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

Honda, Shouhei

J. Math. Soc. Japan 67, No. 1, 69-126 (2015).

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.

Reviewer: Gabjin Yun (Yongin)

MSC:

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

Keywords:

asymptotic cone; Gromov-Hausdorff convergence; harmonic functions; Lipschitz functions; Ricci curvature

Citations:

Zbl 0942.58018; Zbl 0865.53037

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