\documentstyle[11pt,twoside]{article} \pagestyle{myheadings} \sloppy \newtheorem{Thma}{Theorem} \renewcommand{\theThma}{\Alph{Thma}} \newtheorem{Thm}{Theorem}[section] \newtheorem{Lem}[Thm]{Lemma} \newtheorem{Pro}[Thm]{Proposition} \newtheorem{Cor}[Thm]{Corollary} \newtheorem{Def}[Thm]{Definition} \newtheorem{Cnv}[Thm]{Convention} \newenvironment{Pf}[1] {\trivlist\item[]{\it #1\@. }}{\hspace*{\fill}$\Box$\endtrivlist} \newcommand{\mathbb}[1]{{\bf #1}} \newcommand{\operatorname}[1]{\mathop{\rm #1}\nolimits} \newcommand{\text}{\mbox} \renewenvironment{cases}{\left\{\begin{array}{ll}}{\end{array}\right.} \input{figsa} \input psfig.sty \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cL}{\mathcal{L}} \newcommand{\CAT}{\operatorname{CAT}} \newcommand{\diam}{\operatorname{diam}} \renewcommand{\epsilon}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\la}{\langle} \newcommand{\Lip}{\operatorname{Lip}} \newcommand{\ol}{\overline} \newcommand{\ra}{\rangle} \renewcommand{\rho}{\varrho} \newcommand{\sm}{\setminus} \renewcommand{\ss}{\subset} \hyphenation{geo-de-sic Lip-schitz} \begin{document} \title{Curvature-free upper bounds for the smallest area of a minimal surface.} \author{Alexander Nabutovsky and Regina Rotman} \date{May 11, 2004} \maketitle \markboth{A.~Nabutovsky \& R.~Rotman}{bounds for the smallest area of a minimal surface} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{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. \end{abstract} \section{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 $\leq i$ corresponding to a homology class $h\in H_i(M^n)$ 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 $1$.] \label{FillfuncK} 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 $k$, $FH_k:{\bf R_+} \longrightarrow {\bf R_+}$ will be defined as follows: $FH_k(x):=\max_{\{\Sigma_{k}\vert m(\Sigma_{k}) \leq x\}} \min_{\{\Sigma_{k+1}\vert \partial \Sigma_{k+1} = \Sigma_{k}\}} m(\Sigma_{k+1})$, where $\Sigma_k, \Sigma_{k+1}$ are singular chains of dimension $k, k+1$ respectively. \end{Def} 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 $L^{\infty}(X)$ (Recall that the Kuratowski embedding assigns to each $x\in X^n$ 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: \begin{Thm} \label{TheoremA} 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: $$ (1) A(M^n) \leq \frac{(n+1)!}{2} FH(2d(M^n));$$ $$ (2) A(M^n) \leq \frac{(n+2)!}{6} FH(6 FillRad(M^n));$$ $$ (3) A(M^n) \leq \frac{(n+1)!}{6} FH(6(n+1)n^n \sqrt{(n+1)!} vol(M^n)^{\frac{1}{n}}).$$ Here $d(M^n)$ denotes the diameter, $FillRad(M^n)$ denotes the filling radius, and $vol(M^n)$ denotes the volume of $M^n$. If $n=3$, then there exists a non-singular embedded minimal surface such that its area satifies the inequalities (1)-(3). \end{Thm} Formally speaking, ``possibly singular minimal surface" means here ``2-dimensional stationary integral varifold". More generally: \begin{Thm} \label{TheoremB} 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 $k\geq 2$ there exists a non-trivial stationary integral varifold of dimension $k$, such that its mass $V_k$ is bounded from above by $$(1) V_k \leq \frac{(n+1)!}{k!}FH_k(k(FH_{k-1}(...(3FH_2(2d))...))).$$ $$(2) V_k \leq \frac{(n+2)!}{(k+1)!}FH_k((k+1)FH_{k-1}(...(4FH_2(6FillRad M^n))...)).$$ $$(3) V_k\leq \frac{(n+2)!}{(k+1)!}FH_k((k+1)FH_{k-1}(...(4FH_2(6(n+1)n^n \sqrt{(n+1)!}vol(M^n)^{\frac{1}{n}})...))).$$ If $k=n-1$ and $n\leq 7$ 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 $n\geq 9$ we can ensure that the Hausdorff dimension of its singular set does not exceed $k-7$. \end{Thm} {\bf Remarks.} \par\noindent {\bf 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]. \par\noindent {\bf 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 $M^n$. \par\noindent {\bf 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 {\it 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. \medskip \section{Ideas of the proofs} \medskip 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 ~\ref{TheoremA}. 