## Ricci curvature, minimal volumes, and Seiberg-Witten theory

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Venue: | Invent. Math |

Citations: | 27 - 2 self |

### BibTeX

@ARTICLE{Lebrun_riccicurvature,,

author = {Claude Lebrun},

title = {Ricci curvature, minimal volumes, and Seiberg-Witten theory},

journal = {Invent. Math},

year = {},

volume = {145},

pages = {279--316}

}

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### Abstract

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics. 1

### Citations

579 |
Einstein Manifolds
- Besse
- 1987
(Show Context)
Citation Context ...ts Ricci curvature r is a constant multiple of the metric: r = λg. Not every smooth compact oriented 4-manifold M admits such a metric. Indeed, a well-known necessary condition is that M must satisfy =-=[19, 42, 6]-=- the Hitchin-Thorpe inequality 2χ(M) ≥ 3|τ(M)|, where again χ and τ denote the signature and Euler characteristic. Indeed, this is an immediate consequence of (1), since the Einstein condition may be ... |

355 |
de Ven, Compact Complex Surfaces
- Barth, Peters, et al.
- 1984
(Show Context)
Citation Context ...j + k whereas 1 8π2 Ir(X#(j + k)CP2) =c2 1 (X)+(j + k). Thus these manifolds have unequal invariants Ir, as claimed. Note that pairs (X, ˜ X) of the above kind are as common as garden weeds; cf. e.g. =-=[5]-=-. For example, if X is of general type, we can always find a simply connected properly elliptic surface ˜ X with the same geometric genus as X. 5 Sectional Curvature and Volume In the previous section... |

177 |
Topology of 4-Manifolds
- Freedman, Quinn
- 1990
(Show Context)
Citation Context ... −72 �≡ 0mod 16, so, by Rochlin’s theorem, neither is spin. Thus M and N have isomorphic intersection forms by the Minkowski-Hasse classification, and are therefore homeomorphic by Freedman’s theorem =-=[13]-=-. However, N has ample canonical line bundle, and so admits a Kähler-Einstein metric by Yau’s theorem. Thus, although M and N are homeomorphic, one admits Einstein metrics, while the other doesn’t. If... |

170 |
Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der
- Aubin
- 1982
(Show Context)
Citation Context ...choice of u, we therefore attempt to minimize � M F(u) = (6|du|2 + Sg0u2 ) dµg0 �� M u4 dµg0 on the positive sector of the unit sphere in the Sobolev space L2 1(M, g0). Yamabe’s ansatz for doing this =-=[46, 4]-=- is to minimize the functionals � M Fɛ(u) = (6|du|2 + Sg0u2 ) dµg0 �� M u4−ɛ dµg0 � 1/2 � 1/(2− ɛ 2 ) and then take the limit of the minimizers as ɛ ↘ 0. For each ɛ>0, the existence of minimizers foll... |

109 |
Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini
- Schoen
- 1987
(Show Context)
Citation Context ...) � � n(n − 1) ≥−1 � , where our conventions have been chosen so that, by definition, VolK(M) ≥ Vols(M) ≥ 0. One key advantage of the latter definition is that the mature theory of the Yamabe problem =-=[36]-=- allows this minimal volume to be reinterpreted as Vols(M) =inf g � M � � � s− � � � �n(n − 1) � n/2 dµg = � � |Y (M)| n(n−1) 0, Y(M) ≥ 0 � n/2 , Y(M) ≤ 0, where s− = min(s, 0), and where Y (M) is the... |

103 |
Equations du Type Monge-Ampère sur les Variétés Kiihleriennes compactes
- Aubin
- 1976
(Show Context)
Citation Context ...+| 2 � gε dµgε ≤ 1 2 c21(X). To do this, let ˇ X be the pluri-canonical model of X, which carries a KählerEinstein orbifold metric ˇg by an immediate generalization [21, 44] of the Aubin/Yau solution =-=[3, 47]-=- of the c1 < 0 case of the Calabi conjecture. Now � ˇX � 2 sˇg 1 + 24 2 |W+| 2 � ˇg dµˇg = 1 2 c21 ( ˇX) = 1 2 c21 (X) because |W+| 2 ≡ s 2 /24 for any Kähler surface, whereas a Kähler-Einstein surfac... |

