Mathematicians solve 30-year puzzle about geometric surfaces
Researchers have proved a fundamental theorem about minimal surfaces—geometric structures that minimize area, like soap films. The breakthrough, which extends a classic result from complex analysis, could have applications in materials science, manufacturing, and computational geometry where minimal surfaces are used to model efficient structures.
Originaltitel: Finiteness of the number of ends of minimal submanifolds in Euclidean space
<p>We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in ℝ<sup><em>N</em></sup>. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimension<em>n</em> meeting any<em>n</em>-plane passing through the origin in at most<em>k</em> points has no more<em>c(n, N)k</em> ends.</p>