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 ~\ref{TheoremA} Let $f:S^3 \longrightarrow M$ be any diffeomorphism. Assume $S^3$ was triangulated into simplices of diameter smaller than $\delta$. 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 $\sigma_i^k$, (or sometimes $\sigma_{i_0,...,i_k}^k$), where $k \in \{1,2,3,4 \}$. The proof will be by contradiction. Suppose $A(M) > 12 FH(2d+\delta)+5 \delta$. 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]. \realfig{Table1}{minsurf1.eps}{Table 1.}{0.7\hsize} We will begin by extending the map $f: S^3 \longrightarrow M$ to $1$-skeleton of $M$ and then by assigning to each $2$-simplex of $D^4$ a singular $2$-chain on $M$, (see fig. ~\ref{Table1}). To extend to $0$-skeleton, we will assign to the center of the disc $p$, an arbitrary point $\tilde{p} \in M$. 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 $\tilde{p}$ with a vertex $\tilde{v}_{i_1}=f(v_{i_1})$, directed from $\tilde{p}$ to $\tilde{v}_{i_1}$ and denoted $[\tilde{p}, \tilde{v}_{i_1}]$. Now, consider a simplex $\sigma_i^2=[v_{i_0},v_{i_1},v_{i_2}]$, where $v_{i_0}=p$. Its boundary is mapped to the curve $[\tilde{v}_{i_1},\tilde{v}_{i_2}]-[\tilde{v}_{i_0}, \tilde{v}_{i_1}] +[\tilde{v}_{i_0}, \tilde{v}_{i_1}]$ of length $\leq 2d+\delta$. Let $s_i^2$ (also denoted as $s^2_{i_0,i_1,i_2}$) be a singular $2$-chain of the smallest area, such that $\partial s^2_i=f(\partial \sigma^2_i)$. Then its area is $\leq FH(2d+\delta)$. We will assign to $\sigma^2_i$ this surface $s^2_i$. Now we will slightly change our tactics. Consider a $3$-simplex $\sigma_i^3$. There is a preassigned chain $S^2_i=\Sigma_{j=0}^3 (-1)^j s^2_{i_0,...,\hat{i}_j,...,i_3}$ of area $\leq 3FH(2d+\delta)+\delta$ that corresponds to the boundary of this simplex. (We can assume without loss of generality that the area of $s_{i_1,i_2,i_3}^2 \leq \delta$). $S^2_i$ is an element in the space of integral $2$-cycles $Z_2(M^n, {\bf Z})$. Since by our assumption there are no minimal surfaces of ``small'' area, $S^2_i$ can be connected with the zero cycle by a curve that passes through cycles of area $\leq 3FH(2d+\delta)+\delta$, i. e. there exists $h_i^1:[0,1] \longrightarrow Z_2(M,{\bf Z})$, (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 $\sigma_i^3$. Finally, take a $4$-simplex $\sigma_i^4$, (see fig. ~\ref{minsurf2}). Consider its boundary $\partial \sigma_i^4 = \Sigma_{j=0}^4 (-1)^j[v_{i_0},...,\hat{v}_{i_j},...,v_{i_4}]$. Each of its faces $(-1)^j[v_{i_0},...,\hat{v}_{i_j},...,v_{i_4}]$ corresponds to the map $(-1)^j h^1_{i_0,...,\hat{i}_j,...,i_4}(t)$, where $-h^1_{i_0,...,\hat{i}_j,...,i_4}(t)$ is a cycle that is geometrically the same as $h^1_{i_0,...,\hat{i}_j,...,i_4}(t)$, but is oppositely oriented. \realfig{minsurf2}{minsurf2.eps}{The loop $f_i^1(t)$.}{0.6\hsize} {\it \bf Now we will perform the trick that we will use throughout the paper.} Consider the map $f_i^1:[0,1] \longrightarrow Z_2(M, {\bf Z})$, such that $f_i^1(t) = \Sigma_{j=0}^4(-1)^j h_{i_0,...,\hat{i}_j,...,i_4}(1-t)$. 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 $12FH(2d+\delta)+5\delta$. By our assumption $f_i^1(t)$ is contractible in $Z_2(M, {\bf Z})$ along the cycles of area $\leq 12FH(2d+\delta)+5\delta$. (Otherwise the proof of Theorem 4.10 in [P] implies the existence of a minimal surface (=a stationary integral varifold) of area $\leq 12FH(2d+\delta)+5\delta$, which is impossible because of our assumption.) Thus, we obtain a disc $h_i^2:D^2 \longrightarrow Z_2(M, {\bf Z})$. We will assign this disc to the simplex $\sigma_i^4$. \par From the geometric measure theory we know that $$ \pi_k(Z_2(M, {\bf Z})) = H_{k+2}(M).