100 |
Entropies et rigidités des espaces localement symétriques de courbure strictement négative, GAFA 5
- Besson, Courtois, et al.
- 1995
(Show Context)
Citation Context ...ume among all metrics of sectional curvature K ≥−1. For real-hyperbolic manifolds the answer is affirmative, and indeed this holds in all dimensions; however, the proof, due to Besson-Courtois-Gallot =-=[7]-=-, depends not only on a remarkable inequality concerning volume entropy, but also on an optimal application of Bishop’s inequality. For complex-hyperbolic manifolds, the latter breaks down, and the qu... |

89 |
The classification of simply connected manifolds of positive scalar curvature
- Gromov, Lawson
- 1980
(Show Context)
Citation Context ...ial, or else will be completely impossible to calculate in practice. This suspicion would seem to be vindicated by a recent result of Petean [34], who, building upon the earlier work of Gromov-Lawson =-=[17]-=- and Stolz [38], showed that Is(M) = 0 for any simply connected n-manifold, n ≥ 5. Dimension 4, however, turns out to be radically different. Seiberg-Witten theory naturally leads to non-trivial lower... |

76 | Immersed spheres in 4-manifolds and the immersed Thom conjecture
- Fintushel, Stern
- 1995
(Show Context)
Citation Context ...levant copies of H 2 (CP2, Z) so that [c1(X)] + · Ej ≤ 0, j = 1, . . ., k. Let β1, . . .,βℓ be closed curves in M which generate of the fundamental groups of the ℓ relevant copies of S 1 × S 3 . Then =-=[14, 32]-=- there is a spin c structure on M with SWc(M, [β1], · · ·, [βℓ]; H) ̸= 0 and c1(L) = c1(X) − k∑ Ej. j=1 Thus c is a monopole class of (M, H). But one then has (c + 1 )2 ( = [c1(X)] + )2 k∑ + j=1 = ([c... |

65 |
Compact four-dimensional Einstein manifolds
- Hitchin
- 1974
(Show Context)
Citation Context ...ts Ricci curvature r is a constant multiple of the metric: r = λg. Not every smooth compact oriented 4-manifold M admits such a metric. Indeed, a well-known necessary condition is that M must satisfy =-=[19, 42, 6]-=- the Hitchin-Thorpe inequality 2χ(M) ≥ 3|τ(M)|, where again χ and τ denote the signature and Euler characteristic. Indeed, this is an immediate consequence of (1), since the Einstein condition may be ... |

54 |
Simply connected manifolds of positive scalar curvature
- Stolz
- 1992
(Show Context)
Citation Context ...ll be completely impossible to calculate in practice. This suspicion would seem to be vindicated by a recent result of Petean [34], who, building upon the earlier work of Gromov-Lawson [17] and Stolz =-=[38]-=-, showed that Is(M) = 0 for any simply connected n-manifold, n ≥ 5. Dimension 4, however, turns out to be radically different. Seiberg-Witten theory naturally leads to non-trivial lower bounds for Is ... |

53 | Four-manifolds without Einstein metrics
- LeBrun
- 1996
(Show Context)
Citation Context ...nsion 4, however, turns out to be radically different. Seiberg-Witten theory naturally leads to non-trivial lower bounds for Is which, amazingly, are often sharp. Using this, the author has elsewhere =-=[28, 29]-=- computed Is(M) for all complex surfaces M with even first Betti number; it turns out that Is(M) is positive exactly for the surfaces of general type, and for these it is given by the formula Is(M) =3... |

40 |
The Seiberg-Witten Invariants and Symplectic
- Taubes
- 1994
(Show Context)
Citation Context ...is apparently saturated by a larger class of almost-Kähler metrics. This reflects an under-utilized aspect of Taubes’ construction of solutions of the Seiberg-Witten equations on symplectic manifolds =-=[40, 41]-=-, and would appear to be a promising avenue for further research. These estimates also give one new obstructions to the existence for Einstein metrics. Recall that that a smooth Riemannian metric g is... |

