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

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].

### MSC:

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

### Keywords:

Yamabe equation; conformal group; unitary representation