Euclidean Volume Growth for Complete Riemannian Manifolds
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ilan Journal of Mathematics
Euclidean Volume Growth for Complete Riemannian Manifolds Gilles Carron Abstract. We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls. Mathematics Subject Classification (2010). Primary 53C21, 58J35, secondary: 58C40, 58J50 . Keywords. Volume growth.
1. Introduction In this paper, we survey a number of recent results concerning the following question: when does a complete Riemannian manifold (M n , g) has Euclidean volume growth, i.e., we are looking for estimates of the type ∀R > 0 : vol B(x, R) ≤ CRn
(EVG)
where the constant C may depend on the point x or not. We will also obtain some new results and will give several examples that illustrate the optimality of certains of these results. Such an estimate has some important consequences: i) A complete Riemannian surface (M 2 , g) satisfying (EVG) is parabolic. That is to say (M 2 , g) has no positive Green kernel: there is no G : M × M \ Diag −→ (0, ∞) such that Δy G(x, y) = δx (y). We recommend the beautiful and very comprehensive survey on parabolicity written by A. Grigor’yan [19]. In dimension 2, parabolicity is a conformal property and a parabolic surface with finite topological type1 is conformal to a closed surface with a finite number of points removed : there is a closed Riemannian surface (M , g¯), a finite set {p1 , . . . , p } ⊂ M and 2 \ {p1 , . . . , p } −→ R such that (M , g) is isometric to a smooth function f2f: M M \ {p1 , . . . , p }, e g¯ . ii) In higher dimension, the condition (EVG) implies that the manifold is nparabolic. It is a nonlinear analogue of the parabolicity ([12, 22, 23]). 1
that is homeomorphic to the interior of a compact surface with boundary.
G. Carron
iii) According to R. Schoen, L. Simon and S.-T. Yau [28], if a complete stable minimal hypersurface Σ ⊂ Rn+1 with n ∈ {2, 3, 4, 5} satisfies the Euclidean volume growth (EVG), then Σ is an affine hypersurface. In dimension n = 2, M. Do Carmo and C.K. Peng proved that a stable minimal surface in R3 is planar [17]. But nothing is known in higher dimension. ˘ and satisfies the iv) If M n is the universal cover of a closed Riemannian manifold M ˘ is virtually Euclidean volume growth (EVG), then the fundamental group of M nilpotent [21]. v) Another topological implication is that if a complete Riemannian manifold (M n , g) is doubling: there is a uniform constant γ such that for any x ∈ M and R > 0: vol B(x, 2R) ≤ Cγ vol B(x, R), then M has only a finite number of ends, that is to say there is a constant N depending only of γ such that for any K ⊂ M compact subset of M , M \ K has at most N unbounded connected components ([8]). In particular if (M n , g) satisfies a uniform upper and lower Euclidean volume growth: for any x ∈ M and R > 0: θ−1 Rn ≤ vol B(x, R) ≤ θRn , then M has a finite number of ends. vi) In ([30]), G. Tian and J. Viaclovsky have obtained that if (M n , g) is a complete Riemannian manifold such that • ∀x ∈ M, ∀R > 0 : vol B(x, R) ≥ cRn , • Rm (x) = o d(o, x)−2 , then (M n , g) satisfi
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