Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds. (English) Zbl 1074.53031

Kobayashi, Toshiyuki

Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12–19, 2002. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 15-40 (2003).

The paper studies the Yamabe operator \(Y= \Delta- ((n- 2)/4(n- 1))R\) on an \(n\)-dimensional pseudo-Riemannian manifold \((M, g)\), where \(\Delta\) is the Laplacian and \(R\) the scalar curvature and the Yamabe equation \(Yf= 0\). Since the latter is conformally invariant, there is naturally a representation \(w\) of the conformal group of \((M, g)\) on \(\text{Ker\,}Y\), and the following problems arise.
A. When is \(\text{Ker\,}Y\neq 0\)? B. When is \(w\) irreducible? C. Find a suitable scalar product on \(\text{Ker\,}Y\) which leads to a unitary representation. D. Decompose w into irreducibles with respect to the isometry group.
The author develops a general theory for \(A\), \(B\), \(C\), \(D\) and then applies it to the pseudo-Riemannian product manifold \(\mathbb{R}^p\times\mathbb{R}^q\) of signature \((p, q)\) and \(S^p\times S^4\). Sato’s hyperfunctions and stereographic projections are used.
For the entire collection see [Zbl 1014.00011].

Reviewer: J. Eichhorn (Greifswald)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J45	Hyperbolic equations on manifolds

Keywords:

Yamabe equation; conformal group; unitary representation