$$ So, the disc of the form $h_i^2:D^2 \longrightarrow Z_2(M, {\bf Z})$ 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 ~\ref{TheoremA} and ~\ref{TheoremB} given in the next sections are more awkward than this sketch). Now consider $S^4=\Sigma_{i=1}^Q s_i^4$, where $Q$ is the number of simplices of dimension $4$ in the triangulation of $D^4$. $\partial S^4= f_*([S^3])$. Thus, we obtain a contadiction. Therefore, we can conclude that $A(M) \leq 12FH(2d+\delta)+5 \delta$. Now let $\delta$ go to zero. The proof of Theorem ~\ref{TheoremA} will be given in Section 2. The proof of Theorem ~\ref{TheoremB} 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 $f: S^3 \longrightarrow M$ 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 $\delta$). Let $D^4$ be triangulated as a cone over $S^3$. The proof will consist of the following three steps: \noindent {\bf Step 1 (Easy)}. Corresponding to the map $f:S^3 \longrightarrow M$ one can construct a non-contractible map $\tilde{f}:S^1 \longrightarrow Z_2(M, \bf{Z})$, a loop in the space of the integral $2$-cycles on a manifold $M$. We do not have any control over the masses of $f(t), t\in S^1$. \noindent {\bf Step 2 (Main Step)}. The loop $\tilde{f}$ is homotopic to the sum of $Q$ loops $\tilde{g}_i$, where the number $Q$ equals to the number of simplices in the triangulation of $S^3$, and the masses of all 2-cycles $\tilde{g}_i(t), t\in S^1$ satisfy upper bounds as in the right hand side of the inequality (1) in the text of Theorem 1.3. \noindent {\bf 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. \realfig{minsurf3}{minsurf3.eps}{The loop $\tilde{f}$.}{0.5\hsize} {\bf Step 1.} Let $\{ \sigma_i^3 \}_{i=1}^Q$ be the set of simplices that constitute the fundamental class $[S^3]$ of $S^3$. Then $\{ \tilde{\sigma}_i^3 \}_{i=1}^Q$ are the corresponding simplices of $M$. Without any loss of generality we can assume that each $\tilde{\sigma}_i^3$ can be obtained from its boundary $\partial \tilde{\sigma}_i^3$ by contracting it to the ``center'' of the simplex, the point $\tilde{p}_i$ by a homotopy $\tilde{h}_i: [0,1] \longrightarrow M$ along the $2$-spheres $\tilde{h}_i(t)$, (see fig. ~\ref{minsurf3} (a). This figure schematically depicts $\tilde{\sigma}_i^3$ as a simplex of dimension $2$). Then $\tilde{f}$ can be constructed as follows. We will begin with the zero cycle that consists of the sum of the ``centers'' of simplices: $\Sigma_{i=1}^Q \tilde{p}_i$. We will then follow the images of $\partial \tilde{\sigma}_i^3$'s under the area decreasing homotopies $\tilde{h}_i(t)$'s, that is $\tilde{f}(t)=\Sigma_{i=1}^Q \tilde{h}_i(1-t)$. Note that $\tilde{f}(1)= \Sigma_{i=1}^Q \partial \tilde{\sigma}_i^3$, which is also the zero cycle. Thus, we obtain a (non-contractible) loop in the space of $2$-cycles, (see fig. ~\ref{minsurf3} (b). There $f(S^3)$ is schematically depicted as a $2$-dimensional sphere). \realfig{minsruf4}{minsruf4.eps}{}{0.5\hsize} {\bf Step 2.} Each loop $\tilde{g}_i$ is constructed as follows. Consider a disc $D^4$, such that $\partial D^4 = S^3$, triangulated as a cone over $S^3$. Let $p \in D^4$ be the center of this disc. We can assign to this point an arbitrary point $\tilde{p} \in M$, thus extending the map $f:S^3 \longrightarrow M$ 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 $\tilde{p}$ with the vertex $\tilde{v}_i=f(v_i)$. This segment will be directed from $\tilde{p}$ to $\tilde{v}_i$ and denoted as $[\tilde{p}, \tilde{v}_i]$. This extends $f$ to $1$-skeleton of $D^4$. Next consider a $2$-simplex $\sigma_i^2=[p,v_{i_1},v_{i_2}]$, (it will sometimes be denoted as $\sigma_{i_0,i_1,i_2}^2$. Its boundary is mapped to a closed curve $ [\tilde{v}_{i_1}, \tilde{v}_{i_2}]-[\tilde{p}, \tilde{v}_{i_2}] +[\tilde{p}, \tilde{v}_{i_1}]$. Let $s_{i}^2$ (sometimes 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 $\sigma_i^2$, 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 $\sigma^3_i=[p, v_{i_1}, v_{i_2}, v_{i_3}]$. Its boundary $\partial \sigma^3_i=\sum_{j=0}^3 (-1)^j [v_{i_0},...,\hat{v}_{i_j},...,v_{i_3}]$, where $v_{i_0}=p$. For each face $(-1)^j[v_{i_0},...,\hat{v}_{i_j},...,v_{i_3}]$ we have pre-assigned a singular $2$-chain $(-1)^j s^2_{i_0,...,\hat{i}_j,...,i_3}$. Consider $\Sigma_{j=0}^3 (-1)^j s^2_{i_0,...,\hat{i}_j,...,i_3}$. This is a $2$-cycle of area smaller than $3FH(2d+\delta)+\delta$, (without loss of generality we can assume that the area of $s_{i_1,i_2,i_3}^2$ is smaller than $\delta$). Thus, it is an element in $Z_2(M, {\bf Z})$. 