The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (Q6562869)

Let \((M,g)=(\mathbb{R}^n,e^{2u}|dx|^2)\) be a conformally flat manifold where \(n\ge 2\) is an even integer, with \(Q\)-curvature \(Q_g\) satisfying \[ (-\Delta u)^{\frac{n}{2}} u = Q_ge^{nu}. \eqno(1) \] The definitions of the \(Q\)-curvature and the related Paneitz operator, and numerous references to related work, are given in the paper. \((M,g)\) is said to have finite total \(Q\)-curvature if \[ \int_{\mathbb{R}^n} |Q_g|e^{nu}\, dx < \infty. \] The normalized integrated \(Q\)-curvature is given by \[ \alpha_0 = \frac{2}{(n-1)!|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g e^{nu}\, dx, \] where \(|\mathbb{S}^n|\) is the volume of the standard sphere.\N\NUnder the assumption of finite total \(Q\)-curvature, solutions of (1) can be represented as logarithmic potentials \[ u(x)= \frac{2}{(n-1)!|\mathbb{S}^n|} \int_{\mathbb{R}^n} \log\frac{|y|}{|x-y|} Q_g(y) e^{nu(y)}\, dy + C, \] where \(C\) is a constant. Such solutions are called normal solutions; if \(u\) is a normal solution, then the metric \(g=e^{2u}|dx|^2\) is called a normal metric. The volume entropy \(\tau(g)\) of \(g\) is defined by \[ \tau(g) = \lim_{R\rightarrow\infty} \frac{\log\mathrm{Vol}_g(B_R(0))}{\log|B_R(0)|}. \]\N\NThe author's first main result is that if \((M,g)\) as above is complete and conformally flat with finite total \(Q\)-curvature, then \(g\) is normal if and only if the volume entropy \(\tau(g)\) is finite. Furthermore, if \(\tau(g)\) is finite, then \(\tau(g)=1-\alpha_0\).\N\NAs a corollary, if \(\tau(g)\) is finite and \(Q_g\ge 0\), then \(\tau(g)\le 1\) with equality if and only if \(u\equiv C\).\N\NAnother result is a distance comparison theorem which states that if \((M,g)\) as above is conformally flat with finite total \(Q\)-curvature and \(g\) is normal, then for each fixed \(p\in M\) there holds \[ \lim_{|x|\rightarrow\infty} \frac{\log d_g(x,p)}{\log|x-p|} = (1-\alpha_0)^+. \] Moreover, if \(\alpha_0>1\), then \(\mathrm{diam}(\mathbb{R}^n,e^{2u}|dx|^2) < \infty\).\N\NAs a corollary, if \(g\) is normal with \(\alpha_0<1\), then \(g\) is complete. However, if \(\alpha_0=1\), examples of the author show that \(g\) may be complete or incomplete.\N\NThe author also considers the kernel \(\mathcal{PH}(M,g)\) of the GJMS operator \(P_g\) (a higher order analogue of the Paneitz operator) on such manifolds -- these functions are called polyharmonic functions. The result is that under the assumptions of finite total \(Q\)-curvature, completeness of \(g\) and finiteness of \(\tau(g)\), for each \(d\ge 0\) the space \(\mathcal{PH}_d(M,g)\) of polyharmonic functions of polynomial growth at most \(d\) satisfies \[ \dim(\mathcal{PH}_d(M,g)) \le \dim(\mathcal{PH}_{d\tau(g)}(\mathbb{R}^n,|dx|^2)), \eqno(2) \] where \(\mathcal{PH}_d(\mathbb{R}^n,|dx|^2))\) denotes the space of functions \(f\) on \(\mathbb{R}^n\) with \((-\Delta)^{\frac{n}{2}}f=0\) and \(|f(x)|\le C(|x|^d+1)\). Equality holds if \(Q_g\) is nonnegative outside a compact set.\N\NIn the case \(Q_g\ge 0\) everywhere, (2) can be improved to \[ \dim(\mathcal{PH}_k(M,g)) \le \dim(\mathcal{PH}_k(\mathbb{R}^n,|dx|^2)) \] for every integer \(k\ge 1\), with equality if and only if \(u\equiv C\). Finally, \(\tau(g)=0\) if and only if for each \(d\ge 0\) we have \(\mathcal{PH}_d(M,g)=\{\mbox{constant functions}\}\).