A WARPED PRODUCT SPLITTING THEOREM

Christopher B. Croke and Bruce Kleiner^{∗}

January 15, 1991

The celebrated Cheeger-Gromoll splitting theorem (see [ChGr], and also [EsHe, Wu]) implies that a connected, complete Riemannian manifold having nonnegative Ricci curvature and two ends is isometric to a Riemannian product, where one factor is a line. There is a version of this theorem for compact, connected Riemannian manifolds M with two smooth boundary components. It states that if M has nonnegative Ricci curvature and two boundary components, both of which have nonnegative mean curvature with respect to their inward normals, then M is isometric to the Riemannian product of N_{1} with a closed interval (in particular N_{1} is isometric to N_{2}).

In this paper we will prove an analog of this result for manifolds with Ricci curvature ≥ −(n − 1). Since our argument parallels the proof of the splitting theorem for manifolds with nonnegative Ricci curvature, we present the proofs simultaneously by letting δ be either zero or one in what follows.

Theorem 1 Let M a connected, complete, n-dimensional Riemannian manifold with boundary components N_{1} and N_{2}, at least one of which is compact. Suppose Ricci(M) ≥ −(n − 1)δ, the mean curvature of N_{1} is ≥ δ, and the mean curvature of N_{2} is ≥ −δ. Then M is isometric to N_{1} × [0, l] with the warped product metric ds^{2} = e^{2δt}g_{0} + dt^{2}, where g_{0} is the metric on N_{1}. Moreover, g_{0} has nonnegative intrinsic Ricci curvature.

Putting the second boundary component N_{2} at infinity, we get:

∗Supported by an NSF Postdoctoral Fellowship

Theorem 2 (“Cusp”, or “halfspace” rigidity) Let M be a complete, n-dimensional Riemannian manifold with compact, smooth boundary N. Suppose

1. Ricci(M) ≥ −(n − 1)δ.

2. H_{N}, the mean curvature of N with respect to the inward normal, satisfies H_{N} ≥ δ.

3. M is noncompact.

Then M is isometric to N × [0, ∞) with the warped product metric ds^{2} = e^{2δt}g_{0} + dt^{2}, where g_{0} has nonnegative intrinsic Ricci curvature.

Using a slight modification of the proof of theorem 1 we get the following result for any lower bound on the Ricci curvature:

Theorem 3 Let M be a complete, connected Riemannian manifold with two boundary components N_{1} and N_{2}, one of which is compact, and let M̄ be an annular region in a constant curvature model space bounded by two concentric geodesic spheres, N̄_{1} and N̄_{2} of radii r_{1} and r_{2}. Suppose Ricci(M) ≥ Ricci(M̄), and the mean curvature of N_{i} with respect to the inward normal is greater than or equal to the mean curvature of N̄_{i}, i = 1, 2. Then d(N_{1}, N_{2}) ≤ d(N̄_{1}, N̄_{2}) and equality holds only if M is isometric to N_{1} × [r_{1}, r_{2}] with the warped product metric ds^{2} = f^{2}(t)g_{0} + dt^{2}, where g_{0}, the metric on N_{1}, has intrinsic Ricci curvature ≥ Ricci(N̄_{1}), and f(t) is the same as the usual warping function for M̄ in normal polar coordinates.

In general let M be a complete manifold with boundary components N_{1} and N_{2}, one of which is compact. Assume Ricci(M) ≥ (n − 1)k and N_{i} has mean curvature ≥ a_{i}. For given choice of k, a_{1}, a_{2} it is easy to see that one of the following hold:

1. The Riccati equation (see below) implies that no such manifold exists.

2. There are surfaces of revolution satisfying the data and having arbitrarily large distance between boundary components.

3. M satisfies the hypothesis of theorem 1 or theorem 3.

We first recall some simple consequences of the Riccati equation. Let N be a codimension one submanifold of a Riemannian manifold M with Ricci(M) ≥ (n − 1)k with a given normal field ν(x) for x ∈ N. Let N_{t} be the submanifold obtained by exponentiating a distance t in the ν direction, and let ν_{t} be the corresponding normal (i.e if γ_{ν} is the geodesic determined by ν then N_{t} is {γ_{ν(x)}(t) | x ∈ N} and ν_{t}(γ_{ν(x)}(t)) is γ′_{ν(x)}(t)). Fixing a point x ∈ N, we let U(t) denote the second fundamental form of N_{t} with respect to ν_{t} along γ_{ν(x)}(t) and H(t) = trU(t)/(n − 1), be the mean curvature. Taking the trace of the Riccati equation along γ_{ν(x)}(t), we see that as long as U(t) is nonsingular:

(n − 1)H′(t) = trU^{2}(t) + Ricci(ν_{t}).

