[23][24][25] He has had over 44 doctoral students, including Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves.

[26] Schoen has investigated the use of analytic techniques in global differential geometry, with a number of fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.

[29] As a consequence they found some striking geometric results, such as that certain noncompact manifolds do not admit any complete metrics of nonnegative Ricci curvature.

The techniques they developed, making extensive use of monotonicity formulas, have been very influential in the field of geometric analysis and have been adapted to a number of other problems.

In 1979, Schoen and his former doctoral supervisor, Shing-Tung Yau, made a number of highly influential contributions to the study of positive scalar curvature.

By an elementary but novel combination of the Gauss equation, the formula for second variation of area, and the Gauss-Bonnet theorem, Schoen and Yau were able to rule out the existence of several types of stable minimal surfaces in three-dimensional manifolds of positive scalar curvature.

By contrasting this result with an analytically deep theorem of theirs establishing the existence of such surfaces, they were able to achieve constraints on which manifolds can admit a metric of positive scalar curvature.

Schoen and Doris Fischer-Colbrie later undertook a broader study of stable minimal surfaces in 3-dimensional manifolds, using instead an analysis of the stability operator and its spectral properties.

Mikhael Gromov and H. Blaine Lawson obtained similar results by other methods, also undertaking a deeper analysis of topological consequences.

Thus the general case of the positive mass theorem in higher dimensions was left as a major open problem in Schoen and Yau's 1979 work.

In 1984, Schoen settled the cases left open by Aubin's work, the decisive point of which rescaled the metric by the Green's function of the Laplace-Beltrami operator.

In the 1980s, Richard Hamilton introduced the Ricci flow and proved a number of convergence results, most notably for two- and three-dimensional spaces.

[41][42] Although he and others found partial results in high dimensions, progress was stymied by the difficulty of understanding the complicated Riemann curvature tensor.

[43] Simon Brendle and Schoen were able to prove that the positivity of Mario Micallef and John Moore's "isotropic curvature" is preserved by the Ricci flow in any dimension, a fact independently proven by Huy Nguyen.