Asian Journal of Mathematics

On a construction of Burago and Zalgaller

Emil Saucan

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The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \ge 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.

Article information

Asian J. Math., Volume 16, Number 4 (2012), 587-606.

First available in Project Euclid: 12 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B70: Polyhedral manifolds 57R40: Embeddings 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 30C65: Quasiconformal mappings in $R^n$ , other generalizations

$PL$-isometric embedding Burago-Zalgaller construction quasiconformal mapping maximal dilatation local topological index


Saucan, Emil. On a construction of Burago and Zalgaller. Asian J. Math. 16 (2012), no. 4, 587--606.

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