## A class of variational functionals in conformal geometry.(English)Zbl 1154.53019

Let $$(M^n, h_0)$$ be a closed Riemannian $$n$$-manifold. It is known that there is an extension $$g^+$$ on $$X:= [0, \epsilon)\times M$$ for some positive $$\epsilon$$ of the form $g^+ = \frac{1}{r^2} \left(dr^2 + h_{ij}(r,x) dx^i dx^j\right),$ where $$h_{ij}(r, \cdot)$$ is a metric defined on $$M_c := \{r = c\}\subset X$$. In this expression $$(X, r^2 g^+)$$ is compact manifold and $dvol_h(r,x) = \sqrt{\det h_{ij}(r,x)} dx^1 \cdots dx^n.$ Thus one can expand $$\sqrt{\frac{\det h_{ij}(r,x)}{\det h_{ij}(0,x)}}$$ near $$r=0$$ as $\sqrt{\frac{\det h_{ij}(r,x)}{\det h_{ij}(0,x)}} = \sum_{k=0}^\infty v^{(k)}(x,h_0) r^k,$ where $$v^{(k)}(x,h_0)$$ is a curvature invariant of the metric $$h_0 = h_{ij}(0, \cdot)$$ for $$2k \leq n$$. It is known that $$v^{(k)}$$ vanishes for $$k$$ odd and $$2k <n$$; furthermore, when $$n$$ is even, the quantity $$\int_M v^{(n)} dvol_{h_0}$$ is conformally invariant over the conformal class of metrics $$[h_0]$$ [C. R. Graham, in: Proc. 19th Winter School “Geometry and physics”, Srni, Czech Republic, January 9-15, 1999. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 63, 31–42 (2000; Zbl 0984.53020)]. In this paper the authors prove that for any metric $$h$$ on $$M$$ and $$2k \leq n = \dim(M)$$, the functional ${\mathcal F}_k(h) = \frac{\int_M v^{(2k)}(h)\, dvol_h}{\left(\int_M dvol_h\right)^{\frac{n-2k}{n}}}$ is variational within the conformal class when $$2k <n$$; i.e. the critical metric in $$[h]$$ satisfies the equation $$v^{(2k)} = constant.$$ For $$k=1, 2$$ the invariant $$v^{(2k)}$$ turns out to agree, up to a scale, with the well-studied curvature polynomial $$\sigma_k(h)$$, namely
$v^{(2)}(h) = -\frac{1}{2} \sigma_1(h),\quad v^{(4)}(h) = \frac{1}{4}\sigma_2(h),$
where $$\sigma_k$$ is the $$k$$-th elementary symmetric function defined by
$\sigma_k(\lambda) = \sum_{i_1< \cdots < i_k} \lambda_{i_1} \cdots \lambda_{i_k}.$
In particular, it is known that if $$h$$ is locally conformally flat, then
$v^{(2k)}(h) = (-2)^k \sigma_k(h).$
Thus the main theorem proved by the authors is a generalization of a well-known result for the functional
${\mathcal S}_k(g) = \int_M \sigma_k(g) dvol_g$
due to J. A. Viaclovsky [Duke Math. J. 101, No. 2, 283–316 (2000; Zbl 0990.53035)].

### MSC:

 53C20 Global Riemannian geometry, including pinching 58E11 Critical metrics

### Citations:

Zbl 0984.53020; Zbl 0990.53035
Full Text: