A removable singularity condition for maps in \(R^ n\). (English) Zbl 0719.30012

Cristea, Mihai

Stud. Cercet. Mat. 41, No. 5, 379-380 (1989).

In this brief article, the author uses the notation \(D^+f(x)\) for \[ \limsup_{y\to x}\| f(y)-f(x)\| /\| y-x\|. \] With D a domain in \({\mathbb{R}}^ n\), the idea is to study those f: \(D\to {\mathbb{R}}^ n\) having \(D^+f(x)<\infty\) apart from points x lying in a certain exceptional set K, assumed to have \(m_{n-1}(K)=0\). However, he does assume that \(D^+f\), as restricted to \(D\setminus K\), is locally bounded near points of K. Indeed, in his main theorem, where he also assumes that f is continuous, he proves without difficulty that the local bound for \(D^+f\) (on \(D\setminus K)\) is a local Lipschitz constant, and therefore a local bound for \(D^+f\) (on D).
The reviewer feels that the corollaries are somewhat overstated. Denoting the branch set as usual by \(B_ f\), the conclusion of Corollary 2 that the Hausdorff dimension of \(B_ f\) exceeds n-2 (implicitly if \(B_ f\neq \emptyset)\), seems to overlook the standard double-twist example expressed using cylindrical coordinates in \({\mathbb{R}}^ 3\) by (r,\(\Theta\),z)\(\to (r,2\Theta,z)\). For this particular f one has \(D^+f\leq 2\) without exception, yet \(B_ f\) is the entire z-axis. The argument associated to Corollary 2, which the above example does not refute, is basically only that \(B_ f=\emptyset\) if \(m_{n-2}(B_ f)=0\).

MSC:

30C65

Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations