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Harmonic functions on manifolds. (English) Zbl 0928.53030

For an open manifold \(M^n\), given a point \(p\in M^n\), let \(r\) be the distance from \(p\). Define \({\mathcal H}_d(M^n)\) to be the linear space of harmonic functions with order of growth at most \(d\). The main result of this paper is a proof of the following Yau’s conjecture:
Conjecture. For an open manifold with nonnegative Ricci curvature, the space of harmonic functions with polynomial growth of a fixed rate is finite-dimensional.
The authors prove this by giving an explicit bound on the dimension of \({\mathcal H}_d(M^n)\) depending only on \(n\) and \(d\). The proof is a consequence of a more general result which can be stated after recalling some basic inequalities on complete Riemannian manifolds \(M^n\). We say that
(a) \(M^n\) has the doubling property if there exists \(C_D<\infty\) such that for all \(p\in M^n\) and \(r>0\) \[ \text{Vol}(B_{2r}(p))\leq C_D\text{Vol} (B_{r}(p)), \]
(b) \(M^n\) satisfies a uniform Neumann-Poincaré inequality if there exists \(C_N<\infty\) such that for all \(p\in M^n\), \(r>0\) and \(f\in W^{2,1}_{\text{loc}}(M)\) \[ \int_{\dot B_r(p)}(f-{\mathcal A})^2\leq C_Nr^2\int _{B_r(p)}| \nabla f| ^2, \] where \({\mathcal A}=\frac {1}{\text{Vol}(B_r(p))}\int _{B_r(p)}f\).
Then the announced result is given as follows.
Theorem. If \(M^n\) is an open manifold which has the doubling property and satisfies a uniform Neumann-Poincaré inequality, then for all \(d>0\) there exists \(C=C(d,C_D,C_N)<\infty \) such that \(\dim {\mathcal H}_d(M^n)\leq C\).
The authors also provide several very interesting corollaries of the above result.
Reviewer: W.Mozgawa (Lublin)

MSC:

53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
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