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

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

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:
\[ D\omega^J=[\omega^K\omega^J_K],\qquad D\omega^J_K=[\omega^L_K\omega^J_L]+R^J_{KL\bar M}[\omega^L\omega^{\bar M}], \]
\[ dg_{J\bar K}=g_{J\bar L}\omega^{\bar L}_{\bar K}+g_{L\bar K}\omega^L_J. \]

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
\[ \omega^{\bar J}_K=0,\qquad \omega^J_{\bar K}=0. \]

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 \(2n<N\).

Theorem 3. A complex-analytic surface of a unitary space is projected onto the real plane \(R_N(E_J)\) as a strongly minimal surface for some basis \(E_J\).

Analogously to projection onto \(R_N(E_J)\), one can construct a projection onto the imaginary plane \(I_N(E_J)=R_N(iE_J)\). The projections of an analytic surface onto \(R_N(E_J)\) and \(I_N(E_J)\) will be called conjugate strongly minimal surfaces. Mapping \(I_N(E_J)\) onto \(R_N(E_J)\) by the formulas \(y^J(iE_J)\to y^J E_J\), we obtain in \(R_N(E_J)\) a pair of conjugate strongly minimal surfaces. These surfaces are superposable; at corresponding points their tangent planes are parallel, as are their osculating planes of any order. Here there is a complete analogy with conjugate two-dimensional minimal surfaces in three-dimensional Euclidean space.

\(5^\circ\). We shall regard Euclidean space \(\mathcal E_N\) as the real plane \(R_N(E_J)\) in the unitary space \(U_N\).

Theorem 4. If in \(\mathcal E_N=R_N(E_J)\) a strongly minimal surface \(S^*\) is given, then in \(U_N\), with an arbitrary defined by constants, an analytic surface \(S\) can be constructed which is projected onto \(S^*\).

For minimal surfaces in \(\mathcal E_3\), this assertion is given by the Schwarz formulas.

In conclusion I express my deep gratitude to Prof. G. F. Laptev for his guidance and assistance in the work.

Moscow State University
named after M. V. Lomonosov

Received
30 XI 1956

REFERENCES

\(^1\) H. Schwarz, Gesamm. Math. Abhandl., 1, Berlin, 1890, S. 179.
\(^2\) L. P. Eisenhart, Am. J. Math., 34, 215 (1912).
\(^3\) O. Borůvka, Publ. Fac. Sci. Univ. Masaryk, 214, 1 (1935).
\(^4\) G. F. Laptev, Tr. Mosk. matem. obshch., 2 (1953).
\(^5\) P. A. Shirokov, Izv. Kazansk. fiz.-matem. obshch., 25, 86 (1925).
\(^6\) A. Lichnerowicz, Théorie globale des connexions et des groupes d’holonomie, Paris, 1955, p. 227, 251.