Abstract. We prove new adjunction inequalities for embedded surfaces in fourmanifolds with non-negative self-intersection number using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó [11] using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface.

* This research has been partially supported by the SERC. * (M ; Q). A special case is where the bundle Q = P f is the mapping cylinder of a nontrivial SO(3)-bundle π : P → Σ over a Riemann surface Σ for an automorphism f : P → P. The underlying 3-manifold is the mapping cylinder M = Σ h of Σ for the diffeomorphism h : Σ → Σ induced by f. Then the flat connections on P f correspond naturally to the fixed points of the symplectomorphism φ f : M(P) → M(P)

-It was shown that abrupt changes in the large-scale structure of atmospheric flows may lead to the rapid decay of blocking. Analysis of phase diagrams made it possible to identify when sharp changes occurred in the dynamics of the system. The connection of these changes to the decay of blocking was estimated for three blocking events in the Southern Hemisphere. In addition to phase diagrams, enstrophy was used as a diagnostic tool for the analysis of blocking events. From the results of this analysis, four scenarios for the decay mechanisms were determined: (i) decay with a lack of synoptic-scale support, (ii) decay with an active role for synoptic processes, and (iii − iv) either of these mechanisms in the interaction with an abrupt change in the character of the planetary-scale flow.

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible SL2(C) representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an R-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.In this paper we produce an R-tree for any unbounded sequence of irreducible representations of the fundamental group of compact Riemannian manifolds along with an equivariant harmonic map from the universal cover to the tree. The starting point for this is to regard representations of the fundamental group as flat connections on SL 2 (C) bundles, or equivalently, as harmonic maps from the universal cover to H 3 . We first observe that the non-existence of any convergent subsequence of a given sequence of representations is equivalent to the statement that the energy of the corresponding harmonic maps along all such subsequences is unbounded. Then, after rescaling by the energy, the recent work of Korevaar and Schoen [KS2] applies: The harmonic maps pull back the metric from H 3 to a sequence of (pseudo) metrics which, under suitable conditions, have a subsequence that converges pointwise to the pull back of a metric on some non-positive curvature (N P C) space. In our case, the conditions are met thanks to the Lipschitz property of harmonic maps. We then show that the N P C space is an R-tree with length function in the projective class of the Morgan-Shalen limit. In this way, rescaling by energy turns out to be strong enough to give convergence, but at the same time is subtle enough to give a non-trivial limiting length function.The main result, which can also be thought of as an existence theorem for harmonic maps to certain trees, is:The Korevaar-Schoen limit of an unbounded sequence of irreducible SL 2 (C) representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an Rtree. The image of this map is a minimal subtree for the group action, and the projective class of the associated length function is the Morgan-Shalen limit of the sequence. For harmonic maps into hyperbolic manifolds, Corlette [C1], [C2] and Labourie [L] have shown that the existence of a harmonic map implies that the action is semi-simple. A partial analogue of this is our Theorem 5.3 in the last section. This also suggests that there is no obvious generalization for trees of Hartman's result [H] about uniqueness of harmonic maps. Further results in this direction for surface groups are discussed in [DDW].We also note that this paper contains the work of Wolf [W] for surfaces. In the case considered there, sequences of discrete, faithful SL 2 (R)-representations give equivariant harmonic maps H 2 → H 2 . The limiting tree then arises as the leaf space of the measured foliation coming from the sequence of Hopf ...

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