## Harmonic functions on asymptotic cones with Euclidean volume growth.(English)Zbl 1317.53051

Let $$M$$ be a complete $$n$$-dimensional Riemannian manifold with nonnegative Ricci curvature. Then, for any point $$m \in M$$ and a sequence $$\{R_i\}$$ with $$R_i\to \infty$$ as $$i \to \infty$$, the rescaled Riemannian manifolds $$(M, R_i^{-1}d_M, m)$$ converge to an asymptotic cone $$(M_\infty, m_\infty)$$ of $$M$$ with respect to the pointed Gromov-Hausdorff topology. Here $$d_M$$ is the distance function of $$M$$. Introducing the notion of harmonic functions on metric measure spaces due to J. Cheeger [Geom. Funct. Anal. 9, No. 3, 428–517 (1999; Zbl 0942.58018)], we can consider harmonic functions on the asymptotic cone $$(M_\infty, m_\infty)$$ of $$M$$. For $$d \geq 0$$, let $$H^d(M_\infty)$$ be the space of harmonic functions $$f$$ on $$M_\infty$$ satisfying that there exists $$C>1$$ such that $$|f(x)| \leq C(1+r(x)^d)$$, where $$r(x) = d_{M_\infty}(m_\infty, x)$$. In this paper, the author proves that for any $$V>0$$, there exists $$d(n, V)\geq 1$$ such that $C(n)^{-1}V_M d^{n-1}\leq \dim H^d(M_\infty)\leq C(n) V_M d^{n-1}$ holds for every $$n$$-dimensional complete Riemannian manifold $$M$$ of nonnegative Ricci curvature with $$V_M:= \lim_{R\to \infty}\text{vol}B_R(m)/R^n \geq V$$, every $$d \geq d(n, V)$$ and every asymptotic cone $$(M_\infty, m_\infty)$$, where $$C(n)$$ is a positive constant depending only on $$n$$. The author also gives a relationship between harmonic functions with polynomial growth on $$M$$ and that of asymptotic cones and shows that there exists a unique $$d_1 \geq 1$$ such that $$H^d(M) = \{\text{constants}\}$$ for every $$0 < d < d_1$$, $$H^d(M_\infty) = \{\text{constants}\}$$ for every $$0 < d < d_1$$ and for every asymptotic cone $$M_\infty$$, and $$H^{d_1}(\widehat M_\infty) \neq \{\text{constants}\}$$ for some asymptotic cone $$\widehat M_\infty$$.
Second, the author considers the space of harmonic functions with polynomial growth on the metric cone of a compact geodesic space. J. Cheeger and T. H. Colding [Ann. Math. (2) 144, No. 1, 189–237 (1996; Zbl 0865.53037)] shows that $$(M_\infty, m_\infty)$$ is isometric to a metric cone $$(C(X), p)$$ of a compact geodesic space $$X$$ with $$\text{diam} (X) \leq \pi$$. Since, for the $$(n-1)$$-dimensional Hausdorff measure $$H^{n-1}$$, $$X$$ is $$H^{n-1}$$-rectifiable and $$(X, H^{n-1})$$ satisfies a weak Poincaré inequality, there exists the canonical self-adjoint operator (called Laplacian) $$\Delta_X$$ on $$L^2(X)$$. Let $$E_\lambda(X)$$ be the space of functions on $$X$$ spanned by eigenfunctions of $$\Delta_X$$ on $$X$$ associated with the eigenvalue $$\leq \lambda$$. Another main result proved by the author in this paper is the following: For $$d\geq 0$$ $\dim H^d (C(X)) = \dim E_{d(d+n-2)}(X).$ In particular, we have $$\dim H^d(C(X)) < \infty$$, and this property can be considered as a solution of an asymptotic cone’s version of Yau’s conjecture on the growth of harmonic functions.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C43 Differential geometric aspects of harmonic maps

### Citations:

Zbl 0942.58018; Zbl 0865.53037
Full Text:

### References:

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