UDC 513.73

MATHEMATICS

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

AN A PRIORI ESTIMATE OF THE PRINCIPAL RADII OF CURVATURE OF A CLOSED CONVEX HYPERSURFACE DEPENDING ON ITS GAUSSIAN CURVATURE

Let \(F\) be a regular hypersurface in \((n+1)\)-dimensional Euclidean space, and let \(P\) be an arbitrary point on this hypersurface. Take the point \(P\) as the origin of a rectangular Cartesian coordinate system \(x_1, x_2,\ldots,x_n,z\), and the tangent hyperplane at the point \(P\) as the coordinate hyperplane \(z=0\). In these coordinates a neighborhood of the point \(P\) of the hypersurface can be given by the equation

\[ z=f(x_1,x_2,\ldots,x_n). \]

The eigenvalues \(k_i\) of the second differential \(\frac12 d^2 f\) at the point \(P\) are called the principal curvatures, and the reciprocal quantities \(R_i\) are the principal radii of curvature. The total, or Gaussian, curvature of a hypersurface is the product of the principal curvatures.

The purpose of the present note is to establish an a priori estimate for the principal radii of curvature of a closed convex hypersurface in terms of its Gaussian curvature, given as a function of the unit vector of the outward normal. For a two-dimensional surface in three-dimensional space such an estimate was obtained by Miranda \((^1)\) and by the author \((^2)\).

Theorem. Let \(F\) be a closed strictly convex hypersurface in \((n+1)\)-dimensional Euclidean space satisfying the equation

\[ R_1R_2\ldots R_n=\varphi(\nu)>0, \tag{1} \]

where \(R_1,R_2,\ldots,R_n\) are the principal radii of curvature of the hypersurface at the point with outward normal \(\nu\).

Then for the principal radii of curvature \(R_i\) the estimate holds

\[ R_i\leqslant \max_{X,\gamma} \varphi^{1/n} \left( n-1+\frac12\left(\frac{\varphi'}{\varphi}\right)^2-\frac{\varphi''}{\varphi} \right)^{1-1/n}, \]

where differentiation is performed along the arc of a great circle \(\gamma\) issuing from the point \(X\) on the spherical image \(\Omega\) of the hypersurface, and the maximum is taken over all \(X\) and \(\gamma\).

Proof. Consider the function \(w_\gamma(X)\), defined on the unit sphere \(\Omega\) by the equality

\[ w_\gamma(X)=\overline H+\overline H_{\gamma}^{\prime\prime}, \]

where \(\overline H\) is the support function of the hypersurface \(F\), considered on the unit sphere \(\Omega\), and differentiation is performed along the arc of the great circle \(\gamma\) issuing from the point \(X\). The function \(w_\gamma(X)\) is the radius of normal curvature of the projecting cylinder and lies between the largest and smallest radii of normal curvature of the hypersurface at the point with outward normal \(\nu(X)\). If the great circle \(\gamma\) has a principal direction, this function coincides with the corresponding principal radius of curvature. It follows from this that an estimate for the function \(w_\gamma(X)\) is at the same time an estimate for the principal radii of curvature.

Let the function \(w_\gamma(X)\) attain its maximum at a certain point \(X_0\) for some large circle \(\gamma_0\) passing through this point. Introduce a Cartesian coordinate system in space, taking the direction \(\nu(X_0)\) as the direction of the \(z\)-axis, and direct the axes \(x_1, x_2, \ldots, x_n\) parallel to the principal directions of the surface \(F\) at the point with exterior normal \(\nu(X_0)\), with the axis \(x_1\) directed parallel to the tangent to \(\gamma_0\).

The coordinate net \(x_1, x_2, \ldots, x_n\) on the plane \(z=1\) is projected from the origin (the center of the sphere \(\Omega\)) onto a certain curvilinear coordinate net on the sphere \(\Omega\). Define on the hemisphere \(\Omega^+(z>0)\) the function

\[ w(x_1,x_2,\ldots,x_n)=w_{\gamma_1}(x_1,\ldots,x_n), \]

where \(\gamma_1\) is a large circle along which \(x_i=\mathrm{const}\) \((i>1)\). The function \(w(x_1,\ldots,x_n)\) has the same maximum as \(w_\gamma(X)\), and this maximum is attained at the point \(x_1=x_2=\cdots=x_n=0\).

