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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
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