Harmonic maps

In 1976, Schoen and Shing-Tung Yau used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by James Eells and Joseph Sampson to noncompact settings.^[26][27] By identifying a certain interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel conditions on the domain leading to the same conclusion. These rigidity theorems are complemented by their existence theorem for harmonic maps on noncompact domains, as a simple corollary of Richard Hamilton's resolution of the Dirichlet boundary-value problem.^[28] 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.

In two papers from the 1980s, Schoen and Karen Uhlenbeck made a foundational contribution to the regularity theory of energy-minimizing harmonic maps. 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. Fundamental conclusions of theirs include compactness theorems for sets of harmonic maps and control over the size of corresponding singular sets. Leon Simon applied such results to obtain a clear picture of the small-scale geometry of energy-minimizing harmonic maps.^[29]

Later, Mikhael Gromov had the insight that an extension of the theory of harmonic maps, to allow values in metric spaces rather than Riemannian manifolds, would have a number of significant applications, with analogues of the classical Eells−Sampson rigidity theorem giving novel rigidity theorems for lattices. The intense analytical details of such a theory were worked out by Schoen. Further foundations of this new context for harmonic maps were laid out by Schoen and Nicholas Korevaar.

Minimal surfaces, positive scalar curvature, and the positive mass theorem

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.

An inductive argument based upon the existence of stable minimal hypersurfaces allowed them to extend their results to higher dimensions. Further analytic techniques facilitated the application of topological surgeries on manifolds which admit metrics of positive scalar curvature, showing that the class of such manifolds is topologically rich. Mikhael Gromov and H. Blaine Lawson obtained similar results by other methods, also undertaking a deeper analysis of topological consequences.^[30][31]

By an extension of their techniques to noncompact manifolds, Schoen and Yau were able to settle the important Riemannian case of the positive mass theorem in general relativity, which can be viewed as a statement about the geometric behavior near infinity of noncompact manifolds with positive scalar curvature. Like their original results, the argument is based upon contradiction. A more constructive argument, using the theory of harmonic spinors instead of minimal hypersurfaces, was later found by Edward Witten.^[32][33][34]

Schoen, Yau, and Leon Simon identified a simple combination of the Simons formula with the formula for second variation of area which yields important curvature estimates for stable minimal hypersurfaces of low dimensions. In 1983, Schoen obtained similar estimates in the special case of two-dimensional surfaces, making use of the existence of isothermal coordinates. Slightly weaker estimates were obtained by Schoen and Simon, although without any dimensional restriction. Fundamental consequences of the Schoen−Simon estimates include compactness theorems for stable minimal hypersurfaces as well as control over the size of "singular sets." In particular, the Schoen−Simon estimates are an important tool in the Almgren–Pitts min-max theory, which has found a number of applications.

The possible presence of singular sets restricts the dimensions in which Schoen and Yau's inductive arguments can be easily carried out. Meanwhile Witten's essential use of spinors restricts his results to topologically special cases. 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 1988, they settled the problem in arbitrary dimension in the special case that the Weyl tensor is zero; this has been significant in conformal geometry. In 2017, they released a preprint claiming the general case, in which they deal directly with the singular sets of minimal hypersurfaces.

Yamabe problem and conformal geometry

In 1960, Hidehiko Yamabe introduced the "Yamabe functional" on a conformal class of Riemannian metrics and demonstrated that a critical point would have constant scalar curvature.^[35] He made partial progress towards proving that critical points must exist, which was taken further by Neil Trudinger and Thierry Aubin.^[36][37] Aubin's work, in particular, settled the cases of high dimension or when there exists a point where the Weyl tensor is nonzero. 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. This allowed an application of Schoen and Yau's positive mass theorem to the resulting metric, giving important asymptotic information about the original metric. The works of Yamabe, Trudinger, Aubin, and Schoen together comprise the solution of the Yamabe problem, which asserts that there is a metric of constant scalar curvature in every conformal class.

In 1989, Schoen was also able to adapt Karen Uhlenbeck's bubbling analysis, developed for other geometric-analytic problems, to the setting of constant scalar curvature.^[38][39] The uniqueness of critical points of the Yamabe functional, and more generally the compactness of the set of all critical points, is a subtle question first investigated by Schoen in 1991. Fuller results were later obtained by Simon Brendle, Marcus Khuri, Fernando Codá Marques, and Schoen.

Differentiable sphere theorem

In the 1980s, Richard Hamilton introduced the Ricci flow and proved a number of convergence results, most notably for two- and three-dimensional spaces.^[40][41] Although he and others found partial results in high dimensions, progress was stymied by the difficulty of understanding the complicated Riemann curvature tensor.^[42] 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.^[43][44] Brendle and Schoen were further able to relate their positivity condition to the positivity of sectional curvature and of curvature operator, which allowed them to exploit then-recent algebraic ideas of Christoph Böhm and Burkhard Wilking, thereby obtaining a new convergence theorem for Ricci flow.^[45] A special case of their convergence theorem has the differentiable sphere theorem as a simple corollary, which had been a well-known conjecture in the study of positive sectional curvature for the past fifty years.