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**Shadows and convexity of surfaces.**
*(English)*
Zbl 1013.53040

Let \(M\) be a closed connected 2-dimensional manifold, \(f:M\to\mathbb{R}^3\) be a smooth immersion into Euclidean 3-space and \(n:M\to S^2\) the Gauss map induced by \(f\). For every \(u\in S^2\) the shadow, \(S_u\), is defined by \(S_u=\{p \in M:\langle n(P), u\rangle >0\}\), where \(\langle .,. \rangle\) is the standard inner product. The author of the present paper settles the following “Wente’s shadow problem in 3-space”. Does the connectedness of the shadows \(S_u\) imply that \(f\) is a convex embedding? In short, the answer to the problem is yes, provided that either the shadows are simply connected, or \(M\) is a sphere; otherwise, the answer is no. More precisely the author proves the following results:

Theorem 1. \(f\) is a convex imbedding, if and only if, for every \(u\in S^2\), \(S_u\) is simply connected.

Theorem 2. There exists a smooth embedding of the torus, \(f:S^1\times S^1\to\mathbb{R}^3\) such that for all \(u\in S^2\), \(S_u\) is connected.

Theorem 3. If \(M\) is topologically a sphere, and, for every \(u\in S^2\), \(S_u\) is connected, then \(f\) must be a convex imbedding.

For a great circle \(C\subset S^2\), the number of components of \(n^{-1}(S^2-C)\) is called the vision number with respect to the direction perpendicular to \(C\). J. Choe [Arch. Rat. Mech. Anal. 109, 195-212 (1990; Zbl 0695.53045)] conjectured that always exists a direction with respect to which the vision number of \(f:M\to\mathbb{R}^3\) is greater than or equal to \(4-X(M)\), where \(X(M)\) is the Euler characteristic. Theorem 2 gives a counterexample to this conjecture.

Theorem 1. \(f\) is a convex imbedding, if and only if, for every \(u\in S^2\), \(S_u\) is simply connected.

Theorem 2. There exists a smooth embedding of the torus, \(f:S^1\times S^1\to\mathbb{R}^3\) such that for all \(u\in S^2\), \(S_u\) is connected.

Theorem 3. If \(M\) is topologically a sphere, and, for every \(u\in S^2\), \(S_u\) is connected, then \(f\) must be a convex imbedding.

For a great circle \(C\subset S^2\), the number of components of \(n^{-1}(S^2-C)\) is called the vision number with respect to the direction perpendicular to \(C\). J. Choe [Arch. Rat. Mech. Anal. 109, 195-212 (1990; Zbl 0695.53045)] conjectured that always exists a direction with respect to which the vision number of \(f:M\to\mathbb{R}^3\) is greater than or equal to \(4-X(M)\), where \(X(M)\) is the Euler characteristic. Theorem 2 gives a counterexample to this conjecture.

Reviewer: T.Koufogiorgos (Ioannina)