Gromov–Lawson

Gromov–Lawson.

The Gromov–Lawson results are foundational theorems in differential geometry and geometric topology that characterize when a smooth closed manifold admits a Riemannian metric of positive scalar curvature. Developed by Mikhail Gromov and H. Blaine Lawson Jr. in the late 1970s and 1980s, the work connects ideas from Surgery theory, the Atiyah–Singer index theorem, and the topology of spin manifolds to produce existence and nonexistence criteria. These results interact with subsequent developments by Jonathan Rosenberg, Stefan Stolz, and John L. Carr and influence research by Perelman, Cheeger, M. Gromov collaborators, and groups studying scalar curvature problems.

History and Background

Gromov–Lawson arose in the context of efforts by Kazdan and Warner on curvature prescription and by index-theoretic obstructions developed via the Atiyah–Singer index theorem and work of Lichnerowicz, Hirzebruch, and Rosenberg. Influences include surgical classification techniques from Kervaire–Milnor, William Browder, and C. T. C. Wall, and analytic methods from Atiyah, Patodi, and Singer. The original proofs by Gromov and Lawson built on insights of Milnor, Novikov, and Hirzebruch–Riemann–Roch style index theory, synthesizing topology of spin manifolds and geometric constructions inspired by John Nash-type smoothing ideas.

Statements of the Gromov–Lawson Theorems

The main Gromov–Lawson existence theorem asserts that many high-dimensional manifolds admit metrics of positive scalar curvature after performing surgeries of codimension at least three. For closed simply connected manifolds of dimension at least five with appropriate vanishing of Â-genus or with nonobstructed spin manifold structures, a positive scalar curvature metric exists. Complementary nonexistence results use the Lichnerowicz theorem and the Atiyah–Singer index theorem to show that certain spin manifolds with nonzero Â-genus or with nontrivial alpha invariant cannot carry positive scalar curvature metrics. The Gromov–Lawson theorems are thus presented as paired existence and obstruction statements: surgery-based construction tools on one hand, index-theoretic obstructions on the other, tying into invariants studied by Nigel Hitchin and Jonathan Rosenberg.

Techniques and Proof Ideas

Gromov–Lawson techniques combine geometric constructions with topological surgery. The geometric core uses explicit metric surgery and connected-sum constructions inspired by Cheeger–Gromoll and warped product metrics similar to those in work by Karl Schwarzschild-inspired gluing. Topological inputs use surgery theory as developed by C. T. C. Wall, William Browder, and Kervaire–Milnor to reduce manifold topology while preserving control of scalar curvature. Analytical obstructions employ the Atiyah–Singer index theorem and spin Dirac operator techniques from Lichnerowicz and Hitchin to link the existence problem to topological invariants like the Â-genus and the alpha invariant from KO-theory developed further by Stefan Stolz and Jonathan Rosenberg. Later proofs and refinements used techniques from coarse geometry by John Roe and connections to operator K-theory in the work of Nigel Higson and Gennadi Kasparov.

Applications and Consequences

Consequences of Gromov–Lawson include classification results for manifolds admitting positive scalar curvature metrics and new rigidity phenomena studied by M. Gromov, Perelman, and Jeff Cheeger. Their work informed the Stolz conjecture relating positive scalar curvature to string bordism and to analytic assembly maps in the Baum–Connes conjecture context formulated by Paul Baum and Alain Connes. The methods provide tools for constructing metrics in examples studied by Richard Schoen, Shing-Tung Yau, and Michael Anderson, and they underpin nonexistence results used in index-theory studies by Jonathan Rosenberg and Nigel Higson. Impact extends to problems in spin geometry studied by Berline–Getzler–Vergne and to coarse-geometric obstructions examined by Rufus Willett.

Related Results and Extensions

Related developments include Schoen–Yau minimal hypersurface techniques by Richard Schoen and Shing-Tung Yau that produce complementary obstructions in low dimensions, and the Rosenberg index refinement connecting positive scalar curvature to C*-algebraic invariants studied by Baum–Connes researchers. Extensions involve the Stolz positive scalar curvature sequence and work by Thomas Schick, Joachim Ebert, and Carsten Wahl on moduli of metrics, as well as coarse index techniques from John Roe and analytic approaches by Kai Chang and Weiping Zhang. Contemporary research ties Gromov–Lawson themes to large-scale geometry by M. Gromov and to scalar curvature questions in the context of the Novikov conjecture and the Baum–Connes conjecture pursued by Gennadi Kasparov, Alain Connes, and Guoliang Yu.

Category:Riemannian geometry