My research interests have centered on curvature and topology of submanifolds in Euclidean space, and the theory of minimal surfaces in Riemannian manifolds.
In my dissertation at UC Berkeley (1969), I studied isometric immersions of Riemannian products and submanifolds of constant negative curvature in Euclidean space, using Cartan's theory of exteriorly orthogonal forms. The 1972 Pacific J. Math. article, Isometric immersions of space forms in space forms, takes the first steps toward resolving the following conjectured generalization of a theorem of Hilbert: Hyperbolic space of dimension n has no isometric immersion in (2n-1)-dimensional Euclidean space. I later developed a dual theory that studies submanifolds of constant positive curvature and low codimension in Euclidean space in a series of articles, culminating in Euler characters and submanifolds of constant positive curvature (Transactions AMS, 2002), which uses an analog of the Chern-Simons invariant to construct an obstruction to existence of isometric immersions.
One of my early theorems in submanifold theory gives an extrinsic sphere theorem which states: A compact Riemannian manifold of positive sectional curvature and dimension at least three which admits a codimension two isometric immersion in Euclidean space must be homeomorphic to a sphere. The relatively simple proof is presented in a 1978 Proceedings AMS note, Codimension two submanifolds of positive curvature, and is based upon Morse theory of the height function.
In collaboration with Mario Micallef, I gave a proof of a sphere theorem from Riemannian geometry using minimal surfaces, which appeared in the Annals of Math of 1988: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. An interesting fact uncovered by the proof is that the relevant curvature for the study of stability of minimal two-spheres in Riemannian manifolds is isotropic curvature. The argument for the sphere theorem uses Morse theory of the alpha-energy of Sacks and Uhlenbeck to show that a compact simply connected Riemannian manifold with positive isotropic curvature of dimension at least four must be homeomorphic to a sphere. This extends an earlier sphere theorem of Berger, Klingenberg and Toponogov, which was proven using geodesics instead of minimal surfaces.
Recently, my research has focused on developing a partial Morse theory for closed parametrized minimal surfaces in compact Riemannian manifolds which should be analogous to the Morse theory of smooth closed geodesics. The first step in this program is the Bumpy Metric Theorem which states that for generic metrics, all parametrized minimal surfaces are free of branch points and lie on nondegenerate critical submanifolds which are orbits for the identity component G of the group of complex automorphisms of the domain. This is proven in the last chapter of my book, Introduction to global analysis: minimal surfaces in Riemannian manifolds.