We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most $g$, then the filling function in dimension 3 is `almost' linear.
As usual we call vol 2 area and vol 1 length. If M is a simplicial 2manifold or a 2-manifold with a riemannian metric we denote by A(M) the area of M. Similarly if p is a (simplicial or riemannian) path we denote by l(p) the length of p.
We will show that one can give a bound of the Cheeger constant of a surface that depends only on its area. So for example there is a constant c such that any riemannian manifold homeomorphic to the 2-sphere S, which has area 1, has h(S) ≤ c. We state our results both in the simplicial and in the riemannian setting. Our results in the simplicial case are applied in the last section to higher isoperimetric inequalities. We provide explicit bounds but the constants in the theorems below are far from optimal. Proposition 2.3 . Let S be a riemannian manifold or a simplicial complex homeomorphic to the 2-sphere. Then the Cheeger constant, h(S), of S satisfies the inequality:
where A(S) is the area of S.
In general we obtain an upper bound that depends on the genus:
Theorem 2.6. Let S be a closed orientable surface of genus g ≥ 1 equipped either with a riemannian metric or with a simplicial complex structure. Let A(S) be its (simplicial or riemannian) area. Then the Cheeger constant, h(S), of S satisfies the inequality:
One sees easily that the same bound applies to surfaces with boundary (just collapse the boundary curves to points to obtain a closed surface). One can get bounds for non-orientable surfaces too by passing to the orientable double cover.
If (M n , g) is a riemannian manifold of infinite volume the isoperimetric profile function of M n is a function I M : R + → R + defined by:
where Ω ranges over all regions of M n with smooth boundary. One can define similarly an isoperimetric profile function I M : N → N for simplicial manifolds M n . Other functions related to the isoperimetric problem are the filling area, F A 0 , and homological filling area, F A h , functions of M that we define below. For more information on filling invariants and applications we refer the reader to the seminal paper of Gromov [16].
If p is a smooth contractible closed curve in M we define its filling area, F illA 0 (p), as follows: We consider all riemannian discs D such that there is a 1-lipschitz map f : D → X with f | ∂D = p. We define F illA 0 (p) to be the infimum of the areas of this collection of disks. We define now the filling area function of M by: F A 0 (t) = sup p {F illA 0 (p) : l(p) ≤ t} where p ranges over all smooth contractible closed curves of M and D over riemannian disks filling p.
More generally we can consider 1-cycles c (i.e. unions of closed curves) that can be filled by 2-cycles to define the homological filling area function (see sec. 2 for details).
Gromov ([17], ch. 6, see also [10], ch.6) showed the following:
Gromov’s Theorem. Let (M n , g) be a simply connected riemannian manifold. Assume that there is some t 0 such that for all t > t 0 , F A 0 (t) ≤ 1 16π t 2 . Then there is a constant K such that for all t > t 0 , F A 0 (t) ≤ Kt.
Gersten [14] observed that this theorem holds also for homological filling area F A h (see also [17], 6.6E, 6.6F), while Olshanskii [23] gave an elementary proof of Gromov’s theorem (see as well [8], [25], [12], for other proofs).
If the dimension of M is 2 then there is an obvious link between filling area and isoperimetric profile, so from Gromov’s theorem we readily obtain the following:
Corollary. Let (S, g) be a riemannian manifold homeomorphic to the plane. Assume that there is some t 0 such that for all t > t 0 , I S (t) ≥ 4 √ π √ t. Then there is a constant δ > 0 such that for all t > t 0 , I S (t) ≥ δt.
We note that the isoperimetric problem for surfaces has been studied extensively (see [7], [15], [22], [28], [29], [31]).
We see that the ‘gap’ in the filling functions implies a ‘gap’ for the isoperimetric profiles of riemannian planes. It is reasonable to ask whether there are gaps in the isoperimetric profile of other surfaces. Although this does not hold in general we show that this is true for planes with holes or more generally surfaces of finite genus. Theorem 3.5. Let S be a plane with holes equipped either with a riemannian metric or with a simplicial complex structure. Assume that there is some K > 0 such that for all t ∈ [K, 100K], I S (t) ≥ 10 2 √ t. Then there is a constant δ > 0 such that for all t > K, I S (t) ≥ δt.
One obtains as a corollary that the same holds for finite genus surfaces: Corollary 3.6. Let S be a non-compact surface of finite genus equipped either with a riemannian metric or with a simplicial complex structure. Assume that there is some K > 0 such that for all t ∈ [K, 100K], I S (t) ≥ 10 2 √ t. Then there is a constant δ > 0 such that for all t > K, I S (t) ≥ δt.
It is an interesting question whether Gromov’s theorem on filling area has an analogue for higher dimensional filling functions. Our results on Cheeger constants of surfaces can be used to obtain some partial results
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