Shiu-Yuen Cheng

In 1975, Shing-Tung Yau found a novel gradient estimate for solutions of second-order elliptic partial differential equations on certain complete Riemannian manifolds.

As a consequence, Cheng and Yau were able to show the existence of an eigenfunction, corresponding to the first eigenvalue, of the Laplace-Beltrami operator on a complete Riemannian manifold.

[CY76a] In 1916, Hermann Weyl found a differential identity for the geometric data of a convex surface in Euclidean space.

[CY77b] Any strictly convex closed hypersurface in the Euclidean space ℝn + 1 can be naturally considered as an embedding of the n-dimensional sphere, via the Gauss map.

The Minkowski problem asks whether an arbitrary smooth and positive function on the n-dimensional sphere can be realized as the scalar curvature of the Riemannian metric induced by such an embedding.

Luis Caffarelli, Nirenberg, and Joel Spruck later developed more flexible methods to deal with the same problem.