The inequalities trU^{2}(t) ≥ (trU(t))^{2}/(n − 1) = (n − 1)H^{2}(t) and Ricci(ν_{t}) ≥ (n − 1)k yield the inequality:

H′(t) ≥ H^{2}(t) + k.

We note for future reference that if equality holds then U(t) must be scalar and Ricci(ν_{t}) = (n − 1)k. If M̄ is the model space with constant sectional curvature k, N̄ ⊂ M̄ is a totally umbilic hypersurface with constant mean curvature H̄(0) ≤ H(0) with respect to a normal field ν̄, and H̄(t) is the mean curvature of the hypersurface at a distance t from N̄, then we will have H(t) ≥ H̄(t) for every t ≥ 0 provided the hypersurface N_{t} remains nonsingular along γ_{ν}.

In particular, if γ_{ν(x)} is a uniquely minimizing path from N to p = γ_{ν(x)}(t) ∈ N_{t} then we will have Δd_{N}(p) ≤ −(n − 1)H(t) ≤ −(n − 1)H̄(t), where d_{N} is the distance function from N.

Proof of theorem 1: Let d_{i} : M ⟶ R be the distance function from N_{i}, i = 1, 2. We will call a point x in M a shadow point for N_{2} if some minimizing path from N_{1} to x touches N_{2}. We let D_{1} be the set of points in the interior of M which are neither cut points for N_{1} nor shadow points for N_{2}. We define D_{2} similarly, but with the roles of N_{1} and N_{2} interchanged. Note that D_{i} is open for i = 1, 2, and that if x ∈ D_{i} then d_{i} is smooth and satisfies Δd_{i} ≤ (n − 1)(−δ)^{i} near x, by the discussion above.

We now define F : M ⟶ R by setting F = d_{1} + d_{2}, and we let Min_{F} be the set of points in the interior of M where F attains its minimum. Clearly x ∈ Min_{F} if and only if x lies on a minimizing path from N_{1} to N_{2}, so we have Min_{F} ⊆ D_{1} ∩ D_{2}. Therefore F is smooth and satisfies ΔF = Δd_{1} + Δd_{2} ≤ 0 on an open set containing Min_{F}. By the minimum principle for superharmonic functions, Min_{F} is open and closed in Interior(M) so F is constant. Therefore every point in M lies on a unique minimizing segment from N_{1} to N_{2}, d_{1} and d_{2} are smooth on all of M and satisfy Δd_{i} ≡ (n − 1)(−δ)^{i}. This forces the second fundamental form of the hypersurface at a distance t for N_{1} to be scalar, with all principal curvatures equal to δ. Hence the metric on M is of the form ds^{2} = e^{−2δt}g_{0} + dt^{2} where g_{0} is the metric on N_{1}. Finally, since the sectional curvature of M on any two-plane containing the normal to N_{1} is −δ, it follows from the Gauss equations for the hypersurface N_{1} that g_{0} has nonnegative Ricci curvature. □

Proof of theorem 2: Let d_{1} : M ⟶ R be the distance function from N, and let D_{1} be the complement of the cut locus of N in the interior of M. As in the proof of theorem 1 we find that d_{1} is smooth and satisfies Δd_{1} ≤ −(n − 1)δ on the set D_{1}.

Let γ : [0, ∞) ⟶ M be a ray, and let d_{2} : M ⟶ R be the associated Busemann function, d_{2}(x) = lim_{t→∞} d(x, γ(t)) − t. We call a ray η : [0, ∞) ⟶ M satisfying d_{2}(η(t)) = d_{2}(η(0)) − t an asymptote, and we recall that for any x, y ∈ M we have d_{2}(x) ≤ d(x, y) + d_{2}(y) with equality if and only if there is an asymptote starting at x which passes through y.