34 | Algebraic surfaces of general type with small c - Horikawa - 1976 |

29 | Polarized 4-manifolds, extremal K˙ahler metrics, and Seiberg-Witten
- LeBrun
- 1995
(Show Context)
Citation Context ... H ⊂ H 2 (M, R); such metrics will be said to be H-adapted. Assuming there is at least one H-adapted metric, we will then say that H is a polarization of M, andcallthepair(M, H) apolarized 4-manifold =-=[27]-=-. Notice that the restriction of the intersection pairing ⌣: H 2 (M, R) × H 2 (M, R) → R to H is then positive definite, and that H ⊂ H 2 is maximal among subspaces with this property. Let c be a spin... |

27 | Kodaira dimension and the Yamabe problem
- LeBrun
- 1999
(Show Context)
Citation Context ...precisely those of general type; for these, the result is just the ℓ =0case of Theorem 4.3. On the other hand, the analogous assertion for Kodaira dimensions 0 and 1 immediately follows from the fact =-=[29]-=- that any elliptic surface admits sequences of metrics for which Vol(M, g) ↘ 0, but for which s and |W+| are uniformly bounded. In particular, in view of (14), this gives a satisfying new proof of the... |

26 |
The curvature of 4-dimensional Einstein spaces, Global Analysis
- Singer, Thorpe
- 1969
(Show Context)
Citation Context ...oriented 4-manifold decompose as Λ 2 =Λ + ⊕ Λ − , 2swhere Λ ± is the (±1) eigenspace of Hodge star operator ⋆. Thinking of the curvature tensor R as a linear map Λ2 → Λ2 , we thus get a decomposition =-=[37]-=- ⎛ ⎞ ⎜ R = ⎜ ⎝ W+ + s 12 B B ∗ W− + s 12 into irreducible pieces. Here the self-dual and anti-self-dual Weyl curvatures W± are the trace-free pieces of the appropriate blocks. The scalar curvature s i... |

20 |
Volume and Bounded
- Gromov
- 1982
(Show Context)
Citation Context ...-manifold with 2χ(M) =−3τ(M) can admit an Einstein metric only if its a K3 surface. Examples of 4-manifolds with |τ(M)|/χ(M) < 2/3 which do not admit Einstein metrics were first constructed by Gromov =-=[15]-=-, using his simplicial volume invariant; however, this method only works in the presence of an infinite fundamental group. In [28], simply connected examples were first constructed, using Seiberg-Witt... |

20 |
The Seiberg-Witten and Gromov
- Taubes
- 1995
(Show Context)
Citation Context ...is apparently saturated by a larger class of almost-Kähler metrics. This reflects an under-utilized aspect of Taubes’ construction of solutions of the Seiberg-Witten equations on symplectic manifolds =-=[40, 41]-=-, and would appear to be a promising avenue for further research. These estimates also give one new obstructions to the existence for Einstein metrics. Recall that that a smooth Riemannian metric g is... |

20 |
Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type
- Tsuji
- 1988
(Show Context)
Citation Context ...ε→0 + 1 4π2 � � 2 sgε 1 + M 24 2 |W+| 2 � gε dµgε ≤ 1 2 c21(X). To do this, let ˇ X be the pluri-canonical model of X, which carries a KählerEinstein orbifold metric ˇg by an immediate generalization =-=[21, 44]-=- of the Aubin/Yau solution [3, 47] of the c1 < 0 case of the Calabi conjecture. Now � ˇX � 2 sˇg 1 + 24 2 |W+| 2 � ˇg dµˇg = 1 2 c21 ( ˇX) = 1 2 c21 (X) because |W+| 2 ≡ s 2 /24 for any Kähler surface... |

