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The Weyl functional on 4-manifolds of positive Yamabe invariant

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Abstract

It is shown that on every closed oriented Riemannian 4-manifold (M, g) with positive scalar curvature,

$$\begin{aligned} \int _M|W^+_g|^2d\mu _{g}\ge 2\pi ^2(2\chi (M)+3\tau (M))-\frac{8\pi ^2}{|\pi _1(M)|}, \end{aligned}$$

where \(W^+_g\), \(\chi (M)\) and \(\tau (M)\), respectively, denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M. This generalizes Gursky’s inequality [15] for the case of \(b_1(M)>0\) in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky’s inequalities for the case of \(b_2^+(M)>0\) or \(\delta _gW^+_g=0\) and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.

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Notes

  1. For the strong convergence in \(L^2\), we used the Rellich–Kondrakov theorem which holds still on orbifolds. It can be easily derived by using the partition of unity. For a proof, one may refer to [12].

  2. The theorem is originally stated for manifolds, but the proof works well for orbifolds too. If at least one of \(Y(M_1,[g_1])\) and \(Y(M_2,[g_2])\) is positive, then \(Y(M_1\amalg M_2, [g_1\amalg g_2])=\min (Y(M_1,[g_1]),Y(M_2,[g_2]))\).

  3. It’s because any Kähler metric on a 4-orbifold satisfies \(|s|=2\sqrt{6}|W^+|\).

  4. In \(\mathbb Z_2*\mathbb Z_2=\langle a,b|a^2=b^2=1\rangle\), the subgroup generated by \((ab)^d\) for \(d\in \mathbb N\) has index 2d.

  5. When \(k=2\), we have already seen it in the previous subsection. Similarly when \(k\ge 3\), if \(a_i\) denotes a generator of the i-th \(\mathbb Z_2\), then the subgroup generated by \(\{(a_1a_2)^d,a_3,\cdots ,a_k\}\) for \(d\in \mathbb N\) has index 2d.

  6. Many of them are found to have positive Yamabe constant. See [7, 40].

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Acknowledgements

We would like to give special thanks to colleagues and students of KNUE where the first idea of this work was conceived and also researchers at KIAS for sharing fellowship through mathematics. It is our pleasure to express sincere thanks to Claude LeBrun for kind suggestions leading to the improvement of the earlier version of this paper, particularly on the residually finite fundamental group and the Lefschetz fixed point theorem. We are also grateful to the anonymous referee for helpful remarks in examples.

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  1. Department of Mathematics Education, Korea National University of Education, Cheongju, South Korea

    Chanyoung Sung

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Sung, C. The Weyl functional on 4-manifolds of positive Yamabe invariant. Ann Glob Anal Geom 60, 767–805 (2021). https://doi.org/10.1007/s10455-021-09798-x

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