The Keller–Osserman problem

Osserman's most widely cited research article, published in 1957, dealt with the partial differential equation

${\displaystyle \Delta u=f(u).}$

He showed that fast growth and monotonicity of f is incompatible with the existence of global solutions. As a particular instance of his more general result:

There does not exist a twice-differentiable function u : ℝⁿ → ℝ such that

${\displaystyle {\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}\geq e^{u}.}$

Osserman's method was to construct special solutions of the PDE which would facilitate application of the maximum principle. In particular, he showed that for any real number a there exists a rotationally symmetric solution on some ball which takes the value a at the center and diverges to infinity near the boundary. The maximum principle shows, by the monotonicity of f, that a hypothetical global solution u would satisfy u(x) < a for any x and any a, which is impossible.

The same problem was independently considered by Joseph Keller,^[12] who was drawn to it for applications in electrohydrodynamics. Osserman's motivation was from differential geometry, with the observation that the scalar curvature of the Riemannian metric e^2u(dx² + dy²) on the plane is given by

${\displaystyle -e^{-2u}{\Big (}{\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}{\Big )}.}$

An application of Osserman's non-existence theorem then shows:

Any simply-connected two-dimensional smooth Riemannian manifold whose scalar curvature is negative and bounded away from zero is not conformally equivalent to the standard plane.

By a different maximum principle-based method, Shiu-Yuen Cheng and Shing-Tung Yau generalized the Keller–Osserman non-existence result, in part by a generalization to the setting of a Riemannian manifold.^[13] This was, in turn, an important piece of one of their resolutions of the Calabi–Jörgens problem on rigidity of affine hyperspheres with nonnegative mean curvature.^[14]

Non-existence for the minimal surface system in higher codimension

In collaboration with his former student H. Blaine Lawson, Osserman studied the minimal surface problem in the case that the codimension is larger than one. They considered the case of a graphical minimal submanifold of euclidean space. Their conclusion was that most of the analytical properties which hold in the codimension-one case fail to extend. Solutions to the boundary value problem may exist and fail to be unique, or in other situations may simply fail to exist. Such submanifolds (given as graphs) might not even solve the Plateau problem, as they automatically must in the case of graphical hypersurfaces of Euclidean space.

Their results pointed to the deep analytical difficulty of general elliptic systems and of the minimal submanifold problem in particular. Many of these issues have still failed to be fully understood, despite their great significance in the theory of calibrated geometry and the Strominger–Yau–Zaslow conjecture.^[15][16]