SMALL VOLUME ON BIG N-SPHERES

CHRISTOPHER CROKE^{+}

Abstract. We consider Riemannian metrics on the n-sphere for n ≥ 3 such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the 2-dimensional case where Berger has shown that Area ≥ 1/2.

In 1977 Berger [B77] considered the set of Riemannian metrics g on the n-sphere such that d(x, A(x)) ≥ 1 and defined the constants:

h(n) ≡ inf{V ol(g)}.

We will also consider h̄(n) ≡ inf{V ol(g)|A^{∗}g = g}. Where you can think of A as the standard antipodal map or simply set A to be any fixed point free diffeomorphism of S^{n} with A^{2} = Id.

One of Berger’s reasons for his interest in these invariants was their use in getting lower bounds on the volume of balls B(p, R) in a complete Riemannian manifold with R less than half the injectivity radius inj(M). In this case all geodesics in the ball minimize distance. Hence for all r ≤ R the distance in B(p, R) between antipodal points of S(p, r) = ∂B(p, r) is 2r; and thus the intrinsic distance in S(p, r) is even larger. So we see V ol(S(p, r)) ≥ h(n − 1)(2r)^{n−1} and

V ol(B(p, R)) = ∫_{0}^{R} V ol(S(p, r))dr ≥

≥ ∫_{0}^{R} h(n − 1)(2r)^{n−1}dr = 2^{n−1} \frac{h(n − 1)}{n}R^{n}.

Berger noted that h(1) = 2 and proved h(2) ≥ \frac{1}{2} to get a (nonsharp) lower bound V ol(B(p, R)) ≥ 2R^{2} on the volume of 2-balls and V ol(B(p, R)) ≥ \frac{2}{3}R^{3} for 3-balls when R ≤ inj(M)/2.

In this short note we show by relatively easy examples:

Theorem 0.1. h(n) = 0 for n ≥ 3.

This question and some history surrounding it was also discussed in section 6 of [CK03]. In particular, Ivanov (see [I98]) has given examples of a sequence of metrics on S^{3} that Gromov-Housdorff converge to the standard metric but whose volumes go to zero.

Key words and phrases. isoperimetric.

+Supported by NSF grant DMS 02-02536.

The main open question about volume of balls can be stated as a conjecture which is open in all dimensions (even n = 2!).

Conjecture 0.2. If ω_{n} represents the volume of the unit n-sphere:

a) For any r ≤ \frac{inj(M)}{2}, V ol(B(r)) ≥ \frac{ω_{n}}{2} (\frac{2}{π})^{n}r^{n} with equality holding only if the ball is isometric to a hemisphere.

b) For any r ≤ inj(M), V ol(B(r)) ≥ \frac{ω_{n}}{π^{n}} r^{n}, where equality holds if and only if M is isometric to the round sphere of injectivity radius r (i.e. extrinsic radius \frac{r}{π}).

As mentioned above, Berger [B77] gave non-sharp lower bounds for n = 2 and n = 3. The author gave nonsharp lower bounds in all dimensions in [Cr80] and showed that b) was true for the “average” ball (i.e. \frac{1}{V ol(M)} ∫_{M} V ol(B(x, r))dx ≥ \frac{ω_{n}}{π^{n}} r^{n}) for any compact manifold M in any dimension [Cr84]. This followed Berger’s isoembolic inequality V ol(M) ≥ \frac{ω_{n}}{π^{n}} inj(M)^{n}, where equality holds if and only if M is isometric to the round sphere.

As to the question of the exact value of h(2) we have the following interesting

Conjecture 0.3. h(2) = \frac{4}{π} and the round sphere is the only space that achieves this.

The situation for h̄ is quite different. By identifying antipodal points we get a metric on ℝP^{n} such that all non contractible closed curves have length at least 1 (i.e. sys(M) ≥ 1). Hence Pu’s Theorem [Pu52] shows h̄(2) = \frac{4}{π}, while Gromov’s theorem [Gr83] gives us nonsharp positive lower bounds for h̄(n) for all n. The question as to the actual values of h̄(n) for n ≥ 3 is a hard and very interesting one.

One other place where this notion came up is in [Cr02]. There it is shown that for a Riemannian metric on a 3-sphere with d(x, A(x)) ≥ D and L = length of the shortest closed geodesic, we have V ol^{1/3} ≥ const min{L, 2D}. In this note we show that there is no constant such that V ol^{1/3} ≥ constD. It is still an open (and very interesting) question if there is a constant so that V ol^{1/3} ≥ constL.

We now construct the examples on the 3-sphere.

