## 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.
Full Text: