities (\Lambda^J_{\bar k}) form the components of a differential-geometric object in the Kähler manifold, which we shall call the object of analyticity. The equality (\Lambda^J_{\bar k}=0) characterizes complex-analytic surfaces.

Theorem 2. Surfaces of a Kähler manifold along which the object of analyticity is covariantly constant are strongly minimal surfaces in the Kähler manifold deprived of complex structure.

Corollary. Complex-analytic surfaces of a Kähler manifold are strongly minimal in the Kähler manifold deprived of complex structure.

The proof of the theorem is based on the following assertion: if on a surface of a Riemannian space there exists a frame of zero order in which the second fundamental object and the metric tensor of the surface satisfy the relations
[
\Lambda^p_{ij}=-\Lambda^p_{n+i\,n+j},\qquad
\Lambda^p_{i\,n+j}=\Lambda^p_{n+i\,j},\qquad
g_{ij}=g_{n+i\,n+j},\qquad
g_{i\,n+j}=-g_{n+i\,j},
]
then such a surface is strongly minimal.

Remark. Since here everywhere only the local structure of a Kähler manifold is in question, in all arguments it may be ...

replace by Shirokov’s (A)-space ((^5)) or, equivalently, by a pseudoholomorphic space (see, for example, ((^6))).

(4^\circ). Let an (N)-dimensional unitary space (U_N) be given, i.e., a complex linear vector space with a real scalar product, and let (E_J) be a basis in (U_N). By the real plane of the space (U_N) for the given basis we shall mean the set of real linear combinations of the vectors (E_J); denote it by (R_N(E_J)). The mapping

[
z^J E_J \to \frac{z^J+\bar z^J}{2}E_J
]

will be called the projection of the vector (z^J E_J) onto (R_N(E_J)).

A complex-analytic surface of a unitary space is given by the equation (z^J=F^J(p^i)) ((i=1,\ldots,n)), where (F^J) are analytic functions of the complex parameters (p^i). In what follows we shall everywhere assume that (2n