In this paper and in [12], [13], we study the structure of spaces, Y, which arepointed Gromov-Hausdor limits of sequences, f(M n i �pi)g, of complete, connected Riemannian manifolds whose Ricci curvatures

The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of

Abstract. In [HMR1], Holm, Marsden, and Ratiu derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation ˙V (t)+∇U(t)V (t)−α2 [∇U(t)] t ·△U(t) = −grad p(t) where divU = 0, and V = (1 − α2△)U. In this model, the momentum V is transported by the velocity U, with the effect that nonlinear interaction between modes corresponding to length scales smaller than α is negligible. We generalize this equation to the setting of an n dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincaré equation associated with the geodesic flow of the H1 right invariant metric on Ds µ, the group of volume preserving Hilbert diffeomorphisms of class Hs. We prove that the geodesic spray is continuously differentiable from T Ds µ(M) into TTD s µ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [A]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H1 metric on Ds µ is a bounded trilinear map in the Hs topology, from which it follows that solutions to Jacobi’s equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics. 1.

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface,or,more generally,a symplectic 4-manifold with suitable metric,and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles). We show that the lagrangian homology of a Kähler surface (or of a symplectic 4-manifold) is generated by minimizing lagrangian surfaces that are branched immersions except at finitely many singular points. We precisely describe the structure of these singular points. In particular,these singular points are represented by lagrangian cones with an associated local Maslov index. Only those cones of Maslov index 1 or −1 may be area minimizing. The mean curvature of the minimizers satisfies a first-order system of partial differential equations of “Hodge-type”. 1.

This paper is a continuation of the study of some rigidity or non-existence issues discussed in [An1, x6]. The results obtained here also play a signicant role in the approach to geometrization of 3-manifolds discussed in [An4]. Let N be an oriented 3-manifold and consider the functional R 2 (g) = Z N jr g j 2 dV g ; (0.1) on the space of metrics M on N where r is the Ricci curvature and dV is the volume form. The Euler-Lagrange equations for a critical point of R 2 read rR 2 = D Dr +D 2 s 2 R r 1 2 (s jrj 2 ) g = 0; (0.2) s = 1 3 jrj 2 : (0.3) Here s is the scalar curvature, D 2 s the Hessian of s, s = trD 2 s the Laplacian, and R the action of the curvature tensor R on symmetric bilinear forms, c.f. [B, Ch.4H] for further details. The equation (0.3) is just the trace of (0.2). It is obvious from the trace equation (0.3) that there are no non-at R 2 critical metrics, i.e. solutions of (0.2)-(0.3), on compact manifolds N ; this follows immediately by integrating (0.3) over N . Equivalently, since the functional R 2 is not scale invariant in dimension 3, there are no critical metrics g with R 2 (g) 6= 0. To obtain non-trivial critical metrics in this case, one needs to modify R 2 so that it is scale-invariant, i.e. consider v 1=3 R 2 ; where v is the volume of (N; g)

Let Σ be a compact Riemann surface. Any sequence fn: Σ → M of harmonic maps with bounded energy has a “bubble tree limit ” consisting of a harmonic map f0: Σ → M and a tree of bubbles fk: S 2 → M. We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the fn converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps fn when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and {fn} is a Palais-Smale sequence for the harmonic map energy. Consider a sequence of harmonic maps fn: Σ → M from a compact Riemann surface (Σ, h) to a compact Riemannian manifold (M, g) with bounded energy (0.1) E(fn) = 1 2 Σ |dfn | 2 ≤ E0. Such a sequence has a well-known “Sacks-Uhlenbeck ” limit consisting of a harmonic map f0: Σ → M and some “bubbles ” — harmonic maps S 2 → M obtained by a renormalization process. In fact, by following the procedure introduced in [PW], one can modify the Sacks-Uhlenbeck renormalization

In the 1934 paper [L] Leray raised the question of the existence of self-similar solutions of the Navier{Stokes equations ut u +(u r)u+rp=0 div u =0

Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold: see Theorem 1.3. In [16] we proved transversality using an

Abstract. We prove that many features of Thurston’s Dehn surgery theory for hyperbolic 3-manifolds generalize to Einstein metrics in any dimension. In particular, this gives large, infinite families of new Einstein metrics on compact manifolds. 1. Introduction. In this paper, we construct a large new class of Einstein metrics of negative scalar curvature on n-dimensional manifolds M = M n, for any n ≥ 4. Einstein metrics are Riemannian metrics g of constant Ricci curvature, and we will assume the curvature is normalized as (1.1) Ricg = −(n − 1)g,

Submitted by J. L. Lions An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation. 1.