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 $h^1_{i_0,...,i_3}:[0,1] \longrightarrow Z_2(M,{\bf Z})$ that begins with our cycle and ends with the zero cycle. Finally, let us construct $\tilde{g}_i$. Consider a $4$-simplex $\sigma^4_i= [v_{i_0}, v_{i_1},...,v_{i_4}]$. Its boundary, $\partial \sigma^4_i= \Sigma_{j=1}^4 (-1)^j [p,v_{i_1},...,\hat{v}_{i_j},...,v_{i_4}]$. For each face in the boundary $(-1)^j [v_{i_0},v_{i_1},...,\hat{v}_{i_j},..., v_{i_4}]$ there was constructed a map $(-1)^jh^3_{i_0,...,\hat{i}_j,..., i_4}:[0,1] \longrightarrow Z_2(M, {\bf Z})$, where $-h(t)$ is the same cycle as $h(t)$, but taken with an opposite orientation. We will define the map $\tilde{g}_i (t)$ as the sum $\Sigma_{j=0}^4 (-1)^j h^1_{i_0,...,\hat{i}_j,...,i_4}(1-t)$. Note that $\tilde{g}_i(0)$ is the zero cycle represented by $\tilde{p}_{i_0,..., \hat{i}_j,...,i_4}$ and that $\tilde{g}_i(1)$ 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 $\sigma_i^4$ enters twice with the opposite orientation. Thus, $\tilde{g}_i(t)$ is a loop. Area of $\tilde{g}_i(t)$ is bounded from above by $12FH(2d+\delta)+5\delta$. Moreover, $\tilde{f}(t)$ is homotopic to the sum of the above loops. {\bf Step 3.} Since $\tilde{f}(t)$ is homotopic to the sum of the loops $\tilde{g}_i$, (see fig. ~\ref{minsruf4}), 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 $A(M^n) \leq \frac{(n+2)!}{6}FH(6 FillRad (M^n))$. Recall that the filling radius of a Riemannian manifold $M^n$ was introduced by Gromov in [Gr] as follows: \begin{Def}[Filling Radius] \label{FillRad} Let $M^n$ be a Riemannian manifold topologicaly imbedded into an arbitrary metric space $X$. Then its filling radius, denoted $FillRad(M^n \subset X)$, is the infimum of $\epsilon > 0$, such that $M^n$ bounds in the $\epsilon$- neighborhood $N_{\epsilon}(M^n)$, i.e. homomorphism $H_n(M^n, {\bf Z_2}) \longrightarrow H_n(N_\epsilon (M^n), {\bf Z_2})$ induced by the inclusion map vanishes. Let $M^n$ be an abstract manifold. Then its filling radius, denoted $FillRad(M^n)$ will be $FillRad(M^n \subset X)$, where $X=L^\infty(M^n)$, 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 $p \longrightarrow f_p=d(p,q)$. \end{Def} In the same paper Gromov poved the following important inequality relating the filling radius and the volume: \begin{Thm}[(M. Gromov)] \label{VB} $$Fill Rad(M^n)\leq \sqrt{(n+1)!}n^n(n+1)vol(M^n)^{1\over n}.$$ \end{Thm} 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 $(FillRad(M^n)+\delta)$ - neighborhood of $M^n$ in $L^\infty (M^n)$. Let $W$ fill $M^n$ in the $(Fill Rad(M^n)+\delta)$-neighborhood of $M$, (that is $M^n=\partial W$). 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 $\tilde{f}:S^1 \longrightarrow Z_2(M^n,{\bf Z})$; constructing a family of loops $\tilde{g}_i:S^1 \longrightarrow Z_2(M^n, {\bf Z})$; 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. {\bf Step 1.} The first step is analogous to that of (i). {\bf Step 2.} Let $v_i$ be a vertex of $W$. Assign to it a closest vertex in $M^n$, denoted $\tilde{v}_i$. Thus, $d(v_i, \tilde{v}_i) \leq Fill Rad(M^n)+\delta$. This extends the identity map $Id: 2004-006.tex,v 1.1 2004/05/27 17:35:19 levy Exp levy $ on $M^n$ to $0$-skeleton of $W$. Now to any $1$-simplex $[v_{i_1},v_{i_2}] \subset W\setminus M^n$ we can assign a minimal geodesic segment connecting $\tilde{v}_{i_1}$ and $\tilde{v}_{i_2}$ of length smaller than $2 Fill Rad (M^n)+3\delta$. This segment will be denoted as $[\tilde{v}_{i_1}, \tilde{v}_{i_2}]$. This extends $Id: 2004-006.tex,v 1.1 2004/05/27 17:35:19 levy Exp levy $ to $1$-skeleton of $M^n$. Next consider an arbitrary $2$-simplex $\sigma_i^3=[v_{i_1},v_{i_2},v_{i_3}]$. Its boundary is mapped to a curve of length smaller than $6 Fill Rad (M^n)+9 \delta$. Let $s^2_{i_1,i_2,i_3}$ be a singular $2$-chain of the smallest area that has this curve as its boundary. Its area is going to be smaller than $FH(6Fill Rad(M^n)+9\delta)$. The rest of the procedure (including Step 3) is the same as in the proof of (1). