On the left, at the Great Wall near Beijing in 2002. On the right, playing Chopin in the summer of 2007. |

RESEARCH INTERESTS

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, presented in my 1978 Proceedings AMS article, 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 (Annals of Math, 1988). 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. 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."

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.

1973 "The Cartan-Janet theorem for conformal imbeddings" (with Howard
Jacobowitz), Indiana Univ. Math. J. v. 23, pp. 187--203.

1975 "An application of second variation to submanifold theory", Duke Math. J.
v. 42, pp. 191--193.

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.

1978 "Normal characteristic forms: conformal invariance and duality", (with
James White) Math. Z. v. 164, pp. 125--141.

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.

1985 "Minimal disks and compact hypersurfaces in Euclidean space" (with
Thomas Schulte), Proc. Amer. Math. Soc. v. 94, 321--328.

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.

2001 "Lectures on Seiberg-Witten invariants", second edition, Springer,
New York.

2002 "Euler characters and submanifolds of constant positive curvature", Trans.
Amer. Math. Soc., v. 354, pp. 3815--3834.

2006 "Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds",
Trans. Amer. Math. Soc., v. 358, pp. 5193--5256.

2007 "Nondegeneracy of coverings of minimal tori and Klein bottles in Riemannian manifolds",
Pacific J. Math., v. 230, pp. 147--166.

2007 "Second variation of energy for minimal surfaces in Riemannian manifolds",
Matematica Contemporanea, v. 33, pp. 213--241.

2018 **"Introduction to global analysis:
minimal surfaces in Riemannian manifolds",** Graduate Studies in Mathematics
no. 187, Amer. Math. Soc., Providence, RI.

PREPRINTS IN PDF FORMAT

2007 **"Energy growth in minimal surface bubbles",** unpublished manuscript.

2017 **"Minimal two-spheres of low index in manifolds with positive complex sectional curvatures"**
(with Robert Ream), to appear.

2017 **"Closed minimal surfaces of high Morse index in manifolds of negative curvature",**
to appear.