MATHEMATICS

N. V. EFIMOV and E. G. POZNYAK

A GENERALIZATION OF HILBERT’S THEOREM ON SURFACES OF CONSTANT NEGATIVE CURVATURE

(Presented by Academician P. S. Aleksandrov on 26 X 1960)

No. 1. Let \(\Lambda\) be a two-dimensional simply connected manifold on which coordinates \((u,v)\) and a Gaussian metric \(ds^2\) are given, with coefficients \(E,F,G\in C^4(u,v)\). We say that \(\Lambda\) is regularly immersed in three-dimensional Euclidean space \(E_3\) if there are three functions \(L,M,N\in C^2(u,v)\) which, as the coefficients of the second quadratic form of the surface, satisfy throughout \(\Lambda\) the fundamental equations of the theory of surfaces with the given \(ds^2\). Denote by \(K\) the Gaussian curvature of the element \(ds^2\); assuming \(K<0\), put \(k^2=-K\). Suppose that on \(\Lambda\) the inequalities

\[ k^2\geqslant 1,\qquad \left|\frac{k'}{k}\right|\leqslant \frac{1}{\Delta},\qquad \frac{1}{k^{3/2}}\left(\frac{1}{k^{1/2}}\right)''_{g}\leqslant 1-h, \tag{1} \]

hold, where \(k'\) is the derivative in an arbitrary direction; the second derivative is taken along the arc of an arbitrary geodesic; \(h\) and \(\Delta\) are positive constants.

Theorem A. If

\[ \frac{1}{\Delta^2}<\frac{4}{3}\,h \tag{2} \]

and if \(\Lambda\) with the given \(ds^2\) is a complete metric space, then \(\Lambda\) admits no regular immersion in \(E_3\).

Example.

\[ ds^2=du^2+\left(\operatorname{ch}\sqrt{\frac{3}{2}}\,u+a\sin u\right)^2dv^2; \]

for sufficiently small \(a\) all the conditions of Theorem A are fulfilled.

Theorem B. In \(E_3\) there is no complete regular surface whose Gaussian curvature is negative and satisfies conditions (1), (2).

Theorem B follows from Theorem A. Obviously, Theorem B includes, as a special case, the well-known Hilbert theorem on surfaces of constant negative curvature.

No. 2. To prove Theorem A by contradiction, suppose that \(\Lambda\) is regularly immersed in \(E_3\); then on \(\Lambda\) a locally regular asymptotic net is defined. Let \((u,v)\) be local asymptotic coordinates (we regard the family \(v=\mathrm{const}\) as the first); let \(s_1,s_2\) be the natural parameters of the asymptotic curves; and let \(\omega\) be the angle between the asymptotics, \(0<\omega<\pi\).

Lemma 1. The set of values of \(\displaystyle \int_L \sin\omega\,ds\), over the totality of all asymptotic arcs \(L\), cannot be bounded.

The proof is based on the formulas:

\[ \frac{\partial\ln(ek)}{\partial s_2} = \sin\omega\,\frac{\partial Q}{\partial s_1^{*}}, \qquad \frac{\partial\ln(gk)}{\partial s_1} = -\sin\omega\,\frac{\partial Q}{\partial s_2^{*}}, \tag{3} \]

\[ \frac{1}{\rho_1} = -\frac{\partial\omega}{\partial s_1} + \sin\omega\,\frac{\partial Q}{\partial s_2}, \qquad \frac{1}{\rho_2} = \frac{\partial\omega}{\partial s_2} - \sin\omega\,\frac{\partial Q}{\partial s_1}, \tag{4} \]

where \(e^{2}=E,\ g^{2}=G\) in asymptotic coordinates; \(Q=\frac12\ln k\); \(\partial/\partial s_1^{*}\), \(\partial/\partial s_2\) are symbols of differentiation in directions orthogonal to the asymptotics; \(1/\rho_1,\ 1/\rho_2\) are the geodesic curvatures of the asymptotics.

From relations (3), (4) and from the assumption
\[ \int_L \sin\omega\,ds \leq C=\mathrm{const} \]
it follows that each asymptotic of one family intersects every asymptotic of the other, and then a contradiction is obtained with the Gauss–Bonnet formula.

No. 3. There is a proposition, stated in general outline by N. V. Efimov in the note \((^{1})\), which is formulated in detail as follows.

