Prescribing symmetric functions of the eigenvalues of the Ricci tensor. (English) Zbl 1142.53027

Gursky, Matthew J.; Viaclovsky, Jeff A.

Ann. Math. (2) 166, No. 2, 475-531 (2007).

The authors study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor.
Let \((M^n,g)\) be a smooth, closed \(n\)-dimensional Riemannian manifold. The Weyl-Schouten tensor of \(g\) is defined by
\[ A=\frac{1}{n-2}\left (\text{Ric}-\frac{1}{2(n-1)}Rg\right ), \]
where Ric denotes the Ricci tensor and \(R\) is the scalar curvature.
The transformation of \(A\) under conformal deformation \(g\to g_u=e^{-2u}g\) leads to the equation
\[ \sigma_k^{1/k}(g_u^{-1}A_u)=f(x), \]
where \(\sigma_k:\mathbb R^n\to\mathbb R^n\) denotes the elementary symmetric polynomial of degree \(k\), \(A_u\) denotes the Weyl-Schouten tensor of \(g_u\), \(\sigma_k^{1/k}(g_u^{-1}A_u)\) means \(\sigma_k\) applied to the eigenvalues of the (1,1)-tensor \(g_u^{-1}A_u\) obtained by “raising an index” of \(A_u\). The above equation is equivalent to
\[ \sigma_k^{1/k}(A+\nabla^2u+du\otimes du-\tfrac 12| \nabla u| ^2g)=f(x)e^{-2u}.\tag{1} \]
It is worth noting that the problem of solving the above equation with \(f(x)= \text{const}\) is known as the \(\sigma_k\)-Yamabe problem.
The authors call a metric \(g\) \(k\)-admissible, if the eigenvalues of \(A=A_g\) are everywhere in \(\Gamma_k^+\), where \(\Gamma_k^+\) denotes the component of \(\{x\in\mathbb R^n\mid\sigma_k(x)>0\}\) containing the positive cone \(\{x\in\mathbb R^n\mid x_1>0,\dots,x_n>0\}\). In the paper under review the authors are interested in the case \(k>n/2\). The geometric significance of this assumption follows from the fact that a \(k\)-admissible metric with \(k>n/2\) has positive Ricci curvature.
The main result is a general existence theory for solutions of (1).
Theorem. Let \((M^n,g)\) be a closed \(n\)-dimensional Riemannian manifold such that

(i)

\(g\) is \(k\)-admissible with \(k>n/2\),

(ii)

\((M^n,g)\) is not conformally equivalent to the round \(n\)-dimensional sphere.

Then, given any smooth positive function \(f\in C^{\infty}(M)\) there exists a solution \(u\in C^{\infty}(M)\) of (1), and the set of all such solutions is compact in the \(C^m\)-topology for any \(m\geqslant 0\).

Reviewer: Marian Hotloś (Wrocław)

MSC:

53C21

Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

conformal deformation; eigenvalues of Ricci tensor; Weyl-Schouten tensor