MATHEMATICS

Yu. G. Reshetnyak

ON CONFORMAL MAPPINGS OF SPACE

(Presented by Academician M. A. Lavrent'ev on 16 X 1959)

Liouville’s theorem on conformal mappings of space is usually proved for mappings of class \(C^3\) (see, for example, \((^{1,2})\)). From the results of M. A. Lavrent'ev \((^3)\) it follows that it is valid in the class \(C^{1,\alpha}\), \(0<\alpha<1\). In the present paper a proof is given that is free of any assumptions concerning differentiability of the mapping.

In what follows: \(E^n\) is \(n\)-dimensional Euclidean space; \(|\mathbf{x}|\) is the norm of the vector \(\mathbf{x}\in E^n\); \(Q(\mathbf{x}_0,r)\) is the ball \(\{|\mathbf{x}-\mathbf{x}_0|<r\}\); \(S(\mathbf{x}_0,r)\) is the sphere \(\{|\mathbf{x}-\mathbf{x}_0|=r\}\); \(W_p^l(M)\), where \(M\subset E^n\) is an open set, is the set of all functions defined in \(M\) having generalized partial derivatives of order \(l\) in the sense of S. L. Sobolev \((^4)\), summable with power \(p\) on every compact set \(F\subset M\).

For an arbitrary measurable function \(\lambda(x)\geq 0\), put

\[ F_\lambda(\mathbf{x},r)=\int_{S(\mathbf{x},r)}[\lambda(\mathbf{x})]^{(n-1)/2}\,d\sigma, \]

where \(d\sigma\) is the element of area of the sphere \(S(\mathbf{x},r)\),

\[ V_\lambda(\mathbf{x},r)=\iint_{Q(\mathbf{x},r)}[\lambda(\mathbf{x})]^{n/2}\,d\mathbf{x}, \qquad d\mathbf{x}=dx_1\,dx_2\ldots dx_n . \]

If \(\mathbf{x}\in M\subset E^n\), then let \(r_0(\mathbf{x},M)\) be the distance from \(\mathbf{x}\) to the boundary of \(M\). Let \(\mathbf{x}\) be an arbitrary point in \(E^n\). \(U_t(\mathbf{x})\), where \(t\) is a real number, \(0<t<1\), is a family of neighborhoods of the point \(\mathbf{x}\). The family \(U_t(\mathbf{x})\) is called regular if: a) \(U_t(\mathbf{x})\subset U_1(\mathbf{x})\) for all \(t\), and there exists a topological mapping \(\varphi(x)\) of \(U_1(\mathbf{x})\) into \(E^n\) such that \(\varphi[U_t(\mathbf{x})]=Q(0,t)\), \(0<t<1\); b) if \(R(t)\) and \(r(t)\) are the least and the greatest, respectively, of the distances from the point \(\mathbf{x}\) to the points of the boundary of \(U_t(\mathbf{x})\), then as \(t\to 0\), \(R(t)/r(t)\to 1\).

Let \(M\subset E^n\) be an open set. A mapping \(\mathbf{y}(\mathbf{x})\) of \(M\) into \(E^n\) is called conformal if: a) \(\mathbf{y}(\mathbf{x})\) is a topological mapping; b) for every point \(\mathbf{x}\in M\) there is a regular family of neighborhoods which, under the mapping \(\mathbf{y}(\mathbf{x})\), passes into a regular family of neighborhoods of the point \(\mathbf{y}(\mathbf{x})\).

Theorem. Every conformal mapping in \(E^n\) for \(n\geq 3\) is a combination of a finite number of inversions.

To prove the theorem it suffices to establish the analyticity of a conformal mapping. The proof is based on Lemmas 1–7. Fix some domain \(M\) and a conformal mapping \(\mathbf{y}(\mathbf{x})\) of \(M\) into \(E^n\).

Lemma 1. The vector-function \(\mathbf{y}(\mathbf{x})\in W_n^1(M)\). Its partial derivatives \(\partial\mathbf{y}/\partial x_i\), for almost all \(\mathbf{x}\in M\), are mutually orthogonal and have equal lengths.

Set \(\lambda(x)=(\partial y/\partial x_j)^2\).

Lemma 2. For every measurable set \(E\subset M\), the set \(y(E)\) is measurable and its Lebesgue measure is equal to

\[ \iint_E [\lambda(x)]^{n/2}\,dx,\qquad dx=dx_1\cdots dx_n . \]

Lemma 3. For every point \(x\in M\), for almost all \(r\), \(F_\lambda(x,r)\) is equal to the area of the surface \(y[S(x,r)]\) (area is understood in the sense of Lebesgue).