We will let MB_{ϵ} be the ϵ neighborhood in the flat plane ℝ^{2} of a tripod (three unit length line segments from the origin making angles \frac{2π}{3} with each other). As ϵ goes to 0 the area goes to 0 while the length of the boundary goes to 6. This has been known for a while as an example of a metric on a 2-disc whose area goes to 0 but such that on the boundary antipodal points satisfy d_{MB_{ϵ}}(p, A_{ϵ}(p)) ≥ d_{ℝ^{2}}(p, A_{ϵ}(p) ≥ 1. Here A_{ϵ}(p) is the point half way around the boundary. (Note that the non convexity of MB_{ϵ} is not the point here since one could use equilateral triangles in a more and more negatively curved simply connected space form.)

Our examples are just M_{ϵ}^{3} = ∂(MB_{ϵ} × MB_{ϵ}), which as the boundary of a 4-ball is a 3-sphere. Since M_{ϵ} = (∂(MB_{ϵ}) × MB_{ϵ}) ∪ (MB_{ϵ} × ∂(MB_{ϵ})), V ol(M_{ϵ}) = 2Area(MB_{ϵ})L(∂(MB_{ϵ})). Hence lim_{ϵ→0} V ol(M_{ϵ}) = 0.

We need to define the antipodal map Ā_{ϵ} on M_{ϵ}. First extend A_{ϵ} to a continuous homeomorphism A_{ϵ} : MB_{ϵ} → MB_{ϵ} with a single fixed point (the origin in ℝ^{2} say). One can do this simply by mapping the line segment between the origin and p ∈ ∂(MB_{ϵ}) to the line segment between the origin and A_{ϵ}(p). Then let

Ā_{ϵ} = (A_{ϵ} × A_{ϵ})|_{M_{ϵ}}.

For a point (p, q) ∈ ∂(MB_{ϵ}) × MB_{ϵ} ⊂ M_{ϵ} we have

d_{M_{ϵ}}((p, q), Ā_{ϵ}(p, q)) = d_{M_{ϵ}}((p, q), (A_{ϵ}(p), A_{ϵ}(q))) ≥
≥ d_{ℝ^{4}}((p, q), (A_{ϵ}(p), A_{ϵ}(q))) ≥ d_{ℝ^{2}}(p, A_{ϵ}(p)) ≥ 1.

Similarly if (p, q) ∈ MB_{ϵ}×∂(MB_{ϵ}) ⊂ M_{ϵ} then d_{M_{ϵ}}((p, q), Ā_{ϵ}(p, q)) ≥ 1.

This completes the proof of the theorem in the 3 dimensional case after smoothing the metric and Ā_{ϵ}. The higher dimensional cases are handled by spherical suspension. Specifically given a metric g on the n sphere S^{n} we construct a (singular) warped product metric g′ = cos^{2}(t)g + dt^{2} on the n + 1 sphere S^{n+1} = (S^{n} × [−\frac{π}{2}, \frac{π}{2}])/ ∼, where (x, \frac{π}{2}) ∼ (y, \frac{π}{2}) and (x, −\frac{π}{2}) ∼ (y, −\frac{π}{2}) for all x, y ∈ S^{n}. The antipodal map A_{n+1} : S^{n+1} → S^{n+1} is just A_{n+1}(x, t) = (A_{n}(x), −t). It is easy to check that if g has d(x, A_{n}(x)) ≥ π and V ol(g) = V then

D(x, A_{n+1}(x)) ≥ π and V ol(g′) = V ol(g) ∫− \frac{π}{2}^{\frac{π}{2}} cos^{n}(t)dt = \frac{ω_{n+1}}{ω_{n}} V ol(g).

So spherically suspending (and smoothing) a sequence of examples that shows h(n) = 0 gives us a sequence showing that h(n + 1) = 0.

References

[B77] M. Berger, Volume et rayon d’injectivité dans les variétés riemanniennes de dimension 3, Osaka J. Math., 14 (1977), 191- 200.

[Cr80] C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Scient. Ec. Norm. Sup., 4e serie, t.13 (1980), 419-435.

[Cr84] C. Croke, Curvature Free Volume Estimates, Inventiones Mathematicae 76 (1984), 515-521.

[Cr02] C. Croke, the volume and lengths on a three sphere, Comm. Anal. Geom., 10 (2002) no. 3, 467–474.

[CK03] Croke, C.; Katz, M.: Universal volume bounds in Riemannian manifolds, Surveys in Differential Geometry VIII, Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, May 3- 5, 2002, edited by S.T. Yau (Somerville, MA: International Press, 2003.) pp. 109 - 137. See arXiv:math.DG/0302248

[Gr83] Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1-147.

[I98] S. Ivanov Gromov-Hausdorff Convergence and volumes of manifolds, St. Petersburg Math. J., 9(1998) No.5, 945-959

[Pu52] Pu, P.M.: Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71.

Department ofbb: Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA

E-mail address: ccroke@math.upenn.edu