Curvature-free upper bounds for the smallest area of a minimal surface.

Alexander Nabutovsky and Regina Rotman

May 11, 2004

Abstract In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold M^{n} with a trivial first homology group. The first upper bound will be in terms of the diameter of M^{n}, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of M^{n}. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold M^{n} with H_{1}(M^{n}) = ... = H_{k−1}(M^{n}) = {0}, (k > 2). The above results are the first results of such nature.

1 Main results

In this paper we prove an effective version of results of J. Pitts ([P] , section 4) establishing the existence of stationary integral varifolds of all dimensions ≤ i corresponding to a homology class h ∈ H_{i}(M^{n}) for closed Riemannian manifold M^{n}. Namely, we are able to give a priori upper bounds for the smallest volume of a stationary integral varifold of dimension k in M^{n} in terms of the diameter or M^{n} or in terms of the volume of M^{n}. However, we are able to do this only in the situation when all homology groups of M^{n} of dimensions < k vanish. Moreover, we will need the homological filling functions in these dimensions for our estimates. (These homological filling functions will be defined below.)

For readers who are not familiar with Geometric Measure Theory note that stationary integral k-dimensional varifolds are (singular) minimal submanifolds. The set of singular points has k-dimensional Hausdorff measure zero. The set of regular points consists of countably many k-dimensional submanifolds of M^{n} with positive integral multiplicities. The k-dimensional Hausdorff measure of the regular set where each point is counted with its multiplicity is called the mass of the varifold and must be finite. The stationarity of a varifold v has the following formal meaning: Let X be a smooth vector field on M^{n}. Consider the corresponding 1-parametric flow of diffeomorphisms Φ_{t}. Apply these diffeomorphisms to v and consider the mass of the resulting varifold as a function of t. By definition, v is stationary, if t = 0 is a critical point of this function.

We would like also to note that there exists an oriented analog of integral varifolds, namely, integral cycles. Since one can integrate k-forms over integral k-cycles, they can be regarded as a subset of the dual space to the space of k-forms and often are considered with a toplogy of the dual space. The dual space norm of k cycles is called their mass. F. Almgren proved ([A]) that the ith homotopy group of the space of integral k-cycles is isomorphic with the (i + k)th homology group of the ambient manifold. This fact plays a crucial role in the geometric measure theory. Yet integral cycles have the following technical disadvantage in comparison with integral varifolds: the mass is not a continuous function on the space of cycles. Indeed, consider a translate of a cycle z in R^{n} by a small vector. Consider the sum of this translate and −z (that is, z taken with the opposite orientation). If ε ≠ 0 then the mass of the cycle is equal to twice the mass of z. Hoewer, when ε become equal to zero the cycles cancel each other and the mass is equal to zero. If one would forget about the orientations and consider the corresponding varifolds, then this phenomenon cannot take place: the mass is a continuous functionional on spaces of varifolds.

Following an earlier work of F. Almgren J. Pitts developed a version of Morse theory for spaces of cycles. He described how one can start from a non-trivial homology class of the manifold and to assign to it non-trivial stationary integral varifolds in each dimension lower than the dimension of the homology class. He used the Almgren isomorphism between homology groups of the manifold and homotopy groups of the space of cycles here. Note that one gets here merely stationary varifolds instead of minimal cycles precisely because of the mentioned above technical problem with the possible vanishing of the mass of cycles in the limit.

In this paper we provide a quantitative version of Pitts’s results. We combine his technique with our technique from [NR1] to derive upper bounds for the volume of the stationary integral cycles. Our results could, thus, be considred as a multidimensional generalization of results of [NR1]. Yet, they are different from the results of [NR1] in the following three aspects: 1. Here we need to assume that first (k − 1) homology groups of the Riemannian manifold vanish, whenever we did not need any assumptions about the ambient Riemannian manifold in [NR1]; 2. In [NR1] stationary 1-dimensional varifolds had only finitely many pieces (and the number of pieces was controlled in terms of the dimension of the manifold); and 3. Our present estimates involve not only diameter or volume but also defined below homological filling functions. However, we believe that the last limitation is unavoidable and the minimal volume of, say, a minimal hypersurface in a three-dimensional Riemannian manifold diffeomorphic to S^{3} cannot be majorized only in terms of the diameter of the manifold. Our present results can be also compared with the results of [NR2]. There we considered only the case when the ambient manifold is diffeomorphic to S^{3}. Moreover, our estimates required a two-sided bound for the sectional curvature of the manifold. However there we got an upper bound for the area of a minimal surface diffeomorphic to S^{2}, whereas here we provide an upper bound for the area of a minimal surface of an unknown topological type.

In order to state our results we need the notion of homological filling functions. In the definition below we will be considering only singular chains with smooth simplices. The volume of a singular simplex will be defined as the volume of the standard simplex endowed with the pullback measure; the volume of a singular chain Σ_{i}a_{i}σ_{i}, where a_{i} ∈ R and σ_{i} are singular simplices, is defined as Σ|a_{i}| vol(σ_{i}).

Definition 1.1 (Homological filling function.) Let M^{n} be a closed Riemannian manifold of dimension n. Let γ(t) denote a closed piecewise differentiable curve. Then the homological filling function F H = F H_{1} : R_{+} ⟶ R_{+} will be defined as follows: F H(x) = max_{{γ(t)|length(γ(t))≤x}} min_{{Σ_{2}|∂Σ_{2}=γ(t)}} Area(Σ_{2}), where Σ_{2} is a singular 2chain, and Area(Σ_{2}) denotes its area.

Definition 1.2 (Homological filling function in dimension k > 1.) Let M^{n} be a closed Riemannian manifold of dimension n with the trivial k-th homology group. Then the homological filling function of order k, F H_{k} : R_{+} ⟶ R_{+} will be defined as follows: F H_{k}(x) := max_{{Σ_{k}|m(Σ_{k})≤x}} min_{{Σ_{k+1}|∂Σ_{k+1}=Σ_{k}}} m(Σ_{k+1}), where Σ_{k}, Σ_{k+1} are singular chains of dimension k, k + 1 respectively.

Firther, recall that M. Gromov ([Gr]) defined the filling radius of a closed Riemannian manifold X^{n} embedded in a metric space Y as the minimal radius of a neighborhood of X^{n} in Y such that X^{n} bounds in this neighborhood. The filling radius of an abstract closed Riemannian manifold X^{n} is the filling radius of its Kuratowski embedding into L^{∞}(X) (Recall that the Kuratowski embedding assigns to each x ∈ X^{n} the distance function to x. See a more formal and detailed definition of the filling radius below in the next section).

