Mathematicians solve century-old puzzle about curved surfaces
Researchers have extended a foundational theorem about complex functions to work on curved geometric spaces, settling a long-standing theoretical question. The breakthrough has immediate applications in materials science and engineering, where understanding minimal surfaces—the most efficient curved shapes—is critical for designing everything from stronger materials to better optical devices.
Originaltitel: Denjoy-Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces
<p>We extend the well-known Denjoy-Ahlfors theorem about the number of different asymptotic tracts of a holomorphic function to subharmonic functions on arbitrary Riemannian manifolds. We obtain some versions of the Liouville theorem for Aa-harmonic functions without the geodesic completeness requirement on a manifold. Moreover, the upper estimate of the topological index of the height function of a minimal surface in Rn has been established and, as a consequence, a new prove of the Bernstein's theorem has been derived. Other applications to the theory of minimal surfaces are also discussed.</p>