Differential Geometry

The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a collection of critical-exponent Sobolev norms and general Green's operator estimates depending only on first positive eigenvalues or the L^2 norm of the connection's curvature. Such estimates are particularly useful in the gluing construction of solutions to non-linear partial differential equations depending on a degenerating parameter, such as the approximate, reference solution in the anti-self-dual or PU(2) monopole equations. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is optimal in the sense that if a point [A] in the quotient space is known to be just L^2_1-close enough to a reference point [A_0], then the connection A can be placed in Coulomb gauge relative to the connection A_0, with all constants depending at most on the first positive eigenvalue of the covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to manifolds of arbitrary dimension to prove slice theorems which apply when the reference connection is allowed to degenerate.

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their Gauss maps. Further examples are found in conformal geometry, e.g. the curved flats obtained from isothermic surfaces and conformally flat 3-folds in the 4-sphere. Curved flats admit a 1-parameter family of deformations (spectral parameter) which enables us to make contact to integrable system theory. In fact, we give a recipe to construct curved flats (and thus the above mentioned geometric objects) from a hierarchy of finite dimensional algebraically completely integrable flows.

We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bounded morphisms and homotopy operators) to a combinatorial complex with the same coefficients. This is established by using the Witten deformation of the de Rham complex. We also prove that the de Rham complex is chain-homotopy equivalent to the spectrally truncated de Rham complex which is also finitely generated.

The quotients Y=X/conj by the complex conjugation conjX→X for complex rational and Enriques surfaces X defined over $\R$ are shown to be diffeomorphic to connected sums of $\barCP2$, whenever Y are simply connected.

Let L be a lattice in a connected Lie group. We show that besides a few exceptional cases, the deficiency of L is nonpositive.

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and deprolongation which lets us pass back and forth between certain Engel four-manifolds and contact three-manifolds. Every Engel manifold inherits a natural one-dimensional foliation. Its leaves are the fibers of the map from Engel to contact manifold, when this map exists. The foliation has a transverse contact structure and tangential real projective structure. As an application of our investigations, we show that a canonical Engel structure on real projective three-space times an interval corresponds to geodesic flow on the two-sphere, and that a subspace of its Engel deformations corresponds to the space of Zoll metrics on the two-sphere.

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding moduli space of irreducible representations are quadratic. In the course of our proof we exhibit a differential graded Lie algebra of which reflects our deformation problem.

Using a flow first introduced by J.P. Anderson, we obtain some existence theorems for harmonic maps from a noncompact complete Riemannian manifold into a complete Riemannian manifold. In particular, we prove as a corollary a recent result of Hardt and Wolf stating that any quasisymmetric map of the sphere that is sufficiently close to the identity can be extended to a quasiconformal harmonic diffeomorphism of the hyperbolic ball. This version contains a much simpler proof than the first version.

We define extensions of the L 2 -analytic invariants of closed manifolds, called delocalized L 2 -invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spectrum of an odd-dimensional hyperbolic manifold can be recovered from its delocalized L 2 -analytic torsion. There are technical convergence questions.

This is a note for the conference proceedings Topological and Geometrical Problems related to Quantum Field Theory. We summarize our joint work with Dai about eta invariants on manifolds with boundary. Then we apply these results to prove the curvature and holonomy formulas for the natural connection on the determinant line bundle of a family of Dirac operators. These were originally proved by Bismut and the author--the proofs here are much simpler.