In this paper we will prove the following theorems:

Theorem 1.3 Let M^{n} be a closed Riemannian manifold of dimension n with the trivial first homology group. Then the smallest area A(M^{n}) of a possibly singular minimal surface in M^{n} satisfies the following inequalities:

(1)A(M^{n}) ≤ \frac{(n + 1)!}{2}F H(2d(M^{n}));

(2)A(M^{n}) ≤ \frac{(n + 2)!}{6}F H(6F illRad(M^{n}));

(3)A(M^{n}) ≤ \frac{(n + 1)!}{6}F H(6(n + 1)n^{n}\sqrt{(n + 1)!}vol(M^{n})^{\frac{1}{n}}).

Here d(M^{n}) denotes the diameter, F illRad(M^{n}) denotes the filling radius, and vol(M^{n}) denotes the volume of M^{n}. If n = 3, then there exists a nonsingular embedded minimal surface such that its area satifies the inequalities (1)-(3).

Formally speaking, “possibly singular minimal surface” means here “2dimensional stationary integral varifold”. More generally:

Theorem 1.4 Let M^{n} be a closed Riemannian manifold of dimension n with H_{1}(M^{n}) = H_{2}(M^{n}) = ... = H_{k−1}(M^{n}) = {0}. Then for each k ≥ 2 there exists a non-trivial stationary integral varifold of dimension k, such that its mass V_{k} is bounded from above by

(1)V_{k} ≤ \frac{(n + 1)!}{k!}F H_{k}(k(F H_{k−1}(...(3F H_{2}(2d))...))).

(2)V_{k} ≤ \frac{(n + 2)!}{(k + 1)!}F H_{k}((k + 1)F H_{k−1}(...(4F H_{2}(6F illRadM^{n}))...)).

(3)V_{k} ≤ \frac{(n + 2)!}{(k + 1)!}F H_{k}((k+1)F H_{k−1}(...(4F H_{2}(6(n+1)n^{n}\sqrt{(n + 1)!}vol(M^{n})^{\frac{1}{n}})...))).

If k = n − 1 and n ≤ 7 then we can ensure that a non-trivial stationary integral varifold satisfying the inequalities (1)-(3) is a smooth embedded hypersurface; if k = n − 1 and n = 8, then we can ensure that it has only isolated singularities, if k = n − 1 and n ≥ 9 we can ensure that the Hausdorff dimension of its singular set does not exceed k − 7.

Remarks.

1. The third inequalities in Theorems 1.3, 1.4 follow from the second inequalities and the above mentioned upper bound for the filling radius in terms of volume proven by M. Gromov, [Gr].

2. Theorems 1.3, 1.4 provide effective versions of the results by J. Pitts [P], Theorems 4.10, 4.11. Therefore our proofs provide the existence not merely stationary integral varifolds with volume ounded as stated in Theorems 1.3, 1.4 but stationary integral varifolds that are almost minimizing in a small annular neighborhood of every point precisely as in Theorems 4.10, 4.11 in [P]. (See 3.1(2) of [P] for the definition of almost minimizing varifolds.) In particular, our stationary varifolds are stable in a neighborhood of every point but finitely many, and the cardinality of the set of points, where stability is not guaranteed does not exceed some N(n) depending only on the dimension of M^{n}.

3. In view of the last assertion of Theorem 1.3 one can ask for an upper bound of the smallest area of am embedded non-singular minimal hypersurface diffeomorphic to S^{2} in a three-dimensional Riemannian manifold diffeomorphic to S^{3}. The existence of such a surface was proven by F. Smith and L. Simon ([S], see also [CD]). In [NR2] we obtained explicit upper bounds for the area of such a surface. These estimates are given in terms of an upper bound for the diameter of M^{3}, a positive lower bound for its volume and a two-sided bound for the sectional curvature. The methods of [NR2] have very little in common with the methods of the present paper.

2 Ideas of the proofs

This paper extends our earlier paper, (see [NR1]) in which we have found two curvature-free upper bounds for the minimal length of a strongly stationary 1-cycle. Strongly stationary 1-cycle can be considered as homological equivalents of closed geodesics and also as especially nice 1-dimensional equivalents of minimal surfaces with singularities. The proofs in the present paper generalize the proofs in [NR1]. Also they heavily use the results of J.

Pitts ([P]), and can be regarded as a quantitative version of his work. First we will present an informal explanation of the proof of the first estimate of Theorem 1.3.

For the sake of simplicity of the explanation assume that M is diffeomorphic to a round 3-sphere S^{3}. We are going to show that there exists a minimal imbedded surface in S^{3} of area satisfying the upper bounds of Theorem 1.3

Let f : S^{3} ⟶ M be any diffeomorphism. Assume S^{3} was triangulated into simplices of diameter smaller than δ. Let the standard 4-disc D^{4} be triangulated as a cone over S^{3}. A k-simplex [v_{i_{0}}, ..., v_{i_{k}}] of D^{4} will be denoted σ_{i}^{k}, (or sometimes σ_{i_{0},...,i_{k}}^{k}), where k ∈ {1, 2, 3, 4}.

The proof will be by contradiction. Suppose A(M) > 12F H(2d+δ)+5δ. We will show that in that case there exists a singular chain, such that f_{∗}([S^{3}]) bounds. Here [S^{3}] denotes the fundamental class of S^{3}. This trick is based on the obstruction technique first used in the paper [Gr].

Figure 1: Table 1.

We will begin by extending the map f : S^{3} ⟶ M to 1-skeleton of M and then by assigning to each 2-simplex of D^{4} a singular 2-chain on M, (see fig. 1). To extend to 0-skeleton, we will assign to the center of the disc p, an arbitrary point p̃ ∈ M. Next we extend to 1-skeleton by assigning to each arbitrary edge [p, v_{i_{1}}] connecting the center of D^{4} with a vertex v_{i_{1}} and directed from p to v_{i_{1}}, a minimal geodesic that connects p̃ with a vertex ṽ_{i_{1}} = f(v_{i_{1}}), directed from p̃ to ṽ_{i_{1}} and denoted [p̃, ṽ_{i_{1}}]. Now, consider a simplex σ_{i}^{2} = [v_{i_{0}}, v_{i_{1}}, v_{i_{2}}], where v_{i_{0}} = p. Its boundary is mapped to the curve [ṽ_{i_{1}}, ṽ_{i_{2}}]−[ṽ_{i_{0}}, ṽ_{i_{1}}]+[ṽ_{i_{0}}, ṽ_{i_{1}}] of length ≤ 2d+δ. Let s_{i}^{2} (also denoted as s_{i_{0},i_{1},i_{2}}^{2}) be a singular 2-chain of the smallest area, such that ∂s_{i}^{2} = f(∂σ_{i}^{2}). Then its area is ≤ F H(2d + δ). We will assign to σ_{i}^{2} this surface s_{i}^{2}.

