Ricci curvature, minimal volumes, and Seiberg-Witten theory

LeBrun, Claude

doi:10.1007/s002220100148

Cited by 82 publications

(105 citation statements)

References 48 publications

(86 reference statements)

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“…Thus the suprema over the space of all Riemannian metrics of Y [g] and λ g V 2/n g must precisely coincide. Now, there is a substantial literature [5,9,10,14,15,16] concerning manifolds of non-positive Yamabe invariant, and the exact value of the invariant is moreover known for large numbers of such manifolds. By virtue of Theorem A, all of these facts about Y(M ) may therefore immediately be interpreted as instead pertaining toλ(M ).…”

Section: Proposition 1 Suppose That γ Is a Conformal Class On M Whicmentioning

confidence: 99%

Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds

2006

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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.

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Section: Proposition 1 Suppose That γ Is a Conformal Class On M Whicmentioning

confidence: 99%

Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds

2006

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“…For instance, in dimension four, almost-Kähler metrics which saturate the new curvature estimates of LeBrun [63] must satisfy (a) and (b).…”

Section: Curvature Conditions With Respect To the Hermitian Connectionmentioning

confidence: 99%

The curvature and the integrability of almost-Kähler manifolds: A survey

Apostolov¹,

Drăghici²

2003

Symplectic and Contact Topology: Interactions and Perspectives

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“…If a closed manifold has negative Yamabe constant, then it cannot volume collapse with scalar curvature bounded from below; see [Sch89], [LeB01]. In particular, no such manifold can be almost nonnegatively curved.…”

Section: Introductionmentioning

confidence: 99%