Honours Projects for 2009

This project concerns an important conjecture which appears to have been solved recently: It concerns minimal surfaces, in particular minimal submanifolds of spheres. It combines PDE and geometry, though the PDE required is not very much. It does involve some basic spectral theory for the Laplacian on a Riemannian manifold. There are many examples known of submanifolds in spheres which are minimal, in the sense that their mean curvature vanishes (equivalently, they are critical points for the area functional). The simplest are of course the totally geodesic submanifolds (great spheres). The next simplest is the Clifford torus, which is the product S1 × S1 in the three sphere S3 ⊂ R2 × R2. It is known that there are embedded minimal surfaces of every genus in S3 (first proved by Blaine Lawson in 1970), and there are infinitely many immersed minimal tori in S3. The spectral theory comes in with the following observation: A submanifold (of dimension n) of a sphere is minimal if and only if each component of the position vector is an eigenfunction of the Laplacian with eigenvalue n. Lawson conjectured in his 1970 paper that the only embedded minimal torus in S3 is the Clifford torus (there is a related conjecture for higher genus minimal embedded surfaces). Much later Yau conjectured that an embedded minimal submanifold of dimension n has first eigenvalue equal to n (that is, there is no eigenvalue less than n). The latter is false if the submanifold is immersed rather than embedded. The project would aim to discuss: • the construction of minimal surfaces by Lawson; • partial results towards the Yau conjecture by Choi and Wang in 1983 (the first eigenvalue is at least n/2) — it is tempting to think that further progress could be made here by modifying their methods; • the result by Montiel and Ros from 1985 that the only minimal torus in S3 with first eigenvalue equal to 2 is the Clifford torus (the latter implies that the Lawson conjecture is equivalent to the Yau conjecture for tori); • the recent (2007) proof by Pimentel of the Lawson conjecture (this is not yet published so the idea would be to read the preprint carefully and try to decide whether it is correct). References: (1) F. Pimentel, A proof of the Lawson conjecture for minimal tori embedded in S3, arXiv:math/0703136v2 [math.DG] (2) H. B. Lawson, Jr., Complete minimal surfaces in S3, Annals of Mathematics 92 (1970), 335-374 (available online)