Now we will slightly change our tactics. Consider a 3-simplex σ_{i}^{3}. There is a preassigned chain S_{i}^{2} = Σ_{j=0}^{3}(−1)^{j}s_{i_{0},...,î_{j},...,i_{3}}^{2} of area ≤ 3F H(2d + δ) + δ that corresponds to the boundary of this simplex. (We can assume without loss of generality that the area of s_{i_{1},i_{2},i_{3}}^{2} ≤ δ). S_{i}^{2} is an element in the space of integral 2-cycles Z_{2}(M^{n}, Z). Since by our assumption there are no minimal surfaces of “small” area, S_{i}^{2} can be connected with the zero cycle by a curve that passes through cycles of area ≤ 3F H(2d + δ) + δ, i. e. there exists h_{i}^{1} : [0, 1] ⟶ Z_{2}(M, Z), (sometimes denoted h_{i_{0},i_{1},i_{3}}^{1}), such that h_{i}^{1}(0) = S_{i}^{2} and h_{i}^{1}(1) is the 0-cycle. We will assign the above path to the simplex σ_{i}^{3}. Finally, take a 4-simplex σ_{i}^{4}, (see fig. 2). Consider its boundary ∂σ_{i}^{4} = Σ_{j=0}^{4}(−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{4}}]. Each of its faces (−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{4}}] corresponds to the map (−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}^{1}(t), where −h_{i_{0},...,î_{j},...,i_{4}}^{1}(t) is a cycle that is geometrically the same as h_{i_{0},...,î_{j},...,i_{4}}^{1}(t), but is oppositely oriented.

Figure 2: The loop f_{i}^{1}(t).

Now we will perform the trick that we will use throughout the paper. Consider the map f_{i}^{1} : [0, 1] ⟶ Z_{2}(M, Z), such that f_{i}^{1}(t) = Σ_{j=0}^{4}(−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}(1−t). Note that f_{i}^{1}(0) is the zero cycle and that f_{i}^{1}(1) is also the zero cycle represented by 10 pairs of singular chains, where each pair contains two copies of one chain with opposite orientations. Thus, f_{i}^{1}(t) is a loop in the space of 2-cycles. Note also that the area of f_{i}^{1}(t) is bounded from above by 12F H(2d + δ) + 5δ. By our assumption f_{i}^{1}(t) is contractible in Z_{2}(M, Z) along the cycles of area ≤ 12F H(2d + δ) + 5δ. (Otherwise the proof of Theorem 4.10 in [P] implies the existence of a minimal surface (=a stationary integral varifold) of area ≤ 12F H(2d+δ)+5δ, which is impossible because of our assumption.) Thus, we obtain a disc h_{i}^{2} : D^{2} ⟶ Z_{2}(M, Z). We will assign this disc to the simplex σ_{i}^{4}.

From the geometric measure theory we know that

π_{k}(Z_{2}(M, Z)) = H_{k+2}(M).

So, the disc of the form h_{i}^{2} : D^{2} ⟶ Z_{2}(M, Z) corresponds to a singular chain s_{i}^{4} on M. (However the construction of the correspondence is technical. Our intention to avoid the technicalities of this construction is responsible for the fact that the proofs of Theorems 1.3 and 1.4 given in the next sections are more awkward than this sketch). Now consider S^{4} = Σ_{i=1}^{Q}s_{i}^{4}, where Q is the number of simplices of dimension 4 in the triangulation of D^{4}. ∂S^{4} = f_{∗}([S^{3}]). Thus, we obtain a contadiction. Therefore, we can conclude that A(M) ≤ 12F H(2d + δ) + 5δ. Now let δ go to zero.

The proof of Theorem 1.3 will be given in Section 2. The proof of Theorem 1.4 will be given in Section 3. Section 1 will be devoted to the discussion of the regularity of minimal surfaces the area of which we estimate. As we have mentioned before the arguments of the proofs will be somewhat more technical than the explanations above, and will run as follows.

Once again, for the sake of simplicity of the explanation, assume that our manifold M is diffeomorphic to the standard 3-sphere. Let f : S^{3} ⟶ M be a diffeomorphism, and suppose that the triangulation of S^{3} and the induced triangulation on M is very fine, (i.e. the diameter of simplices smaller than δ). Let D^{4} be triangulated as a cone over S^{3}.

The proof will consist of the following three steps: Step 1 (Easy). Corresponding to the map f : S^{3} ⟶ M one can construct a non-contractible map f̃ : S^{1} ⟶ Z_{2}(M, Z), a loop in the space of the integral 2-cycles on a manifold M. We do not have any control over the masses of f(t), t ∈ S^{1}. Step 2 (Main Step). The loop f̃ is homotopic to the sum of Q loops g̃_{i}, where the number Q equals to the number of simplices in the triangulation of S^{3}, and the masses of all 2-cycles g̃_{i}(t), t ∈ S^{1} satisfy upper bounds as in the right hand side of the inequality (1) in the text of Theorem 1.3. Step 3 (An application of [P]). We show that if there is no minimal surface with “small” area than each loop in step 2 can be contracted to a point, thus obtaining a contradiction.

Figure 3: The loop f̃.

Step 1. Let {σ_{i}^{3}}_{i=1}^{Q} be the set of simplices that constitute the fundamental class [S^{3}] of S^{3}. Then {σ̃_{i}^{3}}_{i=1}^{Q} are the corresponding simplices of M.

Without any loss of generality we can assume that each σ̃_{i}^{3} can be obtained from its boundary ∂σ̃_{i}^{3} by contracting it to the “center” of the simplex, the point p̃_{i} by a homotopy h̃_{i} : [0, 1] ⟶ M along the 2-spheres h̃_{i}(t), (see fig. 3 (a). This figure schematically depicts σ̃_{i}^{3} as a simplex of dimension 2).

Then f̃ can be constructed as follows. We will begin with the zero cycle that consists of the sum of the “centers” of simplices: Σ_{i=1}^{Q}p̃_{i}. We will then follow the images of ∂σ̃_{i}^{3}’s under the area decreasing homotopies h̃_{i}(t)’s, that is f̃(t) = Σ_{i=1}^{Q}h̃_{i}(1 − t). Note that f̃(1) = Σ_{i=1}^{Q}∂σ̃_{i}^{3}, which is also the zero cycle. Thus, we obtain a (non-contractible) loop in the space of 2-cycles, (see fig. 3 (b). There f(S^{3}) is schematically depicted as a 2-dimensional sphere).

