H. Blaine Lawson

Minimal surfaces

In 1970 Lawson constructed minimal embeddings of every compact surface into the Euclidean 3-sphere (with the exception of the real projective plane, which cannot be so embedded).^[11] This gave, for example, embedded 3-dimensional cones in Euclidean 4-space of every possible topological type. This also led to interesting periodic surfaces of constant mean curvature in eEuclidean 3-space. His work in this area continued for years.^[12][13][14] One nice result was with Jim Simons^[15] where they showed how to use minimal integral currents for basic riemannnian geometry, and they proved that a stable minimal current (one whose second variation of mass is ≥ 0) in complex projective space, is a positive algebraic cycle.

Foliations

Lawson found codimension-one foliations of higher dimensional spheres,^[16] which answered a long-term question and engendered much subsequent work.

Compact Manifolds of Negative Curvature

Together with S.-T. Yau^[17] Lawson found basic theorems about these manifolds, such as the Splitting Theorem which says that if the fundamental group splits as a product of groups, then the manifold essentially splits as a direct metric product of manifolds. These results were independently found by Detlef Gromoll and Joseph A. Wolf^[18]

Boundaries of Complex Analytic Varieties

Together with F. Reese Harvey^[19][20] Lawson characterized the compact oriented submanifolds of complex Euclidean space which bound complex analytic varieties. These submanifolds could have singularities, and the result has analogues in complex projective space minus a linear subspace of higher codimension. This was a vast geometric generalization of a classical result of S. Bochner.^[21]

Calibrated Geometries

In a 1982 Acta Mathematica paper of F. Reese Harvey and Blaine Lawson^[22] found large classes of submanifolds (even with singularities) that are always homologically volume minimizing. This means that if one takes a compact piece M with boundary, then M has volume less than or equal to the volume of any M' such that M - M' bounds something of higher dimension. This paper was engendered by work of Herbert Federer.^[23] It applied to submanifolds of certain Euclidean spaces, but also to more general manifolds with special geometries. It inspired Robert Bryant to discover G(2) and Spin(7) manifolds,^[24] answering a long-standing question. It turned out the calibrated geometries discovered by Harvey-Lawson play a role in M-theory in modern particle physics. As a result there has been an enormous amount of work in this area.

Manifolds of Positive Scalar Curvature

In a series of three papers^[25][26][27]Mikhael Gromov and Lawson used the Dirac operator and other techniques to prove global results about manifolds with positive scalar curvature ? > 0. The first work in this area was done by Rick Schoen and S.-T. Yau in this area.^[28] Among many things Gromov and Lawson showed that for spin manifolds, the existence of a metric with ? > 0 depends only on the spin cobordism class of the manifold. They conjectured that a necessary and sufficient condition for ? > 0 was a KO-Theory analogue of the Aˆ-invariant. This conjecture was proved by Stephan Stoltz.^[29]

Algebraic Cycles

In his 1989 Annals of Mathematics paper "Algebraic Cycles and Homotopy Theory",^[30] Lawson proved a theorem which showed that the limit of codimension-q algebraic cycles in complex projective n-space Pn is a finite product of spaces which classify integer cohomology in degrees 2, 4, ..., 2q. This basic theorem has analogs on any projective variety, and the homotopy groups of the resulting space gives a new homology theory in algebraic geometry. With Marie-Louise Michelsohn^[31] they showed that the inclusion of the linear cycles in projective space leads to a map from K-theory to cohomology which is the total Chern class. Together with Eric Friedlander, a morphic cohomology was established for algebraic varieties, based on algebraic maps into cycles spaces on Pn, and this cohomology theory was shown to be dual the homology theory mentioned above.^[32][33]

Lawson and Friedlander also proved a Moving Lemma for families of algebraic cycles.^[34]

This cycle theory had many interesting applications in homotopy theory, which was worked out with Michelsohn, Paulo Lima-Filho, Charles Boyer, and Ben Mann.^{[35][36][37][38][39]}

Spin Geometry

Lawson and Michelsohn wrote a book,^[40] published by Princeton Press, which presented the deep index theorems proved by Atiyah and Singer. The book gave the fundamentals of Spin manifolds, K-theory and KO-theory, Clifford algebras and their relation to Bott Periodicity, the construction of Atiyah-Singer-Dirac operators, detailed proofs of various index theorems, and many applications were given. This text has been used worldwide for many years.

Singular Connections and Characteristic Currents

F. Reese Harvey and Lawson considered connections on vector bundles which were not smooth, and so the characteristic forms became singular currents.^[41] This led to a long sequence of interesting results.^{[42] [43] [44] [45]}

Differential Characters

In^{[46] [47]} Lawson, Harvey and Zweck established a Poincaré-Pontryagin Duality for the differential characters of Cheeger and Simons.^{[48] [49]} They also gave a deRham-Federer theory of differential characters, that is, they gave various formulations of the theory using currents and forms. This was generalized to something called spark complexes^[50] which gives rise to differential characters. It gave a theory which established the equivalence of many, quite different spark complexes. This axiomatic approach did not look at a product structure, and so it allowed the deep product of Jeff Cheeger^[51] to be carried over to quite different contexts.

Projective Hulls and the Projective Gelfand Transformation

These results^[52] suggested a complex projective analogue of a theorem of John Wermer,^[53] which was proved in a number of cases.

Potential Theory on Calibrated Manifolds

Harvey and Lawson discovered that while calibrated manifolds do not have analogues of the holomorphic, or pluriharmonic, functions that exist in the Kaehler case, they always have plurisubharmonic functions and, in fact, each manifold has a potential theory (giving rise to these functions) which is, in a sense, dual to the calibrated structure.^[54][55] The plurisubharmonics can be maximized on a domain, subject to boundary constraints, to give solutions to analogues of the Monge-Ampère Equation.

The Dirichlet problem on Riemannian manifolds

Harvey and Lawson eventually realized that this situation on calibrated manifolds has a vast generalization leading to potential theories associated to quite general fully nonlinear differential equations.^{[56][57][58][59][60]} This led to the establishment of viscosity solutions to quite general differential equations on manifolds, and to the discovery of some new geometric differential equations. For example, this gave a new Lagrangian Monge-Ampère operator on Symplectic manifolds with a Gromov metric.^[61] The authors were able to solve the inhomogeneous Dirichlet Problem for inhomogeneous equations,^[62][63] they were able to solve the complex Monge-Ampère equation on almost complex manifolds,^[64] they developed a theory of tangents to subsolutions,^[65][66] and much more.