He earned a Ph.D. in 1955 from Harvard University with the thesis Contributions to the Problem of Type (on Riemann surfaces) supervised by Lars Ahlfors.

[5] Osserman's most widely cited research article, published in 1957, dealt with the partial differential equation (PDE) 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 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.

[14] 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.