Step 2. Each loop g̃_{i} is constructed as follows. Consider a disc D^{4}, such that ∂D^{4} = S^{3}, triangulated as a cone over S^{3}. Let p ∈ D^{4} be the center of this disc. We can assign to this point an arbitrary point p̃ ∈ M, thus extending the map f : S^{3} ⟶ M to the 0-skeleton of D^{4}. Next consider a line segment of the form [p, v_{i}] directed from p to v_{i}. We can assign to it a minimal geodesic segment of length smaller than the diameter d of M joining the point p̃ with the vertex ṽ_{i} = f(v_{i}). This segment will be directed from p̃ to ṽ_{i} and denoted as [p̃, ṽ_{i}]. This extends f to 1-skeleton of D^{4}. Next consider a 2simplex σ_{i}^{2} = [p, v_{i_{1}}, v_{i_{2}}], (it will sometimes be denoted as σ_{i_{0},i_{1},i_{2}}^{2}. Its boundary is mapped to a closed curve [ṽ_{i_{1}}, ṽ_{i_{2}}]−[p̃, ṽ_{i_{2}}]+[p̃, ṽ_{i_{1}}]. Let s_{i}^{2} (sometimes

Figure 4:

denoted as s_{i_{0},i_{1},i_{2}}^{2}) be a singular 2-chain of the smallest area filling this curve. We can assign s_{i}^{2} to simplex σ_{i}^{2}, thus we obtain a map from the 2-skeleton of D^{4} to the space of integral 2-currents. Now let us take an arbitrary 3-simplex σ_{i}^{3} = [p, v_{i_{1}}, v_{i_{2}}, v_{i_{3}}]. Its boundary ∂σ_{i}^{3} = ∑_{j=0}^{3}(−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{3}}], where v_{i_{0}} = p. For each face (−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{3}}] we have pre-assigned a singular 2-chain (−1)^{j}s_{i_{0},...,î_{j},...,i_{3}}^{2}. Consider Σ_{j=0}^{3}(−1)^{j}s_{i_{0},...,î_{j},...,i_{3}}^{2}. This is a 2-cycle of area smaller than 3F H(2d + δ) + δ, (without loss of generality we can assume that the area of s_{i_{1},i_{2},i_{3}}^{2} is smaller than δ). Thus, it is an element in Z_{2}(M, Z). Assuming there is no minimal 2-cycle that locally minimizes the area of area smaller than that, this cycle can be connected with the zero cycle, that is there exists a map h_{i_{0},...,i_{3}}^{1} : [0, 1] ⟶ Z_{2}(M, Z) that begins with our cycle and ends with the zero cycle. Finally, let us construct g̃_{i}. Consider a 4-simplex σ_{i}^{4} = [v_{i_{0}}, v_{i_{1}}, ..., v_{i_{4}}]. Its boundary, ∂σ_{i}^{4} = Σ_{j=1}^{4}(−1)^{j}[p, v_{i_{1}}, ..., v̂_{i_{j}}, ..., v_{i_{4}}]. For each face in the boundary (−1)^{j}[v_{i_{0}}, v_{i_{1}}, ..., v̂_{i_{j}}, ..., v_{i_{4}}] there was constructed a map (−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}^{3} : [0, 1] ⟶ Z_{2}(M, Z), where −h(t) is the same cycle as h(t), but taken with an opposite orientation. We will define the map g̃_{i}(t) as the sum Σ_{j=0}^{4}(−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}^{1}(1−t). Note that g̃_{i}(0) is the zero cycle represented by p̃_{i_{0},...,î_{j},...,i_{4}} and that g̃_{i}(1) is also the zero cycle, because of the cancellations due to the fact that each surface that corresponds to a 2-dimensional face of the simplex σ_{i}^{4} enters twice with the opposite orientation. Thus, g̃_{i}(t) is a loop. Area of g̃_{i}(t) is bounded from above by 12F H(2d + δ) + 5δ. Moreover, f̃(t) is homotopic to the sum of the above loops.

Step 3. Since f̃(t) is homotopic to the sum of the loops g̃_{i}, (see fig.

4), at least one of those loops in not contractible. Therefore, if we try to contract it using a mass decreasing flow described in ch. 4 of [P] (and introduced earlier by F. Almgren), it should get stuck on a critical point, which, as it was shown by J. Pitts in chapter 4 of [P] would be a stationary integral varifold. In ch. 7 of [P] J. Pitts uses the results of [SSY] imply that in the three-dimensional case this stationary varifold will turn out to be an embbedded minimal surface.

Similarly, one can show that A(M^{n}) ≤ \frac{(n+2)!}{6}F H(6F illRad(M^{n})). Recall that the filling radius of a Riemannian manifold M^{n} was introduced by Gromov in [Gr] as follows:

Definition 2.1 (Filling Radius) Let M^{n} be a Riemannian manifold topologicaly imbedded into an arbitrary metric space X. Then its filling radius, denoted F illRad(M^{n} ⊂ X), is the infimum of ε > 0, such that M^{n} bounds in the ε- neighborhood N_{ε}(M^{n}), i.e. homomorphism H_{n}(M^{n}, Z_{2}) ⟶ H_{n}(N_{ε}(M^{n}), Z_{2}) induced by the inclusion map vanishes. Let M^{n} be an abstract manifold. Then its filling radius, denoted F illRad(M^{n}) will be F illRad(M^{n} ⊂ X), where X = L^{∞}(M^{n}), i.e. the Banach space of bounded Borel functions f on M^{n} and the imbedding of M^{n} into X is the map that assigns to each point p of M^{n} the distance function p ⟶ f_{p} = d(p, q).

In the same paper Gromov poved the following important inequality relating the filling radius and the volume:

Theorem 2.2 ((M. Gromov))

F illRad(M^{n}) ≤ \sqrt{(n + 1)!}n^{n}(n + 1)vol(M^{n})^{\frac{1}{n}} .

Now, suppose once again that M^{n} is diffeomorphic to S^{3}. The definition of the Filling Radius implies that M^{n} bounds in the (F illRad(M^{n}) + δ) - neighborhood of M^{n} in L^{∞}(M^{n}). Let W fill M^{n} in the (F illRad(M^{n}) + δ)- neighborhood of M, (that is M^{n} = ∂W). Without loss of generality we can assume that W is a polyhedron. Suppose that W together with M^{n} is endowed with a very fine triangulation. As in (i) the proof will consist of the three steps: constructing a non-contractible loop f̃: S^{1} ⟶ Z_{2}(M^{n}, Z); constructing a family of loops g̃_{i} : S^{1} ⟶ Z_{2}(M^{n}, Z); concluding that one of those loops must be non-contractible, and therefore there exists an almost minimizing integral varifold of area smaller than the area of integral cycles through which this loop passes.