19 |
An ansatz for almost-Kahler, Einstein 4-manifolds
- Armstrong
(Show Context)
Citation Context ... seen, however, whether such metrics can ever be strictly almost-Kähler, in the sense that the almostcomplex structure J fails to be integrable. For example, a beautiful recent result of J. Armstrong =-=[2]-=- asserts the non-existence of compact, Einstein, strictly almost-Kähler 4-manifolds on which ω is everywhere an eigenvector of W+. Thus: Corollary 2.5 Suppose that g is an Einstein metric which satura... |

18 | The Genus of Embedded Surfaces
- Kronheimer, Mrowka
- 1994
(Show Context)
Citation Context ...) if the Seiberg-Witten equations (2–3) have a solution for every H-adapted metric g. This definition will be useful in practice, of course, only because of the existence of Seiberg-Witten invariants =-=[24, 45]-=- . For example, if a spin c structure satisfies [c1(L)] 2 =(2χ +3τ)(M), and if c + 1 �= 0 relative to the polarization H = H + g , then the Seiberg-Witten invariant SWc(M; H) can be defined as the num... |

18 | Einstein metrics and Mostow - LeBrun - 1995 |

16 | Surgery and the Yamabe invariant
- Petean, Yun
- 1999
(Show Context)
Citation Context ...bout two-thirds of the way to the present result. The fact that S1 × S3 handles can be added to a 4-manifold without losing Seiberg-Witten control of the scalar curvature was first observed by Petean =-=[33]-=-, although Seiberg-Witten theory only enters his result in an indirect manner. The search for a direct Seiberg-Witten explanation of this phenomenon then led the author and del Rio [31, 11] to a disco... |

16 | The Yamabe invariant of simply connected manifolds
- Petean
(Show Context)
Citation Context ...t invariants with such soft definitions will either be trivial, or else will be completely impossible to calculate in practice. This suspicion would seem to be vindicated by a recent result of Petean =-=[34]-=-, who, building upon the earlier work of Gromov-Lawson [17] and Stolz [38], showed that Is(M) = 0 for any simply connected n-manifold, n ≥ 5. Dimension 4, however, turns out to be radically different.... |

15 |
An obstruction to the existence of Einstein metrics on 4-manifolds
- Sambusetti
- 1998
(Show Context)
Citation Context ...ks in the presence of an infinite fundamental group. In [28], simply connected examples were first constructed, using Seiberg-Witten estimates for the scalar curvature. Shortly thereafter, Sambusetti =-=[35]-=- showed that the entropy estimates of Besson-Courtois-Gallot [7] allow one to construct examples of arbitrary Euler characteristic and signature, but again with huge fundamental group; see Petean [33]... |

15 |
The Existence of anti-self-dual conformal structures
- Taubes
- 1992
(Show Context)
Citation Context ...ed Riemannian 4-manifold is said to be anti-self-dual if it satisfies the W+ ≡ 0, and that this condition is conformally invariant. Compact anti-self-dual manifolds exist in profusion. Indeed, Taubes =-=[39]-=- has shown that for any smooth compact orientable X 4 , there is an integer k0 such that M = X#kCP2 admits metrics with W+ =0 provided that k ≥ k0. In particular, if we blow up a symplectic 4-manifold... |

12 | Local rigidity of certain classes of almost Kähler 4manifolds
- Apostolov, Armstrong, et al.
(Show Context)
Citation Context ...ctor of W+. Thus: Corollary 2.5 Suppose that g is an Einstein metric which saturates (8), (10), or (11). Then g is actually Kähler-Einstein. Similarly, a recent result of Apostolov-Armstrong-Drăghici =-=[1]-=- implies that an almost-Kähler metric saturating (10) is Kähler iff its Ricci tensor is J-invariant. 3 Einstein Metrics Our first application of the preceding Weyl estimates will be to prove new non-e... |

10 | Einstein metrics and smooth structures
- Kotschick
- 1998
(Show Context)
Citation Context ... 3.4 is the direct descendant of an analogous result in [28], 20swhere, using only scalar curvature estimates, a similar conclusion was proved for k ≥ 2 3c21(X). It was later pointed out by Kotschick =-=[22]-=- that such a result alone suffices to imply the existence of homeomorphic pairs consisting of an Einstein manifold and a 4-manifold which does not admit Einstein metrics; however, the examples that ar... |