Let
\[ 0<\delta<\frac{2(4h\Delta^{2}-3)}{3(8h\Delta^{2}-3)},\qquad M=\frac{4}{3}\left(\frac{2}{3\delta}-1\right)\Delta . \tag{5} \]

If on some asymptotic there is an arc \(L\) for which
\[ \int_L \sin\omega\,ds > M, \]
then for such an asymptotic, for some \(\varepsilon>0\), an \(\varepsilon\)-strip is impossible, \(\varepsilon=\varepsilon(h,\Delta,\delta)\), subject to conditions (1), (2).

Hence, and from Lemma 1, Theorem A follows.

No. 4. We next present the main stages of the proof of assertion No. 3. Put
\[ U=k^{-3/2}\left(\frac{\partial\omega}{\partial s_1}-\sin\omega\,\frac{\partial Q}{\partial s_2}\right). \]

Lemma 2. If
\[ \int_L \sin\omega\,ds > M \]
on some arc \(L\) of the second family of asymptotics, then on \(L\) there is a point where
\[ |U|>(2/3-\delta)h\Delta . \]

Proof. This is obtained by contradiction with the aid of equation (1) of our note \((^{2})\), as a consequence of which
\[ \frac{\partial U}{\partial s_2}\geq \left\{h-\frac{3}{2\Delta}|U|\right\}\sin\omega . \]

Lemma 3. If on an asymptotic of the second family, somewhere
\[ |U|>(2/3-\delta)h\Delta, \]
then in some \(\varepsilon\)-strip of such an asymptotic there is a point where
\[ |U|>\frac{3}{2}h\Delta;\qquad \varepsilon=\varepsilon(h,\Delta,\delta). \]

Proof. Suppose the condition of the lemma is satisfied at some point \(O\) and, consequently, on some segment \(OP\) of the first asymptotic. Through all points of \(OP\) draw asymptotics of the second family in the positive direction of this family, assuming \(U>0\) on \(OP\) (in the opposite case we change the orientations of all asymptotics; for the orientation conditions see the note \((^{2})\)). Denote by \(\Omega\) the region occupied by the drawn asymptotics. Suppose \(U\leq \frac{2}{3}h\Delta\) everywhere in \(\Omega\). We may assume \(ek=1\) on \(OP\). From equation (III) of the note \((^{2})\) we have
\[ \frac{\partial}{\partial s_2}\{(ek)^3U\}\geq h(ek)^3\sin\omega; \tag{6} \]
as a consequence of (3),
\[ \frac{\partial}{\partial s_2}\{(ek)^3U\}\geq \frac{2}{3}h\Delta\,\frac{\partial(ek)^3}{\partial s_2}. \tag{7} \]

Integrating (6) and (7), we find:
\[ (ek)^3>\frac{2-3\delta}{2}>0; \tag{8} \]
\[ k^{-3/2}(ek)^3\frac{\partial\omega}{\partial s_1} >(ek)^3\left[\frac{2}{3}h\Delta-\frac{1}{2\Delta}\right]-\delta h\Delta . \tag{9} \]

From (9), taking into account (8) and (5), we have everywhere in \(\Omega\)

\[ k^{-3/2}\frac{\partial\omega}{\partial s_1}\geq N_1=\mathrm{const}>0. \tag{10} \]

Let \(OT\) be an asymptotic of the second family, going in \(\Omega\) from \(O\); because \(0<\omega<\pi\), and because of (10), every asymptotic of the first family that goes into the domain \(\Omega\) from any point of \(OT\) intersects every asymptotic of the second family passing in \(\Omega\). Therefore all points of \(\Omega\) can be uniquely determined by coordinates \((u,v)\), where \(u\) and \(v\) are single-valued coordinates on the segment \(OP\) and on the ray \(OT\); in particular, \(\omega=\omega(u,v)\).

From (10),

\[ \frac{\partial\omega}{\partial u}\geq (ek)N_1. \tag{11} \]

Let \(O_1P_1\) \((u_0\leq u\leq u_1)\) be some segment lying strictly inside \(OP\); let \(O_1T_1\) be an asymptotic of the second family going in \(\Omega\) from \(O_1\); and let \(\Omega_1\) be the domain \((u_0\leq u\leq u_1,\ v\geq 0)\). From (11) and (8) we have in \(\Omega_1\)

\[ \sin\omega\geq m=\mathrm{const}>0. \tag{12} \]

If \(s_1\) is the length of the arc cut off by \(\Omega_1\) on some asymptotic of the first family, then

\[ s_1=\int_{u_0}^{u_1} e(u,v)\,du<\frac{\pi}{N_1}. \tag{13} \]

We may assume that \(gk^{1/2}=1\) on \(O_1T_1\); then from (3) and (13) we have in \(\Omega_1\)

\[ e^{-\pi/\Delta N_1}\leq gk^{1/2}\leq e^{+\pi/\Delta N_1}. \tag{14} \]

Let \(E_v^*(u)\) be the set of points \(u\in [u_0,u_1]\) at which, for the given \(v\geq 0\), one has
\(e(u,v)k(u,v)\geq \alpha/N_1\) \((\alpha=\mathrm{const}>0)\). From (11) we find \(\operatorname{mes} E_v^*(u)\leq \pi/\alpha\). Consequently, for any \(v\geq 0\), taking \(\alpha\) sufficiently large, we have

\[ \sqrt[8]{\frac{2-3\delta}{2}}<e(u,v)k(u,v)<\frac{\alpha}{N_1}; \]

\[ u\in E_v(u)=[u_0,u_1]-E_v^*(u),\qquad \operatorname{mes} E_v(u)\geq (u_1-u_0)-\frac{\pi}{\alpha}. \]

From equation (III) of note \((^2)\), taking (12) into account,

\[ \frac{\partial}{\partial s_2}\{(ek)^3U\}\geq hm(ek)^3k^{1/2}. \]

Hence, if \(u\in E_v(u)\), then

\[ \left|\,k^{-3/2}(ek)\sin\omega\,\frac{\partial Q}{\partial s_2}\,\right| \leq \frac{\alpha}{N_1}\frac{1}{2\Delta}=R=\mathrm{const}, \]

\[ \frac{\partial\omega}{\partial u}\geq hm\left(\frac{N_1}{\alpha}\right)^2 \int_0^{s_2}(ek)^3k^{1/2}\,ds_2-R. \]

Taking (14) and (8) into account, we find

\[ \int_0^{s_2}(ek)^3k^{1/2}\,ds_2 = \int_0^v (ek)^3gk^{1/2}\,dv \geq e^{-\pi/\Delta N_1}\frac{2-3\delta}{2}\,v. \]

If \(\alpha=2\pi/(u_1-u_0)\), then \(\operatorname{mes} E_v(u)\geq \tfrac12(u_1-u_0)\), and \(\partial\omega/\partial u\geq \lambda v-R\), \(\lambda=\mathrm{const}>0\), \(u\in E_v(u)\). Consequently, for sufficiently large \(v\) we shall have \(\partial\omega/\partial u\geq 2\pi/(u_1-u_0)\), \(u\in E_v(u)\), which contradicts the condition \(0<\omega<\pi\). Thus the assumption \(U\leq \tfrac23 h\Delta\) must be rejected.

Lemma 4. If on an asymptotic of the second family, somewhere
\(|U| > {}^{2}/_{3} h\Delta\), then for such an asymptotic, for some \(\varepsilon > 0\), an \(\varepsilon\)-strip satisfying conditions (1), (2) is impossible; \(\varepsilon = \varepsilon(h,\Delta)\).

Proof. We retain the preceding notation and suppose that
\(U > ({}^{2}/_{3}+\delta)h\Delta\) on \(OP\); here it is important only that \(\delta=\mathrm{const}>0\). Analogously to (9) we find

\[ k^{-3/2}(ek)^3 \frac{\partial \omega}{\partial s_1} > (ek)^3\left[\frac{2}{3}h\Delta-\frac{1}{2\Delta}\right] +\delta h\Delta . \tag{15} \]

Hence \(\partial\omega/\partial s_1 \geqslant N_1=\mathrm{const}>0\), and the conclusions drawn in the preceding proof from (10) are preserved. Then from (15) we have

\[ \frac{\partial\omega}{\partial u} \geqslant (ek)N_1+\frac{1}{(ek)^2}N_2 \qquad (N_1,N_2=\mathrm{const}>0). \tag{16} \]

From (16) follow (13) and (14). From (16) it also follows that

\[ \sqrt{\frac{N_2}{\alpha}}< e(u,v)k(u,v)<\frac{\alpha}{N_1}, \]

\(u\in E_v(u)\), where \(E_v(u)\) is a certain subset of the segment \([u_0,u_1]\), whose measure is
\(\geqslant (u_1-u_0)-2\pi/\alpha\) (\(\alpha\) is a sufficiently large number \(>0\)). After this the proof is completed, in general, analogously to the proof of Lemma 3 (but with the use of Fubini’s theorem for the rectangle \(u_0\leqslant u\leqslant u_1,\ 0\leqslant v\leqslant \mathrm{const}\)).

Moscow State University
named after M. V. Lomonosov

Received
25 X 1960

REFERENCES

1. N. V. Efimov, DAN, 136, No. 6 (1961).
2. N. V. Efimov, E. G. Poznyak, DAN, 137, No. 1 (1961).