Mathematics

Corresponding Member of the Academy of Sciences of the USSR A. V. POGORELOV

ON THE QUESTION OF THE REGULARITY OF A CONVEX SURFACE WITH A REGULAR METRIC IN EUCLIDEAN SPACE

The problem of the regularity of a convex surface with a regular metric in Euclidean space was essentially solved in the author’s paper (¹). Namely, the following theorem was proved.

If a convex surface has a regular (n times differentiable, \(n \geqslant 5\)) metric and positive Gaussian curvature, then the surface itself is regular (at least \(n - 1\) times differentiable). If, moreover, the metric of the surface is analytic, then the surface is analytic.

We say “essentially” solved, since it remained unclear what happens in the cases \(n = 3\) and \(n = 4\). For \(n = 2\) the answer is given by a theorem of A. D. Aleksandrov (²), according to which a convex surface with bounded positive specific curvature, in particular with a twice differentiable metric and positive Gaussian curvature, is smooth, i.e. at least once differentiable. At present, thanks to the works of Nirenberg (³) and Heinz (⁴), the regularity theorem can also be proved in the cases \(n = 3\) and \(n = 4\). We shall show this.

The proof of the theorem on the regularity of a convex surface with a regular metric contained in (¹) consists, in broad outline, of the following. Let \(\Phi\) be a convex surface with a regular metric and positive curvature, and let \(P\) be a point on this surface. To prove the regularity of the surface \(\Phi\) in a neighborhood of \(P\), introduce such a parametrization \(u, v\) in which the metric of the surface has the regularity specified in the hypothesis of the theorem. By a plane parallel to the tangent plane at the point \(P\), cut off from the surface \(\Phi\) a small cap \(\omega\), and denote by \(G_\omega\) the domain in the coordinate plane \(u, v\) corresponding to the cap.

Construct a sequence of closed analytic curves \(\gamma_k\) inside the domain \(G_\omega\), converging to the boundary of the domain, such that the geodesic curvature of each curve \(\gamma_k\) in the metric of the cap, given by the line element

\[ ds^2 = E\,du^2 + 2F\,du\,dv + G\,dv^2 \]

is strictly positive.

Next construct a sequence of analytic metrics

\[ ds_n^2 = E_n\,du^2 + 2F_n\,du\,dv + G_n\,dv^2 \]

in the domain \(G_\omega\), converging to the metric \(ds^2\) of the cap \(\omega\), in the class \(C^n\), i.e. such that the coefficients \(E_k, F_k, G_k\) of the line element \(ds_k^2\) converge to the coefficients \(E, F, G\) of the line element \(ds^2\) together with derivatives up to order \(n\). Without loss of generality one may assume that the geodesic curvature of the curve \(\gamma_k\) in the metric \(ds^k\) is strictly positive.

The metric \(ds_k^2\), considered in the domain \(G_{\omega k}\) bounded by the curve \(\gamma_n\), can be realized by an analytic cap \(\omega_k\) (¹). As \(k \to \infty\), the sequence ...

the sequence of caps \(\omega_k\) converges to the cap \(\bar\omega\), equal to \(\omega\) \((^1)\). Thus, in order to prove the regularity of \(\omega\) in a neighborhood of \(P\), it suffices to prove the regularity of \(\bar\omega\).

Heinz, in paper \((^4)\), proved the following theorem.

Let \(r(u,v)\) be a twice differentiable vector function, defined in a domain \(\Omega\) of the \(u,v\)-plane;

\[ ds^2=E\,du^2+2F\,du\,dv+G\,dv^2 \]

is the linear element of the surface \(\Phi: r=r(u,v)\). Suppose that the coefficients of the form \(ds^2\) are bounded together with their derivatives up to the third order and with the reciprocal of the discriminant of the form by a constant \(a\), that the Gaussian curvature of the surface \(\Phi\) is everywhere greater than \(b>0\), and that the integral mean curvature is bounded by a constant \(c<\infty\). Then, on the set of interior points of \(\Omega\) whose distance from its boundary is not less than \(\rho>0\), one can give an estimate for the moduli of the second derivatives of \(r(u,v)\) depending only on the quantities \(a,b,c,\rho\). Moreover, whatever positive number \(\nu<1\) may be chosen, for the Hölder constants of the second derivatives of \(r\) with exponent \(\nu\) an estimate can be given in terms of the same quantities and the number \(\nu\).

