Communications in Contemporary Mathematics

We will show that in the conformal class of the standard metric g_Sⁿ on Sⁿ, the scaling invariant functional maximizes at g_Sⁿ when n is odd and or . For n odd and , g_Sⁿ is not stable and the functional has no local maximizer. Here Q_2m,g is the 2mth order Q-curvature.

We introduce Morse homology for manifolds with boundary, and construct relative Morse homology sequences.

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Q_k of [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Q_k give rise to conformally invariant quadratic forms Θ_k on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Q_k operators yield a map from conformal structures to Lagrangian subspaces of the direct sum H^k ⊕ H_k (where H_k is the dual of the de Rham cohomology space H^k); in an appropriate sense this generalizes the period map. We couple the Q_k operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Q_k-operators the Q-curvature prescription problem.

An open question is the possibility of defining an integer valued SU(3)-Casson invariant for integral homology 3-spheres which involves counting the irreducible portion of the non-degenerate (perturbed) moduli space of flat SU(3)-connections plus counter-terms associated to only the non-degenerate (perturbed) reducible portion of the moduli space. The obstruction to this is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg–Witten theory also has a similiar obstruction but arising from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg–Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.

Using the spherical trip theorem, we present a new intrinsic proof of Gromoll–Grove diameter rigidity theorem: "If a simply-connected Riemannian manifold has sectional curvature ≥ 1 and diameter , then either it is homeomorphic to a sphere, or it is isometric to one of classic projective spaces".

We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [15] are obtained. The Fourier transforms of solutions with respect to both variables x and t are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results.