Hitchin–Thorpe inequality

In differential geometry the Hitchin–Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric.

Statement of the Hitchin–Thorpe inequality

Let M be a closed, oriented, four-dimensional smooth manifold. If there exists a Riemannian metric on M which is an Einstein metric, then

$\chi (M)\ge \frac{3}{2}|\tau (M)|,$

where Template:Math is the Euler characteristic of Template:Mvar and Template:Math is the signature of Template:Mvar.

This inequality was first stated by John Thorpe in a footnote to a 1969 paper focusing on manifolds of higher dimension. Nigel Hitchin then rediscovered the inequality, and gave a complete characterization of the equality case in 1974; he found that if Template:Math is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of Template:Mvar is zero; if the sectional curvature is not identically equal to zero, then Template:Math is a Calabi–Yau manifold whose universal cover is a K3 surface.

Template:AnchorAlready in 1961, Marcel Berger showed that the Euler characteristic is always non-negative.

Proof

Let Template:Math be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point Template:Mvar of Template:Mvar, there exists a Template:Math-orthonormal basis Template:Math of the tangent space Template:Math such that the curvature operator Template:Math, which is a symmetric linear map of Template:Math into itself, has matrix

$\left(\begin{array}{cccccc}{\lambda }_1& 0& 0& {\mu }_1& 0& 0\\ 0& {\lambda }_2& 0& 0& {\mu }_2& 0\\ 0& 0& {\lambda }_3& 0& 0& {\mu }_3\\ {\mu }_1& 0& 0& {\lambda }_1& 0& 0\\ 0& {\mu }_2& 0& 0& {\lambda }_2& 0\\ 0& 0& {\mu }_3& 0& 0& {\lambda }_3\end{array}\right)$

relative to the basis Template:Math. One has that Template:Math is zero and that Template:Math is one-fourth of the scalar curvature of Template:Mvar at Template:Mvar. Furthermore, under the conditions Template:Math and Template:Math, each of these six functions is uniquely determined and defines a continuous real-valued function on Template:Mvar.

According to Chern-Weil theory, if Template:Mvar is oriented then the Euler characteristic and signature of Template:Mvar can be computed by

$\begin{array}{cc}\chi (M)& =\frac{1}{4{\pi }^2}\underset{M}{\int }({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2+{\mu }_1^2+{\mu }_2^2+{\mu }_3^2)d{\mu }_g\\ \tau (M)& =\frac{1}{3{\pi }^2}\underset{M}{\int }({\lambda }_1{\mu }_1+{\lambda }_2{\mu }_2+{\lambda }_3{\mu }_3)d{\mu }_g.\end{array}$

Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observation

${\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2+{\mu }_1^2+{\mu }_2^2+{\mu }_3^2=\underset{\ge 0}{\underbrace{({\lambda }_1-{\mu }_1{)}^2+({\lambda }_2-{\mu }_2{)}^2+({\lambda }_3-{\mu }_3{)}^2}}+2({\lambda }_1{\mu }_1+{\lambda }_2{\mu }_2+{\lambda }_3{\mu }_3).$

Failure of the converse

A natural question to ask is whether the Hitchin–Thorpe inequality provides a sufficient condition for the existence of Einstein metrics. In 1995, Claude LeBrun and Andrea Sambusetti independently showed that the answer is no: there exist infinitely many non-homeomorphic compact, smooth, oriented 4-manifolds Template:Mvar that carry no Einstein metrics but nevertheless satisfy

$\chi (M)>\frac{3}{2}|\tau (M)|.$

LeBrun's examples are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction only applies to 4-manifolds with infinite fundamental group, but the volume-entropy estimate he uses to prove non-existence only depends on the homotopy type of the manifold.

References

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