Step 1. The first step is analogous to that of (i).

Step 2. Let v_{i} be a vertex of W. Assign to it a closest vertex in M^{n}, denoted ṽ_{i}. Thus, d(v_{i}, ṽ_{i}) ≤ F illRad(M^{n}) + δ. This extends the identity map Id : 2004 − 006.tex, v1.12004/05/2717 : 35 : 19levyExplevy on M^{n} to 0-skeleton of W. Now to any 1-simplex [v_{i_{1}}, v_{i_{2}}] ⊂ W \ M^{n} we can assign a minimal geodesic segment connecting ṽ_{i_{1}} and ṽ_{i_{2}} of length smaller than 2F illRad(M^{n})+3δ. This segment will be denoted as [ṽ_{i_{1}}, ṽ_{i_{2}}]. This extends Id : 2004 − 006.tex, v1.12004/05/2717 : 35 : 19levyExplevy to 1-skeleton of M^{n}. Next consider an arbitrary 2-simplex σ_{i}^{3} = [v_{i_{1}}, v_{i_{2}}, v_{i_{3}}]. Its boundary is mapped to a curve of length smaller than 6F illRad(M^{n})+9δ. Let s_{i_{1},i_{2},i_{3}}^{2} be a singular 2-chain of the smallest area that has this curve as its boundary. Its area is going to be smaller than F H(6F illRad(M^{n}) + 9δ). The rest of the procedure (including Step 3) is the same as in the proof of (1).

3 The proof of Theorem 1.3

In this section we will prove 1.3. We will begin by proving statements (i) and (ii). Statement (iii) follows from (ii).

Proof. We are going to prove the theorem by contradiction. Assume that there is no stationary integral varifold as in the text of the theorem and then prove that it exists.

Suppose M^{n} is a (q − 1)-connected manifold with π_{q}(M^{n}) ≠ {0}. Let f : S^{q} ⟶ M^{n} be a non-contractible map in case (i), and in case (ii) chooser a (singular) W that fills M (in L^{∞}(M^{n})) such that for each x ∈ W dist(x, M^{n}) ≤ (n + 1)n^{n}\sqrt{(n + 1)!}vol(M^{n})^{\frac{1}{n}}. (As it had been already mentioned, in [Gr] M. Gromov proved that such a filling exists.) Let S^{q} be triangulated into simplices σ_{i}^{q} of diameter d(σ_{i}^{q}) ≤ δ for some small δ in case (i). In case (ii) assume that W (and M^{n} = ∂W) has been triangulated into simplices σ_{i}^{n} of diameter at most δ. In both cases let Q denote the number of simplices. The proof will consist of three steps: Step 1. We will begin by constructing a non-contractible map f̃: S^{q−2} ⟶ Z_{2}(M^{n}, Z) in case (i) that corresponds to the original map f : S^{q} ⟶ M^{n} under the Almgren correspondance in case (i) and the map f̃ : S^{n−2} ⟶ Z_{2}(M^{n}, Z) that corresponds to the fundamental class of M^{n} in case (ii). Step 2. We will construct maps g̃_{i} from S^{q−2} in case (i) and from S^{n−2} in case (ii) to the space Z_{2}(M^{n}, Z), i ∈ {1, ..., Q}, where Q is a number of simplices of dimension (q + 1) in the triangulation of D^{(q+1)} in case (i) and it is a number of simplices of dimension (n + 1) in the triangulation of W in case (ii). It will turn out that f̃ is homotopic to the sum of g̃′_{i}s. Therefore, at least one of the maps g̃′_{i}s is not contractible. The most important feature of g̃_{i} is that the mass of g̃_{i}(t) does not exceed the right hand side of (1) (or (2)) for every t, i. Step 3. We can conclude that there exists an almost minimizing integral varifold of area smaller than that of the cycles through which passes the non-contractible map of Step 2.

Step 1. Let us begin by considering simplices σ̃_{i}^{3} = [ṽ_{i_{0}}, ..., ṽ_{i_{3}}] of dimension 3, (we will sometimes denote it as σ̃_{i_{0},...,i_{3}}^{3}, in general, k-simplices will be sometimes denoted as σ̃_{i_{0},...,i_{k}}^{k}). Without loss of generality we can assume that this simplex can be generated by contracting its boundary ∂σ̃_{i}^{3} to the “center” of the simplex p̃_{i}^{3}, (sometimes denoted p̃_{i_{0},...,i_{3}} ) with a homotopy h̃_{i}^{1}(t) = h̃_{i_{0},...,i_{3}}^{1}. This homotopy can be chosen, for example, to be the radial homotopy of the boundary of the (very small and, therefore, almost Euclidean) simplex to its center. Note that we can consider this homotopy as a path in the space of integral 2-cycles that starts with the boundary of a simplex and ends with the zero cycle. Next consider a 4-dimensional simplex σ̃_{i}^{4} = [ṽ_{i_{0}}, ..., ṽ_{i_{4}}]. Its boundary ∂σ̃_{i}^{4} = Σ_{j=0}^{4}(−1)^{j}[ṽ_{i_{0}}, ..., ˆ ṽ_{i_{j}}, ..., ṽ_{i_{4}}]. For each 3-face (−1)^{j}[ṽ_{i_{0}}, ..., ˆ ṽ_{i_{j}}, ..., ṽ_{i_{4}}] we have constructed a path (−1)^{j}h̃_{i_{0},...,î_{j},...,i_{4}}^{1} :

[0, 1] ⟶ Z_{2}(M^{n}, Z). Now, let us construct a map f̃_{i}^{1} : [0, 1] ⟶ Z_{2}(M^{n}, Z) as follows: let f̃_{i}^{1}(t) = Σ_{j=0}^{4}(−1)^{j}h̃_{i_{0},...,î_{j},...,i_{4}}^{1}(1 − t). Then, note that f̃_{i}^{1}(0) is the zero cycle that corresponds to the sum of the “centers” of the 3-faces of σ̃_{i}^{4} and that f̃_{i}^{1}(0) is also the zero cycle due to the cancellation of each pair of the two faces, that have the same geometric image, but are taken with the opposite orientation. Thus, this map is really a “small” loop in the space of the integral 2-cycles. We can contract it there and obtain a “small” disc h̃_{i}^{2} : D^{2} ⟶ Z_{2}(M^{n}, Z), (this map will also be sometimes denoted as h̃_{i_{0},...,i_{4}}^{2}). Geometrically, this discs corresponds to the considered 4-dimensional simplex regarded as the chain filling its boundary. The construction of this disc can be made absolutely explicit: It is just a slicing of the simplex into 2-surfaces that extends the slicing of its boundary. It can be obtained just by coning of the slicing of the boundary. Now we can just proceed by induction until we will get a slicing of each q-dimensional singular simplex of M^{n} obtained from f and the considered very fine triangulation of S^{q} (case (i)) or n-dimensional simplex of M^{n} (case (ii)). As the result, we obtain maps h̃_{i}^{q} of D^{q−2} into Z_{2}(M, Z) (or h̃_{i}^{n} of D^{n−2} into Z_{2}(M^{n}, Z), where the mass of each 2-cycle in the image of each of these maps can be made arbitrarily small.

