Biholomorphic maps determined on the boundary
Annales de l'Institut Fourier, Volume 27 (1977) no. 3, p. 129-133

Let $D$ be a bounded domain in ${\mathbf{C}}^{n}$ such that the boundary $bD$ is topologically ${S}^{2n-1}$ in ${\mathbf{R}}^{2n}$ with a regular point; let $f:\stackrel{˜}{D}\to {\mathbf{C}}^{n}$ be a holomorphic map where $\stackrel{˜}{D}$ is a neighborhood of $\overline{D}$. If $f$ is one-to-one when restricted to $bD$, then $f:D\to f\left(D\right)$ is biholomorphic.

Soit $D$ un domaine borné de ${\mathbf{C}}^{n}$ tel que la frontière $bD$ soit topologiquement égale à ${S}^{2n-1}$ dans ${\mathbf{R}}^{2n}$ contenant un point régulier ; soit $f:\stackrel{˜}{D}\to {\mathbf{C}}^{n}$ une application holomorphe où $\stackrel{˜}{D}$ est un voisinage de $\overline{D}$. Si $f|bD$ est biunivoque $f:D\to f\left(D\right)$ est biholomorphe.

@article{AIF_1977__27_3_129_0,
author = {Mochizuki, Nozomu},
title = {Biholomorphic maps determined on the boundary},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {27},
number = {3},
year = {1977},
pages = {129-133},
doi = {10.5802/aif.664},
zbl = {0352.32016},
mrnumber = {58 \#22681},
language = {en},
url = {http://www.numdam.org/item/AIF_1977__27_3_129_0}
}

Mochizuki, Nozomu. Biholomorphic maps determined on the boundary. Annales de l'Institut Fourier, Volume 27 (1977) no. 3, pp. 129-133. doi : 10.5802/aif.664. http://www.numdam.org/item/AIF_1977__27_3_129_0/

[1] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, New Jersey, 1964. | MR 33 #1797 | Zbl 0129.13102