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Troyanov, M.

Parabolicity of manifolds. (English) Zbl 0991.31008

Sib. Adv. Math. 9, No. 4, 125-150 (1999).
The author presents a rather detailed survey of results concerning the concept of \(p\)-hyperbolicity and \(p\)-parabolicity of Riemannian manifolds and graphs (see related topics in [I. Holopainen, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 74 (1990; Zbl 0698.31010), T. Coulhon and L. Saloff-Coste, Rev. Mat. Iberoam. 9, No. 2, 293-314 (1993; Zbl 0782.53066), V. M. Kesel’man, Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No. 4, 81-83 (1985; Zbl 0583.53037), M. Yamasaki, Hiroshima Math. J. 7, 135-146 (1977; Zbl 0382.90088), V. A. Zorich and V. M. Kesel’man, Funct. Anal. Appl. 30, No. 2, 106-117 (1996; Zbl 0873.53025)]), and proves some new geometric criteria for the \(p\)-hyperbolicity and \(p\)-parabolicity. The material of the paper is supplied with various examples of \(p\)-hyperbolic and \(p\)-parabolic manifolds.
Definition. Let \((M,g)\) be a Riemannian manifold and let \(\Omega \subset M\) be a compact set. For \(1\leq p<\infty\) the \(p\)-capacity of \(D\) in \(\Omega\) is defined as follows: \[ \text{Cap}_p(D,\Omega)=\inf\Bigl\{\int\limits{\Omega} |du|^p: u\in W_0^{1,p}(\Omega)\cap C_0^0(\Omega), \;u|_{D}\geq 1 \Bigr\}, \] where the Sobolev space \(W_0^{1,p}\) is the closure of \(C_0^1(\Omega)\), the space of compactly supported \(C^1\)-functions, with respect to the Sobolev norm \(\|u\|_{1,p}=\|u\|_{L_p}+\|du\|_{L_p}\).
A Riemannian manifold is \(p\)-hyperbolic (\(1\leq p<\infty\)) if it includes a compact set of positive \(p\)-capacity, and \(p\)-parabolic otherwise. Using some properties of the \(p\)-harmonic functions and \(p\)-capacity [see J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, Oxford (1993; Zbl 0780.31001)], the author establishes the following propositions:
Theorem 4.4. Let \(E\) be a subset of a Riemannian manifold \(M\). Then
(a) if \(E\subset M\) has local positive \(p\)-capacity, then \(\Omega=M\setminus E\) is \(p\)-hyperbolic;
(b) if \(M\) is \(p\)-parabolic and \(E\subset M\) is a \(p\)-, then \(\Omega=M\setminus E\) is \(p\)-parabolic.
Corollary 4.1. Let \(M\) be a closed Riemannian manifold and let \(E\subset M\) be of Hausdorff dimension \(s\) (\(0<s<n=\text{dim} (M)\)). Then \(\Omega=M\setminus E\) is \(p\)-parabolic whenever \(1<p<n-s\) and \(p\)-hyperbolic whenever \(p>n-s\).
For a Riemannian manifold \(M\) with warped cylindrical end (i.e. such that there are a compact Riemannian manifold \((N, g_N)\) and a compact set \(D\subset M\) such that \(M\setminus D\) is a of \(N\) and \([1,\infty)\) with Riemannian metric \(dt^2+f^2(t)g_N\)) the author proves the following criterion:
Corollary 5.2. The manifold \(M\) (with warped cylindrical end) is \(p\)-parabolic if and only if \(\int^{\infty}_1 f(t)^{\frac{n-1}{1-p}} dt=\infty\).
Definition. A function \(P\:[0,V)\to \mathbb R\) is an isoperimetric profile for an open set \(\Omega\subset (M,g)\) (\(V=\text{Vol} \Omega\)) if there are constants \(C\), \(\nu\) such that \(\nu<V\) and \(P(\text{Vol} D)\leq C\text{Area} \partial D\) for all compact regions \(D\subset\Omega\) with \(\nu\leq \text{Vol} D\).
Theorem 5.3. Suppose that a domain \(\Omega\subset (M,g)\) admits an isoperimetric profile \(P\:[0,V)\to \mathbb R\) such that \(\int_{\nu}^V\frac{dv}{[P(v)]^{p/(p-1)}} <\infty\), \(1<p<\infty\). Then \(\Omega\) is \(p\)-hyperbolic.
The author also establishes some criteria of the \(p\)-parabolicity via the area and the volume growth of the balls \(B(x_0,r)\) (\(r\to \infty\)) of a complete Riemannian manifold \(M\). He introduces the notion of the parabolic dimension \(d_{\text{par}}\) of a manifold (see the definition below) and proves criteria for \(p\)-parabolicity for manifolds \(M\) with bounded geometry using this notion: a manifold \(M\) is \(p\)-parabolic whenever \(p>d_{\text{par}}\), and \(p\)-hyperbolic whenever \(p<d_{\text{par}}\).
Definition. The parabolic dimension of \(M\) is the number \(d_{\text{par}}(M)=\inf\{p\geq 1: M\) is \(p\)-parabolic}.
Reviewer: A.Greshnov (Novosibirsk)

Cited in 1 Review
Cited in 53 Documents

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
53C20 Global Riemannian geometry, including pinching
31B15 Potentials and capacities, extremal length and related notions in higher dimensions

Keywords:

capacity; \(p\)-parabolicity; Riemannian manifold; \(p\)-hyperbolicity

Citations:

Zbl 0698.31010; Zbl 0782.53066; Zbl 0583.53037; Zbl 0382.90088; Zbl 0873.53025; Zbl 0780.31001
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