Einstein Metrics, Harmonic Forms, & Symplectic Four-Manifolds Claude LeBrun Stony Brook University Sardegna, 2014/9/20 1 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. 2 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. 3 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. \. the greatest blunder of my life!" | A. Einstein, to G. Gamow 4 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. 5 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. As punishment . 6 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. λ called Einstein constant. 7 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. λ called Einstein constant. Has same sign as the scalar curvature j ij s = rj = R ij: 8 Definition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature| i.e. r = λh for some constant λ 2 R. λ called Einstein constant. Has same sign as the scalar curvature j ij s = rj = R ij: volg(B"(p)) "2 4 n = 1 − s + O(" ) cn" 6(n+2) ............. ........... ........... ....... ....... ..... ..... ........... ..... .......... .... .... ..... ... ... ..... ... .. ..... .. .. ......".... ........ ....... .. .. ... ....... ..... .... ... ..... ...... ... .... ........ ........ .................. .................. .. .......................... .. ... ... ... ... .... .... .... r .... ..... ..... ..... ...... ....... ....... ................................ 9 Geometrization Problem: 10 Geometrization Problem: Given a smooth compact manifold Mn, can one 11 Geometrization Problem: Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces? 12 Geometrization Problem: Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces? When n =3, answer is \Yes." 13 Geometrization Problem: Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces? When n =3, answer is \Yes." Proof by Ricci flow. Perelman et al. 14 Geometrization Problem: Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces? When n =3, answer is \Yes." Proof by Ricci flow. Perelman et al. Perhaps reasonable in other dimensions? 15 Recognition Problem: 16 Recognition Problem: Suppose Mn admits Einstein metric h. 17 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? 18 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? 19 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? When n =3, h has constant sectional curvature! 20 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? When n =3, h has constant sectional curvature! 3 3 3 So M has universal cover S , R , H ... 21 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? When n =3, h has constant sectional curvature! 3 3 3 So M has universal cover S , R , H ... But when n ≥ 5, situation seems hopeless. 22 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? When n =3, h has constant sectional curvature! 3 3 3 So M has universal cover S , R , H ... But when n ≥ 5, situation seems hopeless. fEinstein metrics on Sng=∼ is highly disconnected. 23 Recognition Problem: Suppose Mn admits Einstein metric h. What, if anything, does h then tell us about M? Can we recognize M by looking at h? When n =3, h has constant sectional curvature! 3 3 3 So M has universal cover S , R , H ... But when n ≥ 5, situation seems hopeless. fEinstein metrics on Sng=∼ is highly disconnected. When n ≥ 4, situation is more encouraging. 24 Moduli Spaces of Einstein metrics 25 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) 26 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 27 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 4 4 M = T ;K 3; H =Γ; CH2=Γ: 28 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 4 4 M = T ;K 3; H =Γ; CH2=Γ: Berger, Hitchin, Besson-Courtois-Gallot, L. 29 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 4 4 M = T ;K 3; H =Γ; CH2=Γ: Berger, Hitchin, Besson-Courtois-Gallot, L. 30 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 4 4 M = T ;K 3; H =Γ; CH2=Γ: Berger, Hitchin, Besson-Courtois-Gallot,L. 31 Moduli Spaces of Einstein metrics + E (M) = fEinstein hg=(Diffeos × R ) Known to be connected for certain4-manifolds: 4 4 M = T ;K 3; H =Γ; CH2=Γ: Berger, Hitchin, Besson-Courtois-Gallot, L. 32 Four Dimensions is Exceptional 33 Four Dimensions is Exceptional When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature. 34 Four Dimensions is Exceptional When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature. There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds. 35 Four Dimensions is Exceptional When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature. There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds. But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces. 36 Four Dimensions is Exceptional When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature. There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds. But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces. One key question: 37 Four Dimensions is Exceptional When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature. There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds. But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces. One key question: Does enough rigidity really hold in dimension four to make this a genuine geometrization? 38 Symplectic 4-manifolds: 39 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. 40 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. 41 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. 42 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics. 43 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics. Some Suggestive Questions. If (M4;!) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to !)? What if we also require λ ≥ 0? 44 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics. Some Suggestive Questions. If (M4;! ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to !)? What if we also require λ ≥ 0? 45 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics. Some Suggestive Questions. If (M4;! ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to !)? What if we also require λ ≥ 0? 46 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics. Some Suggestive Questions. If (M4;! ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to !)? What if we also require λ ≥ 0? 47 Symplectic 4-manifolds: A laboratory for exploring Einstein metrics. K¨ahlergeometry is a rich source of examples. If M admits a K¨ahlermetric, it of course admits a symplectic form !. On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.