Abstract
Full Text
UDC 517.54
MATHEMATICS
O. V. TITOV
ON QUASICONFORMAL HARMONIC MAPPINGS OF EUCLIDEAN SPACE
(Presented by Academician M. A. Lavrent'ev on 13 III 1970)
Let \(f:\mathbf R^n\to\mathbf R^n\) be a \(C^\infty\) mapping of \(n\)-dimensional Euclidean space \(\mathbf R^n\) into itself, given by \(n\) coordinate functions \(f_1(x),\ldots, f_n(x)\), where \(x=(x_1,\ldots,x_n)\). The mapping \(f\) is called quasiconformal if the inequality
\[ \left(\sum_{i,j=1}^{n}\left|\frac{\partial f_i}{\partial x_j}\right|^2\right)^{1/2}\le C\,|J_f|^{1/n}, \tag{1} \]
holds everywhere, where \(C\) is some constant, and \(J_f\) is the Jacobian of the mapping \(f\).
Combining two theorems of Liouville, one can prove the following assertion: any nonconstant quasiconformal mapping \(f:\mathbf R^{2n}\to\mathbf R^{2n}\) which is at the same time holomorphic in the complex space \(\mathbf C^n\), for \(n>1\), is a nondegenerate linear mapping.
In the present note we study a more general class of quasiconformal mappings \(f:\mathbf R^n\to\mathbf R^n\) that are gradients of harmonic functions. Such mappings will be called harmonic.
Theorem. A harmonic quasiconformal mapping \(f:\mathbf R^3\to\mathbf R^3\) that does not reduce to a constant is nondegenerate linear.
Lemma 1. If a quasiconformal mapping \(f:\mathbf R^3\to\mathbf R^3\) is given by analytic functions \(f_i(x)\) \((i=1,2,3)\) and is different from a constant, then it is a homeomorphism. The Jacobian \(J_f\) therefore cannot assume values of different signs.
Let \(S\) be the set of zeros of \(J_f\). As an analytic set, \(S\) is the union of a finite number of manifolds of class \(C^\infty\) (see \((^1)\)). By virtue of inequality (1), the mapping \(f\) takes each connected component of \(S\) to a point. In the works \((^2,^3)\) it is shown that the full preimage of a point under a quasiconformal mapping \(f\), for any \(\alpha>0\), has zero \(\alpha\)-dimensional Hausdorff measure. Consequently, each connected component of \(S\) consists of a single point, i.e. \(S\) is the union of a finite number of points. The set \(B_f\) of branch points of the mapping \(f\) belongs to \(S\) and, consequently, is also finite.
We show that \(B_f=\varnothing\). If \(B_f\) is nonempty, surround any point \(x\in B_f\) by such a neighborhood \(U\) that it contains no other points of \(B_f\). It is known (see \((^2,^3)\)) that \(f(U)\) is a certain neighborhood of the point \(f(x)\). On the one hand, the fundamental group \(\pi_1(f(U)\setminus f(x))\) is trivial; on the other hand, in \(f(U)\) there certainly are points different from \(f(x)\) whose preimage consists of more than one point, which indicates the nontriviality of \(\pi_1(f(U)\setminus f(x))\). This contradiction proves that \(B_f=\varnothing\).
Thus, the mapping \(f\) is locally homeomorphic, and then, by the theorem of V. A. Zorich \((^4)\), it is homeomorphic globally as well. Finally, it is well known that under these conditions \(f(\mathbf R^3)=\mathbf R^3\) (see \((^5)\) or \((^6)\)). The lemma is proved.
It remains to show that all the functions \(f_i(x)\) \((i=1,2,3)\) in the formulation of the theorem are polynomials. Indeed, consider the mapping—
the mapping \(h=\sigma f\sigma\), where \(\sigma(x)=x|x|^{-2}\) is inversion. Since \(f(\infty)=\infty\), we have \(h(0)=0\), and, by the multidimensional analogue of Mori’s theorem \((^7)\), there exist constants \(C_1\) and \(C_2\) such that, in some neighborhood of zero,
\[ |h(x)|>C_1|x|^{C_2}. \]
Returning to the mapping \(f\), we obtain that for sufficiently large \(|x|\)
\[ |f(x)|<\frac{1}{C_1}|x|^{C_2}. \]
This inequality is certainly also valid if in it \(|f(x)|\) is replaced by \(|f_i(x)|\) \((i=1,2,3)\). Since all \(f_i\) are harmonic functions, it follows from the inequalities obtained that they are polynomials.
It also follows easily from inequality (1) that all degrees of the polynomials \(f_i\) are the same. We shall denote the common value of these degrees by \(m\). We may write
\[ f_i=p_i+q_i\qquad (i=1,2,3), \]
where \(p_i\) are homogeneous polynomials of degree \(m\), while the degrees of \(q_i\) are strictly less than \(m\). It is obvious that the mapping \(p:\mathbf R^3\to\mathbf R^3\) defined by the coordinate functions \(p_i\) \((i=1,2,3)\) is harmonic.
Lemma 2. Let \(f:\mathbf R^3\to\mathbf R^3\) be a quasiconformal homeomorphism, with \(f(0)=0\). Denote by \(m(R)\) and \(M(R)\), respectively, the minimum and maximum of \(|f(x)|\) on the sphere \(|x|=R\). Then there exists a constant \(C\), depending only on \(f\), such that
\[ M(R)\leq C m(R). \tag{2} \]
The proof follows easily from the estimate given by Gehring in \((^8)\) for the modulus of a ring.