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The proof of Theorem ~\ref{TheoremA}} In this section we will prove ~\ref{TheoremA}. We will begin by proving statements (i) and (ii). Statement (iii) follows from (ii). \begin{Pf}{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 $\pi_q(M^n) \neq \{ 0\}$. Let $f:S^q \longrightarrow M^n$ be a non-contractible map in case (i), and in case (ii) chooser a (singular) $W$ that fills $M$ (in $L^\infty(M^n)$) such that for each $x\in W$ $dist (x, M^n)\leq (n+1)n^n\sqrt{(n+1)!}vol(M^n)^{1\over n}$. (As it had been already mentioned, in [Gr] M. Gromov proved that such a filling exists.) Let $S^q$ be triangulated into simplices $\sigma_i^q$ of diameter $d(\sigma_i^q) \leq \delta$ for some small $\delta$ in case (i). In case (ii) assume that $W$ (and $M^n=\partial W$) has been triangulated into simplices $\sigma_i^n$ of diameter at most $\delta$. In both cases let $Q$ denote the number of simplices. The proof will consist of three steps: \noindent {\bf Step 1.} We will begin by constructing a non-contractible map $\tilde{f}:S^{q-2} \longrightarrow Z_2(M^n, {\bf Z})$ in case (i) that corresponds to the original map $f:S^q \longrightarrow M^n$ under the Almgren correspondance in case (i) and the map $\tilde{f}:S^{n-2} \longrightarrow Z_2(M^n, {\bf Z})$ that corresponds to the fundamental class of $M^n$ in case (ii). \noindent {\bf Step 2.} We will construct maps $\tilde{g}_i$ from $S^{q-2}$ in case (i) and from $S^{n-2}$ in case (ii) to the space $Z_2(M^n, {\bf Z}), i \in \{1,...,Q \}$, 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 $\tilde{f}$ is homotopic to the sum of $\tilde{g}_i's$. Therefore, at least one of the maps $\tilde{g}_i's$ is not contractible. The most important feature of $\tilde g_i$ is that the mass of $\tilde g_i(t)$ does not exceed the right hand side of (1) (or (2)) for every $t, i$. \noindent {\bf 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. {\bf Step 1.} Let us begin by considering simplices $\tilde{\sigma}_i^3=[\tilde{v}_{i_0},...,\tilde{v}_{i_3}]$ of dimension 3, (we will sometimes denote it as $\tilde{\sigma}^3_{i_0,...,i_3}$, in general, $k$-simplices will be sometimes denoted as $\tilde{\sigma}^k_{i_0,...,i_k}$). Without loss of generality we can assume that this simplex can be generated by contracting its boundary $\partial \tilde{\sigma}_i^3$ to the ``center'' of the simplex $\tilde{p}_i^3$, (sometimes denoted $\tilde{p}_{i_0,...,i_3}$ ) with a homotopy $\tilde{h}^1_i(t)=\tilde{h}^1_{i_0,...,i_3}$. 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 $\tilde{\sigma}_i^4=[\tilde{v}_{i_0},...,\tilde{v}_{i_4}]$. Its boundary $\partial \tilde{\sigma}_i^4=\Sigma_{j=0}^4 (-1)^j [\tilde{v}_{i_0},...,\hat{\tilde{v}}_{i_j},...,\tilde{v}_{i_4}]$. For each $3$-face $(-1)^j [\tilde{v}_{i_0},...,\hat{\tilde{v}}_{i_j},...,\tilde{v}_{i_4}]$ we have constructed a path $(-1)^j \tilde{h}^1_{i_0,...,\hat{i}_j,...,i_4}:[0,1] \longrightarrow Z_2(M^n, {\bf Z})$. Now, let us construct a map $\tilde{f}_{i}^1:[0,1] \longrightarrow Z_2(M^n, {\bf Z})$ as follows: let $\tilde{f}_i^1 (t)=\Sigma_{j=0}^4 (-1)^j \tilde{h}^1_{i_0,...,\hat{i}_j,...,i_4}(1-t)$. Then, note that $\tilde{f}_i^1(0)$ is the zero cycle that corresponds to the sum of the ``centers'' of the $3$-faces of $\tilde{\sigma}_i^4$ and that $\tilde{f}_i^1(0)$ 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 $\tilde{h}^2_i:D^2 \longrightarrow Z_2(M^n, {\bf Z})$, (this map will also be sometimes denoted as $\tilde{h}_{i_0,...,i_4}^2$). 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 $\tilde h_i^q$ of $D^{q-2}$ into $Z_2(M, {\bf Z})$ (or $\tilde h_i^n$ of $D^{n-2}$ into $Z_2(M^n, {\bf Z})$, 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 $\tilde{f}$ 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 $f$. {\bf Step 2.} We will now construct maps $\tilde{g}_i$s, such that $\tilde{f}=\Sigma_{i=1}^Q \tilde{g}_i$. The beginning of procedure of constructing those maps is somewhat different for case (i) and for case (ii). { \bf 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 $\tilde{p}$. 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 $[\tilde{p}, \tilde{v}_i]$ joining the point $\tilde{p}$ with the corresponding vertex $\tilde{v}_i=f(v_i)$. This extends the map $f$ to $1$-skeleton of $D^{q+1}$. {\bf Case (ii).