Laplacian comparison for Alexandrov spaces

In this paper, we study singular spaces of Ricci curvature bounded below. For Riemannian manifolds, having a lower bound of Ricci curvature is equivalent to an infinitesimal version of the Bishop-Gromov volume comparison condition. Since it is impossible to define the Ricci curvature tensor on Alexandrov spaces, we consider such the volume comparison condition as a candidate of the conditions of the Ricci curvature bounded below.

In Riemannian geometry, the Laplacian comparison theorem is one of the most important tools to study the structure of spaces with a lower bound of Ricci curvature. A main purpose of this paper is to prove a Laplacian comparison theorem for Alexandrov spaces under the volume comparison condition. As an application, we prove a topological splitting theorem of Cheeger-Gromoll type.

Let us present the volume comparison condition. For κ ∈ R, we set

The function s κ is the solution of the Jacobi equation s ′′ κ (r)+κs κ (r) = 0 with initial condition s κ (0) = 0, s ′ κ (0) = 1.

Let M be an Alexandrov space of dimension n ≥ 2. For p ∈ M and 0 < t ≤ 1, we define a subset W p,t ⊂ M and a map Φ p,t : W p,t → M as follows. x ∈ W p,t if and only if there exists y ∈ M such that x ∈ py and d(p, x) : d(p, y) = t : 1, where py is a minimal geodesic from p to y and d the distance function. For a given point x ∈ W p,t such a point y is unique and we set Φ p,t (x) := y. The Alexandrov convexity (cf. §2.2) implies the Lipschitz continuity of the map Φ p,t . Let us consider the following.

Condition BG(κ) at a point p ∈ M: We have

for any x ∈ M and t ∈ ( 0, 1 ] such that d(p, x) < π/ √ κ if κ > 0, where Φ p,t * H n means the push-forward by Φ p,t of the n-dimensional Hausdorff measure H n on M.

If M satisfies BG(κ) at any point p ∈ M, we simply say that M satisfies BG(κ).

The condition, BG(κ), is an infinitesimal version of the Bishop-Gromov inequality. For an n-dimensional complete Riemannian manifold, BG(κ) holds if and only if the Ricci curvature satisfies Ric ≥ (n -1)κ (see Theorem 3.2 of [20] for the ‘only if’ part). We see some studies on similar (or same) conditions to BG(κ) in [7,33,15,16,29,20,38] etc. BG(κ) is sometimes called the Measure Contraction Property and is weaker than the curvature-dimension condition introduced by Sturm [34,35] and Lott-Villani [17]. Any Alexandrov space of curvature ≥ κ satisfies BG(κ). However we do not necessarily assume M to be of curvature ≥ κ. For example, a Gromov-Hausdorff limit of closed nmanifolds of Ric ≥ (n -1)κ, sectional curvature ≥ κ 0 , diameter ≤ D, and volume ≥ v > 0 is an Alexandrov space with BG(κ) and of curvature ≥ κ 0 .

To state the Laplacian comparison theorem, we need some notations and definitions. If M has no boundary, we define M * as the set of non-δ-singular points of M for a number δ with 0 < δ ≪ 1/n. If M has nonempty boundary, we refer to Fact 2.6 below for M * . All the topological singularities of M are entirely contained in M \ M * and M * has a natural structure of C ∞ differentiable manifold. We have a canonical Riemannian metric g on M * which is a.e. continuous and of locally bounded variation (locally BV for short). See §2.2 for more details. We set cot κ (r) := s ′ κ (r)/s κ (r) and r p (x) := d(p, x) for p, x ∈ M. The distributional Laplacian ∆ r p of r p on M * is defined by the usual formula:

∆ r p := -D i ( |g| g ij ∂ j r), on a local chart of M * , where D i is the distributional derivative with respect to the i th coordinate. Then, ∆ r p becomes a signed Radon measure on M * (see §4). An main theorem of this paper is stated as follows.

Theorem 1.1 (Laplacian Comparison Theorem). Let M be an Alexandrov space of dimension n ≥ 2. If M satisfies BG(κ) at a point p ∈ M, then we have Even if M is a Riemannian manifold, ∆ r p is not absolutely continuous with respect to H n on the cut-locus of p (see Remark 5.10). Different from Riemannian, the cut-locus of an Alexandrov space is not necessarily a closed subset. In fact, we have an example of an Alexandrov space for which the singular set and the cut-locus are both dense in the space (cf. Example (2) in §0 of [22]). The Riemannian metric g on M * is not continuous on any singular point and has at most the regularity of locally BV. Therefore, the Laplacian of a C ∞ function does not become a function, only does a Radon measure in general. In particular, considering a Laplacian comparison in the barrier sense is meaningless. In this reason, for Theorem 1.1 a standard proof for Riemannian does not work and we need a more delicate discussion using BV theory.

In [26], Petrunin claims that the Laplacian of any λ-semiconvex function is ≥ -nλ from the study of gradient curves. This implies Corollary 1.2. However we do not know the details. After Petrunin, Renesse [36] proved Corollary 1.2 in a different way under some additional condition. Our proof is based on a different idea from them.

We do not know if the converse to Theorem 1.1 is true or not, i.e., if (1.1) implie

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