We finish by defining the map f̃ as the sum of those maps over all simplices in the triangulation. Note that this map turns out to be a map from the (q − 2)- (Case (i)) or (n − 2)- (Case (ii)) dimensional sphere. Note that by doing so we lose any control over the mass of 2-cycles in the image of f.

Step 2. We will now construct maps g̃_{i}s, such that f̃ = Σ_{i=1}^{Q}g̃_{i}. The beginning of procedure of constructing those maps is somewhat different for case (i) and for case (ii).

Case (i). Take a point p, the center of the disc D^{q+1}, which has been triangulated as a cone over S^{q}. Assign to this point an arbitrary point in a manifold, that will be denoted p̃. Note that this extends the map f to 0-skeleton of D^{q+1}. Next consider an edge of the form [p, v_{i}]. We will assign to it a minimal geodesic segment [p̃, ṽ_{i}] joining the point p̃ with the corresponding vertex ṽ_{i} = f(v_{i}). This extends the map f to 1-skeleton of D^{q+1}.

Case (ii). Let v_{i} be an arbitrary vertex of W. Let ṽ_{i} be a vertex of M^{n} that minimizes the distance between v_{i} and M^{n}. Then d(v_{i}, ṽ_{i}) ≤ F illRadM^{n} +δ. We will assign ṽ_{i} to the vertex v_{i}. This extends the identity map Id : M^{n} ⟶ M^{n} to W. Now consider an arbitrary edge of W of the form [v_{i}, v_{j}]. We can assign to this edge a minimal geodesic segment joining the corresponding vertices ṽ_{i} and ṽ_{j}. It will be denoted [ṽ_{i}, ṽ_{j}]. This segment will have the length of at most 2F illRadM^{n} + 3δ.

The rest of the procedure will be the same in the two cases. Consider an arbitrary 2-simplex of the form σ_{i}^{2} = [v_{i_{0}}, v_{i_{1}}, v_{i_{2}}] of D^{q+1}, where (v_{i_{0}} denotes here and later the center of the disc p) in Case (i). Its boundary is mapped to a closed curve of length at most 2d + δ in Case (i) (of 6F illRadM^{n} + 9δ in Case (ii)). Let s_{i}^{2} = s_{i_{0},i_{1},i_{2}}^{2} be a singular surface (=singular 2-chain) of area smaller than F H(2d + δ) in Case (i) (F H(6F illRadM^{n} + 6δ) in Case (ii)). We will assign this surface to the simplex σ_{i}^{2}.

Next consider a 3-simplex σ_{i}^{3} = [v_{i_{0}}, ..., v_{i_{3}}]. Its boundary corresponds to the singular surface Σ_{j=0}^{3}(−1)^{j}s_{i_{0},...î_{j},...,i_{3}}. Its area is ≤ 3F H(2d + δ) + δ̃, (since we can assume that all simplices of f(S^{q}) are small) in Case (i) (≤ 4F H(6F illRadM^{n} + 6δ) in Case (ii)). This surface can be considered as an element in the space of integral 2-cycles on M^{n}. Now, either there exists a minimal 2-cycle that locally minimize the mass of non-zero mass smaller than that of this surface, or this surface, viewed as a point in Z_{2}(M^{n}, Z) can be connected with the zero cycle with a path h_{i}^{1} : [0, 1] ⟶ Z_{2}(M^{n}, Z).

Here we are using the proof of Theorem 4.10 in [P]. This theorem does notamention the mass of the stationary varifold but its proof implies that the mass does not exceed the maximal mass of cycles in the image of the considered map of the sphere.)

Therefore, we can establish a correspondence between the 3- simplices and the above maps.

Now consider a 4-simplex σ_{i}^{4} = [v_{i_{0}}, ..., v_{i_{4}}]. For each face (−1)^{j}[v_{i_{0}}, ...., v̂_{i_{j}}, ...., v_{i_{4}}] there was preassigned a map (−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}^{1} :

[0, 1] ⟶ Z_{2}(M^{n}, Z). Now, let f_{i}^{1}(t) = Σ_{j=0}^{1}(−1)^{j}h_{i_{0},...,î_{j},...,i_{4}}^{1}(1 − t). Each f_{i}^{1}(t) is an integral 2-cycle of area ≤ 4 · 3F H(2d + δ) + 2δ in Case (i) ( ≤ 5 · 4F H(6F illRadM^{n} + 9δ)). Note that f_{i}(0) is the zero cycle and that f_{i}(1) is also the zero cycle due to the cancellations of the pairs of surfaces that enter with the opposite orientation. Thus, in reality, f_{i}^{1}(t) is a loop in the space of the integral 2-cycles. Since we have assumed that there are no almost minimizing integral varifolds of “small” area the map f_{i}^{1}(t) is contractible over the disc h_{i}^{2}(s). Therefore, we can assign this map of the disc to a 4-simplex σ_{i}^{4}.

Now suppose that for each simplex of dimension k we have constructed the corresponding map h_{i}^{k−2} : D^{k−2} ⟶ Z_{2}(M^{n}, Z). Let us consider an arbitrary (k+1)-simplex σ_{i}^{k} = [v_{i_{0}}, ..., v_{i_{k+1}}]. Consider a face in the boundary of this simplex (−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{k+1}}]. To this face there corresponds a map h_{i_{0},....,î_{j},...,i_{k+1}}^{k−2} : D^{k−2} ⟶ Z_{2}(M^{n}, Z). Now consider a map f_{i}^{k−2} :

D^{k−2} ⟶ Z_{2}(M^{n}, Z), such that f_{i}^{k−2}(r, θ) = Σ_{j=0}^{k+1}(−1)^{j}h_{i_{0},....,î_{j},...,i_{k+1}}^{k−2}(1 −

r, θ). Note that f_{i}^{k−2}(1, θ) is the zero cycle. Therefore, in reality this map is a map from S^{k−2} to the space of cycles. By our assumption this map is contractible, thus we obtain h_{i}^{k−1} : D^{k−1} ⟶ Z_{2}(M^{n}, Z). We will assign this map to simplex σ_{i}^{k−1}.