} Let $v_i$ be an arbitrary vertex of $W$. Let $\tilde{v}_i$ be a vertex of $M^n$ that minimizes the distance between $v_i$ and $M^n$. Then $d(v_i, \tilde{v}_i) \leq Fill Rad M^n+\delta$. We will assign $\tilde{v}_i$ to the vertex $v_i$. This extends the identity map $Id: M^n \longrightarrow M^n$ 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 $\tilde{v}_i$ and $\tilde{v}_j$. It will be denoted $[\tilde{v}_i, \tilde{v}_j]$. This segment will have the length of at most $2Fill Rad M^n + 3\delta$. The rest of the procedure will be the same in the two cases. Consider an arbitrary $2$-simplex of the form $\sigma_i^2= [v_{i_0}, v_{i_1}, v_{i_2}]$ 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 $2d+\delta$ in Case (i) (of $6 Fill Rad M^n + 9 \delta$ 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 $FH(2d + \delta)$ in Case (i) ($FH(6 Fill Rad M^n+6 \delta)$ in Case (ii)). We will assign this surface to the simplex $\sigma_i^2$. Next consider a $3$-simplex $\sigma_i^3=[v_{i_0},...,v_{i_3}]$. Its boundary corresponds to the singular surface $\Sigma_{j=0}^3 (-1)^j s_{i_0,...\hat{i}_j,...,i_3}$. Its area is $\leq 3FH(2d+\delta)+\tilde{\delta}$, (since we can assume that all simplices of $f(S^q)$ are small) in Case (i) ($\leq 4FH(6 Fill Rad M^n + 6 \delta)$ 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 $Z_2(M^n, {\bf Z})$ can be connected with the zero cycle with a path $h_i^1:[0,1] \longrightarrow Z_2(M^n, {\bf Z})$. 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 $\sigma_i^4=[v_{i_0},...,v_{i_4}]$. For each face $(-1)^j [v_{i_0},....,\hat{v}_{i_j},....,v_{i_4}]$ there was preassigned a map $(-1)^j h_{i_0,...,\hat{i}_j,...,i_4}^1: [0,1] \longrightarrow Z_2(M^n, {\bf Z})$. Now, let $f_i^1(t)=\Sigma_{j=0}^1 (-1)^j h^1_{i_0,..., \hat{i}_j,...,i_4}(1-t)$. Each $f_i^1(t)$ is an integral $2$-cycle of area $\leq 4 \cdot 3 FH(2d+\delta)+2\delta$ in Case (i) ( $\leq 5 \cdot 4 FH(6 Fill Rad M^n + 9\delta)$). 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 $\sigma_i^4$. Now suppose that for each simplex of dimension $k$ we have constructed the corresponding map $h_i^{k-2}:D^{k-2} \longrightarrow Z_2(M^n, {\bf Z})$. Let us consider an arbitrary $(k+1)$-simplex $\sigma_i^k=[v_{i_0},...,v_{i_{k+1}}]$. Consider a face in the boundary of this simplex $(-1)^j [v_{i_0},..., \hat{v}_{i_j},...,v_{i_{k+1}}]$. To this face there corresponds a map $h^{k-2}_{i_0,....,\hat{i}_j,...,i_{k+1}}:D^{k-2} \longrightarrow Z_2(M^n, {\bf Z})$. Now consider a map $f_i^{k-2}:D^{k-2} \longrightarrow Z_2(M^n, {\bf Z})$, such that $f_i^{k-2}(r, \theta)= \Sigma_{j=0}^{k+1} (-1)^j h_{i_0,....,\hat{i}_j,...,i_{k+1}}^{k-2}(1-r, \theta)$. Note that $f_i^{k-2}(1,\theta)$ 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 $h^{k-1}_i: D^{k-1} \longrightarrow Z_2(M^n, {\bf Z})$. We will assign this map to simplex $\sigma_i^{k-1}$. 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 $\tilde{g}_i$. The sum of these maps is homotopic to the map $f$ constructed in Step 1. {\bf Step 3.} We can conlude that one of the maps $\tilde{f}_i$ 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]. \end{Pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of Theorem ~\ref{TheoremB}} The proof is similar to that of Theorem ~\ref{TheoremA}. \begin{Pf}{Proof} The proof is by contradiction. We will assume that there is no stationary varifold as in Theorem 1.4. As in the ~\ref{TheoremA} we are going to discuss cases (i) and (ii) at the same time. Theorem will be proved in three steps. \noindent {\bf Step 1.} Let $f: S^q \longrightarrow M^n$ be a non-contractible map in Case (i) and let $Id: 2004-006.tex,v 1.1 2004/05/27 17:35:19 levy Exp levy $ be the identity map in Case (ii). We are going to construct a non-contractible map $\tilde{f}: S^{q-k} \longrightarrow Z_k(M^n, {\bf Z})$ in Case (i) and the map $\tilde{f}:S^{n-k} \longrightarrow Z_k(M^n, {\bf Z})$ in Case (ii). (Here and below $Z_k(M^n, {\bf Z})$ denotes the space of integral $k$-cycles on $M^n$.) \noindent {\bf Step 2.