9 |
Einstein-Kähler V -metrics on open Satake V -surfaces with isolated quotient singularities
- Kobayashi
- 1985
(Show Context)
Citation Context ...ε→0 + 1 4π2 � � 2 sgε 1 + M 24 2 |W+| 2 � gε dµgε ≤ 1 2 c21(X). To do this, let ˇ X be the pluri-canonical model of X, which carries a KählerEinstein orbifold metric ˇg by an immediate generalization =-=[21, 44]-=- of the Aubin/Yau solution [3, 47] of the c1 < 0 case of the Calabi conjecture. Now � ˇX � 2 sˇg 1 + 24 2 |W+| 2 � ˇg dµˇg = 1 2 c21 ( ˇX) = 1 2 c21 (X) because |W+| 2 ≡ s 2 /24 for any Kähler surface... |

8 |
The ”total scalar curvature” as a symplectic invariant and related results
- Blair
- 1991
(Show Context)
Citation Context ...4πc1(L) · [ω], since by assumption equality holds in (8). It thus follows that � 1 2 (s + s∗ )dµ ≥ 4πc1(L) · [ω], with equality iff ω everywhere belongs to the lowest eigenspace of W+. However, Blair =-=[8]-=- has shown that � 1 2 (s + s∗ )dµ =4πc1 · [ω] (12) for any almost-Kähler 4-manifold. We thus conclude that ω is everywhere in the lowest eigenspace of W+. If we instead have equality in (11) or (10), ... |

8 |
Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein
- Bourgignon
- 1981
(Show Context)
Citation Context ... V = Vol(M, g) = � |( 2 3 sg +2wg)−| 3 �2/3 dµ ≥ 32π 2 (c + 1 ) 2 , (8) M dµg is the total volume of (M, g). Proof. Any self-dual 2-form ψ on any oriented 4-manifold satisfies the Weitzenböck formula =-=[9]-=- (d + d ∗ ) 2 ψ = ∇ ∗ ∇ψ − 2W+(ψ,·)+ s 3 ψ, where W+ is the self-dual Weyl tensor. It follows that � � (−2W+)(ψ,ψ) ≥ (− s 3 )|ψ|2 � dµ − |∇ψ| 2 dµ, M M so that � − 2w|ψ| M 2 � ≥ (− M s 3 )|ψ|2 � dµ − ... |

7 |
Remarks concerning the conformal deformation of metrics to constant scalar curvature
- Trudinger
- 1968
(Show Context)
Citation Context ...hich solves 6∆g0u + Sg0u = cɛu 3−ɛ , (9) where cɛ is the infimum of Fɛ on the positive sector of the unit sphere of L2 1(M, g0). The convergence as ɛ ↘ 0 then follows from an observation of Trudinger =-=[43]-=-. Namely, since S ≤ s for any metric, and since there is nothing to prove if S = s, our hypothesis that there is no metric of positive scalar curvature in γ allows us to assume that that c0 =infcɛ < 0... |

6 |
On the geography of surfaces – simply connected minimal surface of positive index
- Chen
- 1987
(Show Context)
Citation Context ...al number with 8 ≤|q| < 1. Then there are 23 smooth, compact, simply connected 4-manifolds with τ/χ = q which do not admit Einstein metrics. Proof. For m any even integer bigger than 17 million, Chen =-=[10]-=- has constructed a simply connected minimal complex surface X of general type (in fact, a hyperelliptic fibration) with τ(X) =m and χ(X) =4m. If we now blow up such a surface at k points, where k ≥ 11... |

6 |
On the Ricci-Curvature of a Complex Kähler
- Yau
- 1978
(Show Context)
Citation Context ...+| 2 � gε dµgε ≤ 1 2 c21(X). To do this, let ˇ X be the pluri-canonical model of X, which carries a KählerEinstein orbifold metric ˇg by an immediate generalization [21, 44] of the Aubin/Yau solution =-=[3, 47]-=- of the c1 < 0 case of the Calabi conjecture. Now � ˇX � 2 sˇg 1 + 24 2 |W+| 2 � ˇg dµˇg = 1 2 c21 ( ˇX) = 1 2 c21 (X) because |W+| 2 ≡ s 2 /24 for any Kähler surface, whereas a Kähler-Einstein surfac... |

