Harmonic sections of polynomial growth
Starting from a conjecture of S.-T. Yau and a series of more general statements due to \textit{J. H. Colding} and \textit{W. P. Minicozzi} (see, for instance, [Math. Res. Lett. 3, 241-246 (1996; Zbl 0849.53034)]), the author considers a complete manifold \(M\) satisfying the weak volume comparison condition and the mean value inequality. For a vector bundle \(E\) over \(M\), let \(S_d(M,E)\) be a linear subspace of sections of \(E\), such that all \(u\in S_d(M,E)\) satisfy (a) \(\Delta |u|^2 \geq 0\); (b) \(|u|(x)\leq O(\rho^d(x))\) as \(x\to\infty\). Then the dimension of \(S_d(M,E)\) is finite. Moreover, a sharp estimate of the dimension of the space of harmonic functions which have at most polynomial growth of degree \(d\) is established. Interesting geometric applications are given. We want to mention estimates of the dimensions of the space of Killing vector fields of polynomial growth on a complete Ricci flat manifold and of the space of holomorphic sections of polynomial growth of a Hermitian vector bundle over a complete Kähler manifold, respectively.
scientific article; zbMATH DE number 1019784