} We will construct maps $\tilde{g}_i$ from $S^{q-k}$ (or $S^{n-k}$) to the space of $k$-dimensional cycles on $M^n$, such that the map $\tilde{f}$ of Step 1 will be homotopic to their sum. The masses of $k$-cycles $\tilde g_i(t)$ do not exceed the right hand side in the inequality (i) (or (ii)) in Theorem 1.4. \noindent {\bf Step 3.} We will conclude that, one of the maps $\tilde g_i$ 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. {\bf Step 1.} Consider an arbitrary simplex $\tilde{\sigma}^{k+1}_i= \tilde{v}_{i_0},...,\tilde{v}_{i_{k+1}}]$, (in general an arbitrary simplex of dimension $l$ will be sometimes denoted as $\sigma^l_{i_0,...,i_l}$ 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 $\tilde{h}_i^1:[0,1] \longrightarrow Z_k(M^n,{\bf Z})$ that connects its boundary with a point $\tilde{p}^{k+1}_i$, (the homotopy will be sometimes denoted as $\tilde{h}^1_{i_0,...,i_{k+1}}$, and the point will be sometimes denoted as $\tilde{p}^{k+1}_{i_0,...,i_{k+1}}$). We can define a correspondence between $\tilde{\sigma}_i^{k+1}$ and the map $h_i^1$. Now consider a $(k+2)$-dimensional simplex $\tilde{\sigma}_i^{(k+2)}= [\tilde{v}_{i_0},...,\tilde{v}_{i_{k+2}}]$. Each face $(-1)^j [\tilde{v}_{i_0},...,\hat{\tilde{v}}_{i_j},...,\tilde{v}_{i_{k+2}}]$ of its boundary corresponds to the map $(-1)^j \tilde{h}^1_{i_0,...,\hat{i}_j, ...,i_{k+2}}$. Define a new map $\tilde{f}_i^1:[0,1] \longrightarrow Z_k(M^n,{\bf Z})$ by letting $\tilde{f}_i^1(t)= \Sigma_{j=0}^{k+2} (-1)^j \tilde{h}^1_{i_0,...,\hat{i}_j,...,i_{k+2}}(1-t)$. Note that $\tilde{f}_i^1(0)=\tilde{f}_i^1(1)$ 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 $\tilde{h}^2_i:D^2 \longrightarrow Z_k(M^n, {\bf Z})$, and so we can define a correspondence between simplex $\tilde{\sigma}_i^{k+2}$ and the map $\tilde{h}^2_i$. Now suppose we have constructed maps $\tilde{h}_i^{l-k}:D^{l-k} \longrightarrow Z_k(M^n, {\bf Z})$ that correspond to $l$-dimensional simplices $\tilde{\sigma}^l_i$. Consider an arbitrary $(l+1)$-dimensional simplex $\tilde{\sigma}^{l+1}_i=[\tilde{v}_{i_0},...,\tilde{v}_{i_{l+1}}]$. Define the map $\tilde{f}_i^{l-k}:D^{l-k} \longrightarrow Z_k(M^n, {\bf Z})$ by letting $\tilde{f}_i^{l-k}(r,\theta)=\Sigma_{j=0}^{l+1}(-1)^j \tilde{h}^{l-k}_{i_0,...,\hat{i}_j,...,i_{l+1}}(1-r, \theta)$. This map corresponds to the boundary of $\tilde{\sigma}_i^{l+1}$. Note that $\tilde{f}_i^{l-k}(1,\theta)$ 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 $\tilde{h}_i^{l+1-k}:D^{l+1-k} \longrightarrow Z_k(M^n, {\bf Z})$ 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 $\tilde{h}^{q-k}_i$s in case (i) and the maps $\tilde{h}_i^{n-k}$s in case (ii). Take the sum of those maps over all the simplices in the triangulation to obtain sphere $\tilde{f}:S^{q-k} \longrightarrow Z_k(M^n, {\bf Z})$ in the space of integral $k$-cycles in Case (i) and $\tilde{f}:S^{n-k} \longrightarrow Z_k(M^n, {\bf Z})$ in Case (ii). {\bf Step 2. (The main step.)} We will begin by extending the map $f:S^q \longrightarrow M^n$ to the $2$-skeleton of $D^3$ (Case (i)) or by extending the map $Id: 2004-006.tex,v 1.1 2004/05/27 17:35:19 levy Exp levy $ to the $1$-skeleton of $W$ (Case (ii)). In both of those cases the procedure is identical to that of Step 2 in the proof of Theorem ~\ref{TheoremA}. After that we will establish a correspondence between simplices $\sigma_i^l$, where $2 \leq l \leq k$ and singular chains of the corresponding dimension on $M^n$. Let us consider an arbitrary $2$-simplex $\sigma_i^2=[v_{i_0}, v_{i_1}, v_{i_2}]$. Its boundary is mapped to a curve of length $\leq 2d+\delta$ (Case (i)) or of length $\leq 6 Fill Rad M^n + 9\delta$ (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 $\leq FH_2(l_c)$. Now consider an arbitrary $3$-simplex $\sigma_i^3$. Its boundary is assigned a $2$-cycle $S_i^2=\Sigma_{j=0}^3 (-1)^j s_{i_0,..,\hat{i}_j,...,i_3}^2$ of area $\leq 3FH_2(l_c)+\delta$ in Case (i) and of area $\leq 4FH_2(l_c)$ in Case (ii). Let $s_i^3$ be a singular $3$-chain of smallest area that has $S_i^2$ as its boundary. Its volume is at most $FH_3(3FH_2(l_c)+\delta)$ in case (i) and at most $FH_3(4FH_2(l_c))$ in case (ii). Next suppose that for each arbitrary simplex $\sigma^l_i$, where $l \leq k-1$ we have constructed a chain $s_i^l$ of volume $\leq FH_l(lFH_{l-1}((l-1)FH_{l-2}(.....) +\delta)+\delta)$ in Case (i) and $\leq FH_l((l+1)FH_{l-1}(lFH_{l-2}((l-1)...)))