Les vari�t�s de dimension 4 � signature non nulle dont la courbure est harmonique sont d'Einstein

Ricci curvature, minimal volumes, and Seiberg-Witten theory

“…Nonetheless, we can at least say the following: Proposition 5.6 Suppose that M is a complex surface of general type, and suppose that the mixed minimal volume Vol K,s (M) is actually achieved by some metric g. Then either Vol K,s (M) > 9 4 Vol s (M), or else (M, g) is a complex-hyperbolic manifold, normalized so that s ≡ −8.…”

“…Suppose that g is a metric with K + s 12 ≥ −2 and total volume V = 9 4 Vol s (M). Thus, letting X denote the minimal model of M, and choosing a spin c structure on M as in the proof of Proposition 3.2, by virtue of (11).…”

Compact self-dual Hermitian surfaces

“…The classification (up to conformal equivalence) of the compact self-dual manifolds is a very difficult problem which has been solved so far under additional curvature or topological assumptions [5,6,7,8,9,10,13,15,16,17,19,21,23,24,25,28]. The main purpose of the present paper is to obtain a classification of the compact self-dual Hermitian surfaces.…”

“…Chen [8] (cf. also J. P. Bourguignon [5] and A.Derdzinski [9]) that any compact self-dual Kähler surface is one of that listed in the cases (i)-(v) of Theorem 1 .…”

Curvature Functionals, Optimal Metrics, and the Differential Topology of 4-Manifolds

“…Recall that b + (M ) is exactly the dimension of the space of harmonic self-dual 2-forms on M . However, any self-dual 2-form ϕ on any Riemannian 4-manifold satisfies the Weitzenböck formula [13] …”