John Douglas Moore
My research interests have centered on curvature and topology of submanifolds in Euclidean space, and
the theory of minimal surfaces in Riemannian manifolds.
One of my early theorems shows that compact
Riemannian manifolds which have positive sectional curvature and admit codimension two isometric immersions in
Euclidean space must be homeomorphic to spheres. The proof is presented in a Proceedings AMS note,
Codimension two submanifolds of positive curvature,
that appeared in 1978, and is based upon Morse theory of the height function.
Later, in collaboration with Mario Micallef, I gave a proof of the 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 surfaces 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.