We will continue in the above manner until we construct maps f_{i}^{q−2} in Case (i) and maps f_{i}^{n−2}s in Case (ii). We will call these maps g̃_{i}. The sum of these maps is homotopic to the map f constructed in Step 1.

Step 3. We can conlude that one of the maps f̃_{i} is not contractible. Therefore, another application of Theorem 4.10 of [P] (or, more precisely, of its proof) implies that there exists a non-trivial stationary integral varifold with the mass as claimed in Theorem 1.3.

If n = 3, then the regularity assertion follows from Theorem 7.12 in [P]. □

4 Proof of Theorem 1.4

The proof is similar to that of Theorem 1.3.

Proof. The proof is by contradiction. We will assume that there is no stationary varifold as in Theorem 1.4. As in the 1.3 we are going to discuss cases (i) and (ii) at the same time. Theorem will be proved in three steps. Step 1. Let f : S^{q} ⟶ M^{n} be a non-contractible map in Case (i) and let Id : 2004 − 006.tex, v1.12004/05/2717 : 35 : 19levyExplevy be the identity map in Case (ii). We are going to construct a non-contractible map f̃ : S^{q−k} ⟶ Z_{k}(M^{n}, Z) in Case (i) and the map f̃ : S^{n−k} ⟶ Z_{k}(M^{n}, Z) in Case (ii). (Here and below Z_{k}(M^{n}, Z) denotes the space of integral k-cycles on M^{n}.) Step 2. We will construct maps g̃_{i} from S^{q−k} (or S^{n−k}) to the space of kdimensional cycles on M^{n}, such that the map f̃ of Step 1 will be homotopic to their sum. The masses of k-cycles g̃_{i}(t) do not exceed the right hand side in the inequality (i) (or (ii)) in Theorem 1.4. Step 3. We will conclude that, one of the maps g̃_{i} constructed on Step 2 is not contractible, and therefore the results of J. Pitts imply the existence of a stationary varifold as in Theorem 1.4 thereby obtaining a contradiction.

Here is a detailed description of the first two steps of the proof.

Step 1. Consider an arbitrary simplex σ̃_{i}^{k+1} = ṽ_{i_{0}}, ..., ṽ_{i_{k+1}}], (in general an arbitrary simplex of dimension l will be sometimes denoted as σ_{i_{0},...,i_{l}}^{l} to keep track of the vertices that generate it). This simplex lies in the (k + 1)-skeleton of f(S^{q}) in Case (i) and in the (k + 1)-skeleton of M^{n} in Case (ii). Without any loss of generality we can assume that this simplex is generated by a volume decreasing homotopy h̃_{i}^{1} : [0, 1] ⟶ Z_{k}(M^{n}, Z) that connects its boundary with a point p̃_{i}^{k+1}, (the homotopy will be sometimes denoted as h̃_{i_{0},...,i_{k+1}}^{1}, and the point will be sometimes denoted as p̃_{i_{0},...,i_{k+1}}^{k+1}). We can define a correspondence between σ̃_{i}^{k+1} and the map

h_{i}^{1}. Now consider a (k + 2)-dimensional simplex σ̃_{i}^{(k+2)} = [ṽ_{i_{0}}, ..., ṽ_{i_{k+2}}]. Each face (−1)^{j}[ṽ_{i_{0}}, ..., ˆ ṽ_{i_{j}}, ..., ṽ_{i_{k+2}}] of its boundary corresponds to the map (−1)^{j}h̃_{i_{0},...,î_{j},...,i_{k+2}}^{1}. Define a new map f̃_{i}^{1} : [0, 1] ⟶ Z_{k}(M^{n}, Z) by letting

f̃_{i}^{1}(t) = Σ_{j=0}^{k+2}(−1)^{j}h̃_{i_{0},...,î_{j},...,i_{k+2}}^{1}(1 − t). Note that f̃_{i}^{1}(0) = f̃_{i}^{1}(1) and is the zero cycle, thus the newly constructed map is a loop in the space of k-cycles. Since all the simplices in the triangulation of S^{q} or M are small, this loop is contractible. The contraction amounts to slicing the (k + 2)-dimensional simplex into k-cycles, so that this slicing extends the slicing of the boundary.

(It can be explicitly defined using a coning of the slicing of the boundary.) That allows us to obtain a disc h̃_{i}^{2} : D^{2} ⟶ Z_{k}(M^{n}, Z), and so we can define a correspondence between simplex σ̃_{i}^{k+2} and the map h̃_{i}^{2}.

Now suppose we have constructed maps h̃_{i}^{l−k} : D^{l−k} ⟶ Z_{k}(M^{n}, Z) that correspond to l-dimensional simplices σ̃_{i}^{l}. Consider an arbitrary (l +1)- dimensional simplex σ̃_{i}^{l+1} = [ṽ_{i_{0}}, ..., ṽ_{i_{l+1}}]. Define the map f̃_{i}^{l−k} : D^{l−k} ⟶ Z_{k}(M^{n}, Z) by letting f̃_{i}^{l−k}(r, θ) = Σ_{j=0}^{l+1}(−1)^{j}h̃_{i_{0},...,î_{j},...,i_{l+1}}^{l−k}(1 − r, θ). This

map corresponds to the boundary of σ̃_{i}^{l+1}. Note that f̃_{i}^{l−k}(1, θ) is the zero cycle, and thus it is a map from the (l − k)-dimensional sphere to the space of integral k-cycles. This sphere is contractible over the disc h̃_{i}^{l+1−k} : D^{l+1−k} ⟶ Z_{k}(M^{n}, Z) in the space of “small” integral k-cycles. This contraction amounts to extending the slicing of the boundary of the considered (l + 1)-dimensional simplex to its interior, and can be explicitly constructed just by the coning. We continue in the above manner until we construct the maps h̃_{i}^{q−k}s in case (i) and the maps h̃_{i}^{n−k}s in case (ii). Take the sum of those maps over all the simplices in the triangulation to obtain sphere f̃: S^{q−k} ⟶ Z_{k}(M^{n}, Z) in the space of integral k-cycles in Case (i) and f̃: S^{n−k} ⟶ Z_{k}(M^{n}, Z) in Case (ii).

Step 2. (The main step.) We will begin by extending the map f : S^{q} ⟶ M^{n} to the 2-skeleton of D^{3} (Case (i)) or by extending the map Id : 2004 − 006.tex, v1.12004/05/2717 : 35 : 19levyExplevy to the 1skeleton of W (Case (ii)). In both of those cases the procedure is identical to that of Step 2 in the proof of Theorem 1.3. After that we will establish a correspondence between simplices σ_{i}^{l}, where 2 ≤ l ≤ k and singular chains of the corresponding dimension on M^{n}. Let us consider an arbitrary 2-simplex σ_{i}^{2} = [v_{i_{0}}, v_{i_{1}}, v_{i_{2}}]. Its boundary is mapped to a curve of length ≤ 2d + δ (Case (i)) or of length ≤ 6F illRadM^{n} + 9δ (Case (ii)). In the future we will denote it l_{c}.