We call x ∈ Interior(M) a shadow point for N if some asymptote leaving x touches N. The set of shadow points of N is closed, and we let D_{2} be its complement in Interior(M). We now claim that d_{2} satisfies Δd_{2} ≤ (n − 1)δ weakly on the set D_{2}. By definition, this means that for any x ∈ D_{2} and any ϵ > 0 there exists a smooth function ϕ_{x,ϵ} defined on a neighborhood of x which satisfies ϕ_{x,ϵ} ≥ d_{2}, ϕ_{x,ϵ}(x) = d_{2}(x), and (Δϕ_{x,ϵ})(x) ≤ (n − 1)δ + ϵ. To obtain such a supporting function near x ∈ D_{2}, we take an asymptote η : [0, ∞) ⟶ M starting at x and consider the function d_{η,t} : M ⟶ R given by

d_{η,t}(p) = d(p, η(t)) − t + d_{2}(x) ≥ d_{2}(p) − d_{2}(η(t)) − t + d_{2}(x) = d_{2}(p).

Since η is a ray which doesn’t touch N, d_{η,t} is smooth near x = η(0). Hence we may apply the reasoning from the beginning of the proof of theorem 1 to the geodesic spheres centered at η(t), concluding that (Δd_{η,t})(x) ≤ (n−1)δ+f(t) where f(t) → 0 as t → ∞.

We define F : M ⟶ R by F = d_{1}+d_{2}, and let Min_{F} be the set of interior minima of F. Set a_{0} = min(F|_{N}). Pick x ∈ M, and find a y ∈ N satisfying d(x, y) = d(x, N) = d_{1}(x). Then F (x) = d_{1}(x) + d_{2}(x) = d(y, x) + d_{2}(x) ≥ d_{2}(y) ≥ a_{0}, and we can have F (x) = a_{0} only if y ∈ N ∩ B_{a_{0}}(x) and x lies on an asymptote η : [0, ∞) ⟶ M starting from y. We therefore see that x ∈ Min_{F} if and only if x lies on an asymptote starting from N ∩ d_{2}^{−1}(a_{0}), so Min_{F} is nonempty and is contained in D_{1} ∩ D_{2}. Our functio n F satisfies ΔF ≤ 0 weakly on an open set containing Min_{F}, so we may apply Calabi’s maximum principle (see [Cal]) to conclude that Min_{F} is an open subset of the interior of M. Hence F is constant, and theorem 2 now follows using the same reasoning as in the proof of theorem 1.□

Proof of theorem 3: We will assume that r_{1} < r_{2} and let H̄(r) represent the mean curvature of the ball of radius r in M̄ with respect to the outward normal. Let γ be a minimizing geodesic from N_{1} to N_{2}, say of length L. Then since N_{2} lies outside of but is tangent to the boundary of the L−neighborhood of N_{1} the Ricatti equation forces the mean curvature of N_{2} with respect to the outward normal to be ≥ H̄(r_{1} + L). Hence, by our assumption on the mean curvature on N_{2}, H̄(r_{2}) ≥ H̄(r_{1} + L). Hence the fact that H̄ is increasing says that L ≤ r_{2} − r_{1}.

We now assume that d(N_{1}, N_{2}) = d(N̄_{1}, N̄_{2}) = r_{2} −r_{1}. This is proved just as in theorem 1 once we see that on D_{1} ∩ D_{2}, Δ(d_{1} + d_{2}) ≤ 0. The Riccati equation tells us that for x in D_{1} ∩ D_{2}, Δd_{1}(x) ≤ −(n − 1)H̄(r_{1} + d_{1}(x)) and Δd_{2}(x) ≤ (n − 1)H̄(r_{2} − d_{2}(x)). Now the monotonicity of H̄, along with the fact that d_{1}(x) + d_{2}(x) ≥ r_{2} − r_{1}, yields Δ(d_{1} + d_{2})(x) ≤ (n − 1)(−H̄(r_{1} + d_{1}(x)) + H̄(r_{1} + d_{1}(x))) = 0 .□

References

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[Cal] E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian Geometry, Duke Math. J., 25, (1958), 45-56.

[ChGr] J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative curvature, 6, (1971), 119-128.

[Wu] H. Wu, An elementary method in the study of nonnegative curvature, Acta Math., 142, (1979), 57-78.