Mathematical Research Letters

Volume 23 (2016)

Number 4

Yamabe invariants and the $\mathrm{Pin}^-(2)$-monopole equations

Pages: 1049 – 1069



Masashi Ishida (Mathematical Institute, Tohoku University, Sendai, Japan)

Shinichiroh Matsuo (Graduate School of Mathematics, Nagoya University, Nagoya, Japan)

Nobuhiro Nakamura (Department of Mathematics, Osaka Medical College, Osaka, Japan)


We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a “twisted” version of the Seiberg–Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature.

2010 Mathematics Subject Classification

53C21, 53C25, 53C44, 57R57

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