Denote by \(h(x_1,x_2,\ldots,x_n)\) the values of the support function of the hypersurface \(F\) on the hyperplane \(z=1\). The function \(w\) is expressed in terms of the function \(h\), and for it one obtains the value

\[ w=\frac{\partial^2 h}{\partial x_1^2}\, \frac{(1+x_1^2+\cdots+x_n^2)^{3/2}}{1+x_2^2+\cdots+x_n^2}. \]

The product of the principal radii of curvature is also expressed in terms of the function \(h\):

\[ R_1R_2\cdots R_n=(1+x_1^2+\cdots+x_n^2)^{n/2+1}\left|\partial^2h/\partial x_i\partial x_j\right|. \]

On the right-hand side of the equality there stands the determinant of the second derivatives of the function \(h\). We shall agree to denote differentiation with respect to \(x_i\) by the corresponding indices. Then

\[ w=h_{11}\frac{(1+x_1^2+\cdots+x_n^2)^{3/2}}{1+x_2^2+\cdots+x_n^2}, \tag{2} \]

\[ (1+x_1^2+\cdots+x_n^2)^{n/2+1}\left\|h_{ij}\right\|=\varphi. \tag{3} \]

With our choice of coordinate system, at the point \(X_0\), i.e. for \(x_1=x_2=\cdots=x_n=0\), we shall have

\[ h_{ii}=R_i,\qquad h_{ij}=0\quad \text{for } i\ne j. \]

Differentiating equality (2) at the point \(X_0\), where \(w\) attains a maximum, we obtain

\[ w_i=h_{11i}=0, \tag{4} \]

\[ w_{11}=(h_{11})_{11}+3R_1\leq 0,\qquad w_{ii}=(h_{11})_{ii}+R_1\leq 0,\quad i\ne 1. \tag{5} \]

Differentiating equality (3) at the point \(X_0\) with respect to \(x_1\), we shall have

\[ \sum_i h_{11}\cdots (h_{ii})_1\cdots h_{nn}=\varphi \]

or

\[ \varphi\sum_i \frac{(h_{ii})_1}{h_{ii}}=\varphi_1. \tag{6} \]

Differentiating equation (3) at the point \(X_0\) twice with respect to \(x_1\), we obtain

\[ (n+2)\varphi+\varphi\sum_i\frac{(h_{ii})_{11}}{h_{ii}} +\varphi\sum_{i\ne j}\frac{(h_{ii})_1}{h_{ii}}\frac{(h_{jj})_1}{h_{jj}} =\varphi_{11}. \tag{7} \]

Taking into account equality (6), we conclude that

\[ \sum_{i\ne j}\frac{(h_{ii})_1}{h_{ii}}\frac{(h_{jj})_1}{h_{jj}} \leq \frac{1}{2}\left(\frac{\varphi_1}{\varphi}\right)^2 . \tag{8} \]

By the property of the maximum of the function \(w\) at the point \(X_0\), from inequality (5) we obtain

\[ \frac{(h_{11})_{11}}{h_{11}}\leq -3,\qquad \frac{(h_{ii})_{11}}{h_{ii}}\leq -\frac{R_1}{R_i}\quad \text{for } i\ne 1. \tag{9} \]

From equality (7) and inequalities (8) and (9) we conclude that

\[ (n-1)\varphi-\varphi R_1\sum_{i>1}\frac{1}{R_i} +\frac{1}{2}\frac{\varphi_1^2}{\varphi}\geq \varphi_{11}. \]

Hence, noting that

\[ \sum_{i>1}\frac{1}{R_i}\geq (n-1)\left(\frac{1}{R_2}\cdots \frac{1}{R_n}\right)^{1/(n-1)} =(n-1)\left(\frac{R_1}{\varphi}\right)^{1/(n-1)}, \]

we finally obtain the required estimate

\[ R_1\leq \varphi^{1/n} \left(n-1+\frac{1}{2}\left(\frac{\varphi_1}{\varphi}\right)^2 -\frac{\varphi_{11}}{\varphi}\right)^{1-1/n}. \]

Differentiation with respect to \(x_1\) at the point \(X_0\) can be replaced by differentiation along the arc of the circle \(\gamma_0\).

The theorem is proved.

Institute of Low Temperatures
of the Academy of Sciences of the Ukrainian SSR

Received
12 V 1967

CITED LITERATURE

\(^{1}\) C. Miranda, Rend. Semin. mat. Roma, 4 (1939).
\(^{2}\) A. V. Pogorelov, Mat. Sb., 31 (73), No. 1 (1952).