Applying this theorem to the sequence of caps \(\omega_k\) and passing to the limit as \(k\to\infty\), we conclude that the limiting cap \(\bar\omega\) must belong to the class \(C^{2+\nu}\), \(0<\nu<1\), i.e., the vector function \(r(u,v)\) defining it has second derivatives satisfying a Hölder condition with any exponent \(\nu,\ 0<\nu<1\).

The further regularity of the limiting cap \(\bar\omega\) is obtained by means of Nirenberg’s theorem on the character of the regularity of a twice differentiable solution of an elliptic equation with regular coefficients, applying it to Darboux’s equation. According to this theorem, from the twice differentiability of the surface \(\bar\omega\) and the \(n\)-fold differentiability of its metric it follows that the surface \(\bar\omega\) is differentiable at least \(n-1\) times, and the \((n-1)\)-st derivatives of the function \(r(u,v)\) defining the surface satisfy a Hölder condition with any exponent \(\lambda,\ 0<\lambda<1\).

Thus, if \(n\geq 3\), then a convex surface with a metric of class \(C^n\) and positive Gaussian curvature belongs at least to the class \(C^{\,n-1+\alpha}\) for any \(\alpha,\ 0<\alpha<1\).

An analogous result also holds for \(n=2\). It is obtained from another theorem of Heinz \((^5)\). Heinz proved that if a twice differentiable function \(z(x,y)\) in a domain \(\Omega\) of the \(xy\)-plane satisfies the condition

\[ 0<\alpha< z_{xx}z_{yy}-z_{xy}^2<\beta<\infty, \]

then on the set of interior points of \(\Omega\) whose distance from its boundary is not less than \(d>0\), the first derivatives of the function \(z(x,y)\) satisfy a Hölder condition with exponent \(\nu=\sqrt{\beta/\alpha}\) and with a constant depending only on \(\alpha,\beta,d,\) and

\[ \delta=d\max_{\Omega}(z_x^2+z_y^2)^{1/2}.' \]

Applying this theorem to the caps \(\omega_k\) converging to \(\bar\omega\), we conclude that in a neighborhood of the point \(P\) the first derivatives of the functions \(z_k(x,y)\) defining these surfaces uniformly satisfy a Hölder condition with any exponent \(\nu<1\). (For a small neighborhood, \(\beta/\alpha\) is arbitrarily close to one, in view of the continuity of the Gaussian curvature and the smoothness of the limiting surface.) Passing to the limit as \(k\to\infty\), we conclude that the surface \(\omega=\bar\omega\) also has this property.

Thus, we arrive at a complete solution of the question of the regularity of a convex surface with a regular metric in the following form.

Theorem. If a convex surface has a regular metric of class \(C^n\) (\(n \geq 2\)) and positive Gaussian curvature, then the surface itself belongs at least to the class \(C^{\,n-1+\alpha}\) for any \(\alpha\), \(0<\alpha<1\). If the metric of the surface is analytic, then the surface is analytic.

Physical-Technical Institute
of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Received
18 V 1961

References

\(^{1}\) A. V. Pogorelov, Bending of Convex Surfaces, 1951.
\(^{2}\) A. D. Aleksandrov, DAN, 36, No. 7 (1942).
\(^{3}\) L. Nirenberg, Comm. Pure and Appl. Math., 6, 337 (1953).
\(^{4}\) E. Heinz, Math. Zs., 74, 129 (1960).
\(^{5}\) E. Heinz, Math. Zs., 72, No. 2, 107 (1959).