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Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds. (English) Zbl 1127.53036

The purpose of the paper is to prove a Hitchin-Thorpe type inequality for non-compact 4-dimensional Einstein manifolds. For a 4-dimensional Einstein manifold \(M\), let \(\chi(M)\) denote the Euler number and \(\tau(M)\) the signature. The Hitchin-Thorpe inequality [J. Differ. Geom. 9, 435–441 (1974; Zbl 0281.53039)] states that if \(M\) is a compact oriented 4-dimensional Einstein manifold, then \(\chi(M)\geq{3\over 2}| \tau(M)| \) with equality if and only \(M\) is either flat or its universal cover is a K3 surface.
Now let \(M\) be a complete, but non-compact, 4-dimensional Einstein manifold which is asymptotic to a fibered cusp or a fibered boundary at infinity. Also in the fibered boundary case, first assume that the dimension of the fibres is positive. The authors then prove that \(\chi(M)\geq{3\over 2}| \tau(M)+{1\over 2}a\)-\(\lim\eta| \) where \(a\)-\(\lim\eta\) is the adiabatic limit of the \(\eta\) invariant of the boundary of the compactification. Moreover, equality holds if and only if \(M\) is a Calabi-Yau manifold. Continuing, the authors treat the fibered boundary case where the fiber is a single point yielding a more involved inequality but one for which equality characterizes asymptotically conical Calabi-Yau manifolds.
The paper is well written and contains a nice history and explanation of the problem.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citations:

Zbl 0281.53039
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References:

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