$ in Case (ii). Consider an arbitrary $(l+1)$-simplex $\sigma_i^l$. For each face $(-1)^j[v_{i_0},...,\hat{v}_{i_j},...,v_{i_{l+1}}]$ in its boundary there exists a preassigned singular chain $(-1)^js_{i_0,...,\hat{i}_j,...,i_{l+1}}$. Consider a cycle $S^{l}_i= \Sigma_{j=0}^{l+1} (-1)^j s_{i_0,...,\hat{i}_j,...,i_{l+1}}$. Find a singular $(l+1)$-chain of $s_i^{l+1}$, such that $\partial s_i^{l+1}=S_i^l$. This chain will be assigned to $\sigma_i^l$. 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 $\sigma_i^{k+1}$. Each face in its boundary $(-1)^j [v_{i_0,...,\hat{i}_j,...,i_{k+1}}]$ corresponds to the singular chain $(-1)^j s_{i_0,...,\hat{i_j},...,i_{k+1}}$. Consider the following cycle $S_i^{k+1}=\Sigma_{j+0}^{k+1}(-1)^js_{i_0,...,\hat{i}_j,...,i_{k+1}}$. This is an element of $Z_k(M^n,{\bf Z})$ of volume $\leq (k+1)FH_k(kFH_{k-1}((k-1)...)+\delta)+\delta$ in Case (i) and of volume $\leq (k+2)FH_k((k+1)FH_{k_1}(k...))$. 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 $h^1_i:[0,1] \longrightarrow Z_k(M^n, {\bf Z})$. We will assign this path to the above simplex $\sigma^{k+1}_i$. Now suppose we have constructed the maps $h_i^{l-k}:D^{l-k} \longrightarrow Z_k(M^n, {\bf Z})$ corresponding to simplices $\sigma_i^l$, where $l \leq q-1$ in Case (i) and $\leq n-1$ in Case (ii). Consider an arbitrary $(l+1)$-dimensional simplex $\sigma_i^{l+1}$. Each face $(-1)^j[v_{i_0},...,\hat{v}_{i_j},...,v_{i_{l+1}}]$ corresponds to a map $(-1)^j h^{l-k}_{i_0,...,\hat{i}_j,...,i_{l+1}}(r,\theta)$. Define a new map $f_i^{k-l}(r,\theta)=\Sigma_{j=0}^{l+1} (-1)^j h^{l-k}_{i_0,...,\hat{i}_j,..., i_{l+1}}(1-r, \theta)$. Note that $f(1,\theta)$ 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 $h_i^{l-k+1}:D^{l-k+1} \longrightarrow Z_k(M^n, {\bf Z})$. We continue in the above manner until we construct maps $f^{q-k}_i$ in case (i) or the maps $f^{n-k}_i$ in case (ii). Those are our maps $\tilde{g}_i's$. Note that $\tilde{f}$ is homotopic to their sum. \end{Pf} {\bf 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \small \begin{thebibliography}{BaGSc} \bibitem[A]{A} {\sc F. ~Almgren}, The homotopy groups of the integral cycle groups, Topology 1(1962), 257-289. \bibitem[CD]{CD} {\sc T. ~Colding, C. ~De Lellis}, The min-max construction of minimal surfaces, Surv. in Diff. Geom., 8(2003), 75-107. \bibitem[Gr]{Gr} {\sc M. ~Gromov}, Filling Riemannian Manifolds, J.\ Diff.\ Geom. 18(1983), 1-147. \bibitem[NR1]{NR1} {\sc A. ~Nabutovsky, R. ~Rotman}, Volume, diameter and the minimal mass of a stationary 1-cycle, to appear in GAFA. \bibitem[NR2]{NR2} {\sc A. ~Nabutovsky, R. ~Rotman}, The area of a minimal embedded 2-sphere in a manifold diffeomeorphic to $S^3$, IMRN, 2003, no. 39, 2121-2129. \bibitem[P]{P} {\sc J. ~Pitts}, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes, 27, Princeton Univ. Press, 1981. \bibitem[Si]{Si} {\sc 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. \bibitem[SS]{SS} {\sc L. ~Simon, R. ~Schoen}, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34(1981), 741-797. \bibitem[SSY]{SSY} {\sc L. ~Simon, R. ~Schoen, S.T. ~Yau}, Curvature estimates for minimal hypersurfaces", Acta Math. 134(1975), 276-288. \bibitem[S]{S} {\sc F. ~Smith}, On the existence of embedded minimal 2-sphere in the 3-sphere, endowed with arbitrary metric, Ph. D. Thesis, University of Melbourne, Melbourne. %\bibitem[G3]{G3} {\sc M. ~Gromov}, Systoles and intersystolic inequalities, Ac%tes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), 291-362, Sem%in, Congr., 1, Soc. Math. France, Paris, 1996. %\bibitem[G3]{G3} %{\sc M. ~Gromov}, Metric structures for Riemannian and Non-riemannian %spaces, Birkhauser Verlag, 1999. \end{thebibliography} \normalsize \bigskip \begin{tabbing} \hspace*{7.5cm}\=\kill 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 \end{tabbing} \end{document}