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

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 $$m\geqslant 0$$.

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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