Applying the isoperimetric inequality in \(E^n\), we obtain the following lemma:

Lemma 4. For each point \(x\in M\), for all \(r\in[0,r_0(x)]\), the inequality

\[ [F_\lambda(x,r)]^n \ge n^{\,n-1}\omega_{n-1}[V_\lambda(x,r)]^{n-1}, \]

holds, where \(\omega_{n-1}\) is the area of the unit sphere in \(E^n\).

Let \(\alpha_h(Z)\) be the Sobolev averaging kernel \({}^{(4)}\). Let \(M_h\) be the set of all \(x\in M\) for which \(r_0(x,M)\ge h\). Put

\[ \lambda_h(x)=\left\{\iint_{|z|\le h} [\lambda(x+z)]^{n/2}\alpha_h(z)\,dz\right\}^{2/n}. \]

The function \(\lambda_h(x)\) is defined and positive in \(M_h\), and there has all derivatives of arbitrary order.

Lemma 5. For every point \(x\in M\), if \(h<r_0(x)\), for all \(r\in[0,r_0(x)-h]\) the isoperimetric inequality

\[ [F_{\lambda_h}(x,r)]^n-n^{\,n-1}\omega_{n-1}[V_{\lambda_h}(x,r)]^{n-1}\ge 0 \]

is satisfied.

Lemma 6. Let \(\lambda(x)>0\) be a function defined in \(M\) and having continuous derivatives there up to and including the second order. Then for all \(x\in M\)

\[ \lim_{r\to0} \frac{[F_\lambda(x,r)]^n-n^{\,n-1}\omega_{n-1}[V_\lambda(x,r)]^{n-1}} {r^2[F_\lambda(x,r)]^n} = \frac{2n-2}{n^2-4}\, \frac{\Delta[\lambda(x)]^{(n-2)/4}}{[\lambda(x)]^{(n-2)/4}}, \]

where \(\Delta\) is the Laplace operator, \(n>2\).

From Lemmas 5 and 6 it follows that, for every point \(x\in M\), when \(h<r_0(x)\),
\(\Delta[\lambda_h(x)]^{(n-2)/4}\ge0\), i.e., the function \([\lambda_h(x)]^{(n-2)/4}\) is subharmonic. Hence, passing to the limit as \(h\to0\), we obtain that the function \([\lambda(x)]^{(n-2)/4}\) is subharmonic. Thus \(\lambda(x)\) is bounded above on every compact set \(A\subset M\). Taking into account that the mapping inverse to \(y(x)\) is also conformal, we obtain that the function \(1/\lambda(x)\) is bounded on every compact set \(A\subset M\). Therefore, for every compact set \(A\subset M\), there exist constants \(\lambda_0\) and \(\lambda_1\) such that

\[ 0<\lambda_0\le \lambda(x)\le \lambda_1<\infty \]

for all \(x\in A\).

Lemma 7. If a subharmonic function \(u(x)\), defined in \(M\), is bounded on every compact set \(A\subset M\), then \(u(x)\in W^1_2(M)\).

It follows from Lemma 7 that \(\lambda(x)\in W^1_2(M)\).

We take the components \(y_1,y_2,\ldots,y_n\) of the vector \(y(x)\) as coordinates in \(M\). The function \(y_k\), \(k=1,2,\ldots,n\), satisfies Laplace’s equation in the coordinates \((y_1,\ldots,y_n)\) and, consequently, minimizes the Dirichlet integral for the given boundary values. In the coordinates \((x_1,\ldots,x_n)\) this integral

is equal to

\[ \iint_{Q(x_0,r)} \sum_{i=1}^{n} \left( \frac{\partial u}{\partial x_i} \right)^2 [\lambda(x)]^{n/2-1}\, dx . \tag{1} \]

Thus, \(y_k(x)\) minimizes the integral (1) for the given boundary values. Hence, \(y_k(x)\) is a generalized solution of equation (5)

\[ \sum_{i=1}^{n} \frac{\partial}{\partial x_i} \left\{ [\lambda(x)]^{(n-1)/2} \frac{\partial y_k}{\partial x_i} \right\} = 0 . \]

Using the already established properties of the functions \(\lambda(x)\) and \(y_k(x)\), \(k=1,2,\ldots,n\), it is not difficult from this first to establish that \(y_k(x)\in W_2^2(M)\), and then, by the usual methods \({}^{5}\) of the theory of partial differential equations, to prove the analyticity of \(y_k(x)\).

Received
9 X 1959

REFERENCES

\({}^{1}\) V. F. Kagan, Foundations of the Theory of Surfaces, 2, Moscow, 1948.
\({}^{2}\) I. A. Schouten, D. J. Struik, Introduction to the New Methods of Differential Geometry, 2, Moscow, 1948.
\({}^{3}\) M. A. Lavrent’ev, DAN, 95, No. 5, 925 (1954).
\({}^{4}\) S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics, Leningrad, 1950.
\({}^{5}\) O. A. Ladyzhenskaya, Uspekhi Mat. Nauk, 13, issue 6 (1956).