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Conformal geometry on four manifolds – Noether lecture. (English) Zbl 1454.53033

Sirakov, Boyan (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume I. Plenary lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 119-146 (2018).
The paper presents in a concise way a summary of many results which have come out of research on the geometry of four manifolds. In Part 1, a brief description of the prescribing of Gaussian curvature problem in compact surfaces and the Yamabe problem is given. Next the quadratic curvature polynomial \(\sigma_2\) on compact closed 4-manifolds is introduced. This appears as part of the integrand of the Gauss-Bonnet-Chern formula. Its algebraic structure is discussed. Some variational approaches to the study of curvature are examined, and as a geometric application. results to characterize the diffeomorphism type of \((S^4, g_c)\) and \(( \mathbb CP^{2}, g_{FS})\) in terms of the size of the conformally invariant quantity. In Part 3, the discussion is extended to compact 4-manifolds with boundary. A third-order pseudo-differential operator \(P_3\) and third-order curvature \(T\) on the boundary of the manifolds is introduced. Next, attention is given to the class of conformally compact Einstein (CEE) four-manifolds. Some recent research on the problem of filling in a given 3-dimensional manifold as the conformal infinity of a CCE manifold is surveyed. Finally, some partial results on compactness of CCE manifolds is discussed. The author suggests they are a key step toward an existence theory for CCE manifolds.
For the entire collection see [Zbl 1436.00059].

MSC:

53C18 Conformal structures on manifolds
57-06 Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes
53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry
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