Let s_{i}^{2} be a singular surface of smallest area that has this curve as its boundary. We will assign this surface to this simplex. The area of this surface will be ≤ F H_{2}(l_{c}). Now consider an arbitrary 3-simplex σ_{i}^{3}. Its boundary is assigned a 2-cycle S_{i}^{2} = Σ_{j=0}^{3}(−1)^{j}s_{i_{0},..,î_{j},...,i_{3}}^{2} of area ≤ 3F H_{2}(l_{c}) + δ in Case (i) and of area ≤ 4F H_{2}(l_{c}) in Case (ii). Let s_{i}^{3} be a singular 3chain of smallest area that has S_{i}^{2} as its boundary. Its volume is at most F H_{3}(3F H_{2}(l_{c})+δ) in case (i) and at most F H_{3}(4F H_{2}(l_{c})) in case (ii). Next suppose that for each arbitrary simplex σ_{i}^{l}, where l ≤ k − 1 we have constructed a chain s_{i}^{l} of volume ≤ F H_{l}(lF H_{l−1}((l − 1)F H_{l−2}(.....) + δ) + δ) in Case (i) and ≤ F H_{l}((l + 1)F H_{l−1}(lF H_{l−2}((l − 1)...))) in Case (ii). Consider an arbitrary (l + 1)-simplex σ_{i}^{l}. For each face (−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{l+1}}] in its boundary there exists a preassigned singular chain (−1)^{j}s_{i_{0},...,î_{j},...,i_{l+1}}. Consider a cycle S_{i}^{l} = Σ_{j=0}^{l+1}(−1)^{j}s_{i_{0},...,î_{j},...,i_{l+1}}. Find a singular (l + 1)-chain of s_{i}^{l+1}, such that ∂s_{i}^{l+1} = S_{i}^{l}. This chain will be assigned to σ_{i}^{l}. We should continue in the above manner until we reach the k-skeleton of D^{q} or respectively of W.

Now consider an arbitrary (k + 1)-simplex σ_{i}^{k+1}. Each face in its boundary (−1)^{j}[v_{i_{0},...,î_{j},...,i_{k+1}}] corresponds to the singular chain (−1)^{j}s_{i_{0},...,î_{j},...,i_{k+1}}. Consider the following cycle S_{i}^{k+1} = Σ_{j+0}^{k+1}(−1)^{j}s_{i_{0},...,î_{j},...,i_{k+1}}. This is an element of Z_{k}(M^{n}, Z) of volume ≤ (k + 1)F H_{k}(kF H_{k−1}((k − 1)...) + δ) + δ in Case (i) and of volume ≤ (k+2)F H_{k}((k+1)F H_{k_{1}}(k...)). By our assumption this cycle can be connected with the zero cycle with a path that will only pass through the cycles of smaller volume. Let us denote this path by h_{i}^{1} : [0, 1] ⟶ Z_{k}(M^{n}, Z). We will assign this path to the above simplex σ_{i}^{k+1}. Now suppose we have constructed the maps h_{i}^{l−k} : D^{l−k} ⟶ Z_{k}(M^{n}, Z) corresponding to simplices σ_{i}^{l}, where l ≤ q − 1 in Case (i) and ≤ n − 1 in Case (ii). Consider an arbitrary (l + 1)-dimensional simplex σ_{i}^{l+1}. Each face (−1)^{j}[v_{i_{0}}, ..., v̂_{i_{j}}, ..., v_{i_{l+1}}] corresponds to a map (−1)^{j}h_{i_{0},...,î_{j},...,i_{l+1}}^{l−k}(r, θ).

Define a new map f_{i}^{k−l}(r, θ) = Σ_{j=0}^{l+1}(−1)^{j}h_{i_{0},...,î_{j},...,i_{l+1}}^{l−k}(1 − r, θ). Note that

f(1, θ) is the zero cycle, and thus, this is a map from S^{l−k} to the space of k-cycles. By our assumption this map is contractible and this is how we obtain h_{i}^{l−k+1} : D^{l−k+1} ⟶ Z_{k}(M^{n}, Z). We continue in the above manner until we construct maps f_{i}^{q−k} in case (i) or the maps f_{i}^{n−k} in case (ii). Those are our maps g̃′_{i}s. Note that f̃ is homotopic to their sum.

□

Acknowledgements: A part of this work was done at MSRI, Berkeley during our visit in April-May, 2004. We would like to thank MSRI for its warm hospitality.

References

[A] F. Almgren, The homotopy groups of the integral cycle groups, Topology 1(1962), 257-289.

[CD] T. Colding, C. De Lellis, The min-max construction of minimal surfaces, Surv. in Diff. Geom., 8(2003), 75-107.

bounds for the smallest area of a minimal surface 19

[Gr] M. Gromov, Filling Riemannian Manifolds, J. Diff. Geom. 18(1983), 1-147.

[NR1] A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, to appear in GAFA.

[NR2] A. Nabutovsky, R. Rotman, The area of a minimal embedded 2sphere in a manifold diffeomeorphic to S^{3}, IMRN, 2003, no. 39, 2121-2129.

[P] J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes, 27, Princeton Univ. Press, 1981.

[Si] L. Simon, Survey lectures on minimal submanifolds, in “Seminar on Minimal submanifolds”, ed. by E. Bombieri, Ann. Math. Studies, 103, Princeton University Press, 1983, pp. 3-52.

[SS] L. Simon, R. Schoen, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34(1981), 741-797.

[SSY] L. Simon, R. Schoen, S.T. Yau, Curvature estimates for minimal hypersurfaces”, Acta Math. 134(1975), 276-288.

[S] F. Smith, On the existence of embedded minimal 2-sphere in the 3sphere, endowed with arbitrary metric, Ph. D. Thesis, University of Melbourne, Melbourne.

A. Nabutovsky R. Rotman Department of Mathematics Department of Mathematics The Pennsylvania State University, The Pennsylvania State University, University Park, PA 16802, University Park, PA 16802, USA USA e-mail: nabutov@math.psu.edu e-mail: rotman@math.psu.edu

and and Department of Mathematics Department of Mathematics University of Toronto University of Toronto Toronto, Ontario M5P 3G3 Toronto, Ontario M5P 3G3 Canada Canada e-mail: alex@math.toronto.edu e-mail: rina@math.toronto.edu