5 | Four-dimensional Einstein manifolds and beyond - LeBrun |

2 |
Four-manifolds with δW
- Gursky
(Show Context)
Citation Context ... it is an L 3 , rather than an L 2 , estimate. Fortunately, however, we will be able to extract an L 2 estimate by means of a conformal rescaling trick, the general idea of which is drawn from Gursky =-=[18]-=-: Lemma 2.2 Let (M, γ) be a compact oriented 4-manifold with a fixed smooth conformal class of Riemannian metrics. Suppose, moreover, that γ does not contain a metric of positive scalar curvature. The... |

2 |
Instantons Gravitationelles et Singularités de
- Kronheimer
- 1986
(Show Context)
Citation Context ...etrics on the minimal model X by modifying the orbifold metric ˇg, without introducing substantial amounts of extra volume, Ricci curvature, or self-dual Weyl curvature. Indeed, for each Γj there are =-=[23]-=- gravitational instanton metrics on the minimal resolution of C 2 /Γj, which is precisely obtained by replacing the origin with a set of (−2)curves as above. These gravitational instanton metrics are ... |

2 |
Some Remarks on the Gauss-Bonnet
- Thorpe
- 1969
(Show Context)
Citation Context ...ts Ricci curvature r is a constant multiple of the metric: r = λg. Not every smooth compact oriented 4-manifold M admits such a metric. Indeed, a well-known necessary condition is that M must satisfy =-=[19, 42, 6]-=- the Hitchin-Thorpe inequality 2χ(M) ≥ 3|τ(M)|, where again χ and τ denote the signature and Euler characteristic. Indeed, this is an immediate consequence of (1), since the Einstein condition may be ... |

1 |
Seiberg-Witten Invariants of Non-Simple Type and Einstein Metrics, e-print math.DG/0002243, available at http://xxx.lanl.gov
- Guerra
(Show Context)
Citation Context ...d equality is possible only if k = ℓ =0. Proof. The proof is a direct extension of the computations in [30], although for ℓ>0 we now use the holonomy-constrained Seiberg-Witten invariant of [32]; cf. =-=[11, 31]-=-. 17sNow notice that we may assume that (2χ +3τ)(X) > 0, since otherwise there is nothing to prove. This has the pleasant consequence that any Seiberg-Witten invariant of X is independent of polarizat... |

1 |
On Some 4-Dimensional Almost Kähler
- Drăghici
- 1995
(Show Context)
Citation Context ...stant, ω an eigenvector of W+ at each point, and c + 1 ∝ [ω]. Since c1 = c + 1 , the almost-complex structure satisfies c1 ∝ [ω], so that our almost-Kähler manifold is monotonic in the terminology of =-=[12]-=-. Since equality holds in (22), we also have K + s 12 s w ≡ + 6 2 , 36sand this, for starters, tells us that W− ≡ 0 by (21). Inspection of (20) then shows that B ∗ (ω) ≡ 0, so that ω is in fact an eig... |

1 |
On the Deformation of Riemannian Structures
- Yamabe
- 1960
(Show Context)
Citation Context ...choice of u, we therefore attempt to minimize � M F(u) = (6|du|2 + Sg0u2 ) dµg0 �� M u4 dµg0 on the positive sector of the unit sphere in the Sobolev space L2 1(M, g0). Yamabe’s ansatz for doing this =-=[46, 4]-=- is to minimize the functionals � M Fɛ(u) = (6|du|2 + Sg0u2 ) dµg0 �� M u4−ɛ dµg0 � 1/2 � 1/(2− ɛ 2 ) and then take the limit of the minimizers as ɛ ↘ 0. For each ɛ>0, the existence of minimizers foll... |