Boundary constructions for CR manifolds and Fefferman spaces (Q2929602)

The aim of this PhD thesis is to discuss the Cartan boundary of strictly pseudoconvex CR hypersurfaces and of their Fefferman spaces. On such a CR hypersurface \(M\) the author considers the Fefferman structure, described by a triple \((M,T_{10},\theta)\), where \(\theta\) is a suitable section of the characteristic real line bundle \(H^0M\) of \(M\). The Fefferman space \((F,[h_\theta])\) is an \(S^1\)-principal bundle over \(M\), whose total space \(\mathcal{F}\) is the quotient of the canonical bundle \(\mathcal{K}\) modulo the action of \(\mathbb{R}_+\). We recall that, if \(\mathcal{I}\) is the ideal of complex-valued alternate forms vanishing on \(\overline{T_{10}}\), then \(\mathcal{K}=\mathcal{I}^{n+1}\), where \(n\) is the CR dimension of \(M\), is a complex line bundle on \(M\). The class \([h_\theta]\) of the Fefferman metric \(h_\theta\) is a conformal invariant of \(M\).NEWLINENEWLINEChapters 2 to 6 cover several special topics, ranging from basic notions of CR geometry, to Cartan and parabolic geometries, touching on Tanaka's construction of a tower of principal bundles associated to a \(\mathbb{Z}\)-graded nilpotent Lie algebra fixing a generalized contact structure. Under the Serre's condition, this construction yields a \(P\)-frame bundle. This is thoroughly discussed in Chapter 3, and applied in the next chapter to the special example of CR hypersurfaces modeled on the real projective quadric of any Witt index. Chapter 6 discusses Cartan boundaries from various perspectives. These results are applied, in the last part of this work, to the case of CR manifolds and its Fefferman spaces. Among various interesting results, an inclusion is proved of the Cartan boundary of \(M\) into the Cartan boundary of its Fefferman space.NEWLINENEWLINEThe thesis is well written and contains beautiful results and ideas.