Curriculum Vitae

John Douglas Moore

Birthplace: Chicago, Illinois, USA
Birthdate: July 11, 1943
Citizenship: USA

EDUCATION

BA, University of California, Berkeley, CA 1965
PhD, University of California, Berkeley, CA 1969

PROFESSIONAL EXPERIENCE

Assistant Professor, University of California, Santa Barbara, CA 1969-1975
Associate Professor, University of California, Santa Barbara, CA 1975-1981
Professor, University of California, Santa Barbara, CA 1981-present
Chair, Department of Mathematics, University of California, Santa Barbara, CA 1999-2004.




RESEARCH INTERESTS

My research interests center on curvature and topology of submanifolds in Euclidean space, and applications of minimal surfaces to Riemannian geometry. For example, one of my 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, first presented in 1978, is based upon Morse theory of the height function.

In collaboration with Mario Micallef, I gave a proof of the sphere theorem from Riemannian geometry in 1988 using minimal surfaces. The proof shows that the relevant curvature for the study of stability of minimal surfaces in Riemannian manifolds is the isotropic sectional curvature. In fact, our argument shows 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, I have been studying topological conditions on a smooth manifold which may ensure that the manifold must contain infinitely many geometrically distinct minimal surfaces of a fixed genus when it is given a generic metric. My approach is based upon Morse theory for a perturbed version of the energy (the alpha-energy introduced by Sacks and Uhlenbeck). The first step in this program consisted of establishing a bumpy metric theorem for parametrized closed minimal surfaces in compact Riemannian manifolds. This theorem states that for generic metrics, oriented parametrized closed minimal surfaces lie on nondegenerate critical submanifolds with the same dimension as the group of conformal transformations on the domain.


SELECTED PUBLICATIONS

1971 "Isometric immersions of Riemannian products," J. Differential Geometry v. 5, pp. 159--168.
1972 "Isometric immersions of space forms in space forms," Pacific J. Math. v. 40, pp. 157--166.
1977 "Conformally flat submanifolds of Euclidean space," Math. Ann. v. 225, pp. 89--97.
1977 "Submanifolds of constant positive curvature I," Duke Math. J. v. 44, pp. 449--484.
1978 "Codimension two submanifolds of positive curvature," Proc. Amer. Math. Soc. v. 70, pp. 72--74.
1980 "On equivariant isometric embeddings," (with Roger Schlafly), Math. Z. v. 173, pp. 125--141.
1985 "On stability of minimal spheres and a two-dimensional version of Synge's theorem," Arch. Math. v. 44, pp. 278--281.
1986 "Compact Riemannian manifolds with positive curvature operators," Bull. Amer. Math. Soc. v. 14, pp. 279--282.
1988 "Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes," (with Mario Micallef) Annals of Math. v. 127, pp. 199--227.
1990 "On the number of minimal two-spheres of small area in manifolds with curvature bounded above," Math. Ann. v. 288, pp. 323--343.
1996 "On extendability of isometric immersions of spheres," Duke Math. J. v. 85, pp. 685--699.
2002 "Euler characters and submanifolds of constant positive curvature," Trans. Amer. Math. Soc., v. 354, pp. 3815--3834.

PREPRINTS IN PDF FORMAT

2006 "Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds," Trans. Amer. Math. Soc., to appear.
2006 "Self-intersections of closed parametrized minimal surfaces in generic Riemannian manifolds," to appear.
2006 "Nondegeneracy of coverings of minimal tori and Klein bottles in Riemannian manifolds," to appear.