We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta [5, 4], we compute these invariants in many cases that were previously intractable. In particular, we are now able to calculate the Yamabe invariant for many connected sums of complex surfaces.

Abstract. In his study of Ricci flow, Perelman introduced a smooth-manifold invariant calledλ. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals +∞ whenever the Yamabe invariant is positive. Mathematics Subject Classification (2000). Primary 53C21; Secondary 58J50.Keywords. Scalar curvature, Ricci flow, conformal geometry, Perelman invariant, Yamabe problem.Let M be a smooth compact manifold of dimension n ≥ 3. Perelman's celebrated work on Ricci flow [12,13] led him to consider the functional which associates to every Riemannian metric g the least eigenvalue λ g of the elliptic operator 4∆ g + s g , where s g denotes the scalar curvature of g, and ∆ = d * d = −∇ · ∇ is the positive-spectrum Laplace-Beltrami operator associated with g. In other words, λ g can be expressed in terms of Raleigh quotients aswhere the infimum is taken over all smooth, real-valued functions u on M .One of Perelman's remarkable observations is that the scale-invariant quantity λ g V 2/n g is non-decreasing under the Ricci flow, where V g = M dµ g denotes the total volume of (M, g). This led him to consider the differential-topological invariant obtained by taking the supremum of this quantity over the space of all Riemannian metrics [13,6]: * Supported in part by NSF grant DMS-0604735.

Abstract. We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant [4,3], in conjunction with curvature estimates previously proved by the second author [17]. These methods also allow one to easily construct many examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist.

Let M be a closed Riemannian manifold with a family of Riemannian metrics g ij (t) evolving by a geometric flow ∂ t g ij = −2S ij , where S ij (t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equationwhere a and γ are constants and S = g ij S ij is the trace of S ij . Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover lead to new Harnack inequalities for a variety of geometric flows 1 .