Lemma 3. Let \(f:\mathbf R^3\to\mathbf R^3\) be a homeomorphism given by polynomials, and let \(f(0)=0\). Take some ray \(l\) issuing from the origin \(0\), and a point \(x\in l\) such that \(|x|=r\). If \(s_l(r)\) is the length of the image of the segment \([0,x]\) under the mapping \(f\), then there exists a constant \(C(l)\), depending possibly on \(l\), such that for every \(r>1\)
\[ s_l(r)\leq C(l)|f(x)|. \tag{3} \]
The proof is elementary.
Without loss of generality we may assume that for the original mapping \(f\), \(J_f\geq 0\) and \(f(0)=0\). Let \(J_f|_l\) be the restriction of the Jacobian \(J_f\) to some ray \(l\) issuing from the origin. Then, in the notation of Lemma 3, \(J_f|_l\) is a polynomial in \(r\) of degree \(k_l\).
The number \(k_l\) does not depend on \(l\). Indeed, for rays \(l_j\) \((j=1,2)\) such that \(k_{l_1}<k_{l_2}\), take points \(x_j\) lying on \(l_j\) at distance \(r>1\) from the origin \((j=1,2)\). Using (2) and (3), we obtain
\[ s_{l_1}(r)\geq m(r)\geq C M(r)\geq C|f(x_2)|\geq \frac{C}{C(l_2)}\,s_{l_2}(r). \]
But
\[ s_{l_j}(r)=\int_0^r \mu_{l_j}(t)\,dt\qquad (j=1,2), \]
where \(\mu_l(t)\) are the coefficients of local stretching in the corresponding directions. From the geometric definition of quasiconformality there follows the existence of constants \(C_1\) and \(C_2\) such that
\[ C_1\bigl(J_f|_{l_j}\bigr)^{1/3}\leq \mu_{l_j}\leq C_2\bigl(J_f|_{l_j}\bigr)^{1/3}\qquad (j=1,2). \]
Hence \(k_{l_1}\geq k_{l_2}\), which contradicts the inequality \(k_{l_1}<k_{l_2}\).
Denote by \(k\) the common value of the \(k_l\), and let us show that \(k = 3(m - 1)\). On the sphere \(|x| = 1\) there exists a point \(x\) such that \(p(x) \ne 0\). It is easy to find constants \(M_1(x)\) and \(M_2(x)\) such that, for \(r > 1\),
\[ M_1(x)r^m \le |f(rx)| \le M_2(x)r^m . \tag{4} \]
If \(l\) is the ray determined by the vector \(x\), then, using (3) and (4), we obtain
\[ \int_0^r (J_f|_l)^{1/3}\,dt \ge \frac{1}{C_2}|f(rx)| \ge \frac{M_1(x)}{C_2}r^m = \frac{mM_1(x)}{C_2}\int_0^r [t^{3(m-1)}]^{1/3}\,dt . \]
It follows that \(k \ge 3(m - 1)\). Similarly one proves that \(k \le 3(m - 1)\).
Let us show that everywhere on the unit sphere the Jacobian \(J_p\) of the mapping \(p\), defined above by the homogeneous components of highest degree of the polynomials \(f_i\) \((i = 1, 2, 3)\), does not vanish. Indeed, if at some point \(x\), \(|x| = 1\), one had \(J_p(x) = 0\), then
\[ \lim_{r\to\infty}\frac{J_f(rx)}{r^{3(m-1)}} = \lim_{r\to\infty}\det\left( \frac{\partial p_i/\partial x_j|_{rx}+\partial q_i/\partial x_j|_{rx}}{r^{m-1}} \right) = \lim_{r\to\infty}\det\left( \left.\frac{\partial p_i}{\partial x_j}\right|_x + \frac{\partial q_i/\partial x_j|_{rx}}{r^{m-1}} \right) =0, \]
and this contradicts the equality \(k = 3(m - 1)\).
In view of the compactness of the unit sphere, there exists a constant \(C\) such that, for \(|x| = 1\),
\[ \left( \sum_{i,j=1}^3 \left|\frac{\partial p_i}{\partial x_j}\right|^2 \right)^{1/2} \le C|J_p|^{1/3}. \tag{5} \]
Since on the right and on the left in (5) there are homogeneous functions of degree \(m - 1\), this inequality is valid everywhere in \(\mathbf R^3\), i.e., \(p\) is quasiconformal.
By Lemma 1 the mapping \(p\) is a homeomorphism, but, according to a result of Levy [9], the Jacobian of a harmonic homeomorphism in \(\mathbf R^3\) does not vanish at \(0\). This is possible only when the degree of homogeneity of \(J_p\) is zero, i.e., \(m = 1\). The theorem is proved.
Remark. All the results presented, except Levy’s theorem [9], are also valid for the \(n\)-dimensional \((n \ge 3)\) case. The question of the nonvanishing of the Jacobian of a harmonic homeomorphism in \(\mathbf R^n\) \((n > 3)\) remains open.
The author expresses gratitude to B. V. Shabat for posing the question and for great assistance in the work.
Moscow State University
named after M. V. Lomonosov
Received
7 III 1970
References
- R. Thom, in: Collection, Singularities of differentiable mappings, 1968.
- Yu. G. Reshetnyak, Siberian Mathematical Journal, 9, No. 2 (1968).
- Yu. G. Reshetnyak, ibid., 8, No. 3 (1967).
- V. A. Zorich, Mathematical Collection, 74 (116), No. 3 (1967).
- B. V. Shabat, Dokl. Akad. Nauk SSSR, 132, No. 5 (1960).
- Ch. Loewner, J. Math. Mech., 8 (1959).
- P. Caraman, Homeomorfisme cvasiconforme n-dimensionale, București, 1968.
- F. W. Gehring, Trans. Am. Math. Soc., 101, No. 3 (1961).
- H. Levy, Ann. Math., 88, No. 3 (1968).