Local study of curvature operators on the hyperbolic space
Concerning the existence of Riemannian metrics on a manifold with prescribed Ricci tensor, several results are well known. In particular, \textit{D. DeTurck} [Ann. Math. Stud. 102, 525-537 (1982; Zbl 0478.53031)] and \textit{R. Hamilton} [Publ., Math. Sci. Res. Inst. 2, 47-72 (1984; Zbl 0557.53018)] investigated this problem for open sets of \(\mathbb{R}^n\) and open neighbourhoods of the standard metric on the unit sphere \(S^n\), respectively. In this paper the author considers, as a model of the hyperbolic space \(H^n(-1)\), the unit ball \(B\) in \(\mathbb{R}^n\) endowed with the metric \(H_0=\rho^{-1}E\), \(E\) being the Euclidean metric and \(\rho\) the function on \(B\) defined by \(\rho(x)={1\over 2}(1-|x|^2)\). To any metric \(H\) on \(B\) there are associated the Ricci tensor \(\text{Ricci}(H)\) and the Einstein tensor \(\text{Ein}(H) =\text{Ricci} (H)-{1\over 2}\text{Scal}(H)H\), \(\text{Scal}(H)\) being the scalar curvature of \(H\). Moreover, given real numbers \(s,\alpha\), \(0<\alpha<1\), and an integer \(k>0\), \(\Lambda^s_{k, \alpha} (B)\) denotes the Banach space of the \(k\)-times continuously differentiable functions \(u\) on \(\overline B\) with finite norm \(\|u\|^{(s)}_{k,\alpha}\) the norm being suitably defined in terms of the derivatives of \(u\) and of the Euclidean distance; \(\Lambda^s_{k,\alpha}(B,{\mathcal S}_2)\) is the space of the symmetric (0;2)-tensors on \(B\) whose local components, in Euclidean coordinates, are in \(\Lambda^s_{k,\alpha}(B)\). Then, one has: Theorem A. Given integers \(n,k,n\geq 10\), \(k\geq 2\), \(\alpha,s\) real numbers, \(0<\alpha<1\), \(2n<s(n-1-s)\), \(r\in \Lambda^{s-2}_{k+2, \alpha}(B,{\mathcal S}_2)\) near the zero, then the equation \(\text{Ricci} (H_0+ h)=\text{Ricci} (H_0)+r\) has a unique solution \(h\in\Lambda^{s-2}_{k+2,\alpha} (B,{\mathcal S}_2)\) near the zero and the correspondence \(r\mapsto h\) defines a smooth map between neighbourhoods of the zero in \(\lambda^{s-2}_{k+2,\alpha}(B,{\mathcal S}_2)\). For the proof, the author applies a theorem of isomorphism involving the Laplacian of the metric \(H_0\). This allows to study the asymptotic behaviour of the solutions of the Ricci equation, too. The Ricci equation in the two-dimensional case is investigated in details. Moreover, concerning the Einstein equation, the following result is stated. Theorem B. Given integers \(n,k,n\geq 3,k\geq 1\), \(\alpha,s\) real numbers, \(0<s<n-1\), and \(e\in \Lambda^{s-2}_{k+2, \alpha}(B, {\mathcal S}_2)\) near the zero, then the equation \(\text{Ein} (H_0+h)= \text{Ein} (H_0)+e\) has a unique solution \(h\) in \(\lambda^{s-2}_{k+2, \alpha} (B,{\mathcal S}_2)\) near the zero and the correspondence \(e\mapsto h\) defines a smooth map between neighbourhoods of the zero in \(\Lambda^{s-2}_{k+2, \alpha}(B, {\mathcal S}_2)\).
scientific article; zbMATH DE number 1344438
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