My research interests have centered on submanifolds of Euclidean space satisfying conditions on curvature and Morse theory of parametrized minimal surfaces in Riemannian manifolds. Here is an introduction to some of the topics treated:
SUBMANIFOLDS OF EUCLIDEAN SPACE
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. Some of the results are included in the 1972 Pacific J. Math. article, Isometric immersions of space forms in space forms, which takes the first steps toward resolving a conjectured generalization of a theorem of Hilbert: Hyperbolic space of dimension n has no isometric immersion in (2n-1)-dimensional Euclidean space.
A dual theory for submanifolds of constant positive curvature and low codimension in Euclidean space is presented in my article Submanifolds of constant positive curvature I, Duke Math. J. (1977). Submanifolds of constant positive curvature in (2n-1)-dimensional Euclidean space can have two types of points, nonumbilics and weak umbilics. There are two corresponding types of local isometric immersions, but compact submanifolds of constant positive curvature cannot have nonumbilics by an argument which extends that of Liebmann's theorem for surfaces. Subsequent articles explore various consequences of this result, while similar techniques are used to study conformally flat submanifolds in the 1977 Math. Annalen article Conformally flat submanifolds of Euclidean space.
For an introduction to my work on curvature and topology of submanifolds, I recommend the 1978 Proceedings AMS note, Codimension two submanifolds of positive curvature, which uses Morse theory of height functions to give an extrinsic sphere theorem that 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. Finally, let me mention a 1980 article in Math. Zeitschrift with Roger Schlafly, On equivariant isometric embeddings, which gives an equivariant version of Nash's imbedding theorem.
PARTIAL MORSE THEORY OF PARAMETRIZED MINIMAL SURFACES IN RIEMANNIAN MANIFOLDS
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.