## 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.

### MSC:

 53C40 Global submanifolds 53A05 Surfaces in Euclidean and related spaces

Zbl 0695.53045
Full Text: