\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 $<k$ 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
$\Phi_t$.
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.
\par
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 $i$th 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 $\epsilon\not = 0$
then the mass of the cycle is equal to twice the mass of $z$.
Hoewer, when $\epsilon$ 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.
\par
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.
\par
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 $\Sigma_ia_i\sigma_i$,
where $a_i\in R$ and $\sigma_i$ are singular simplices, is defined as
$\Sigma\vert a_i\vert\ vol(\sigma_i)$.
\par
\begin{Def}[Homological filling function.] \label{Fillfunc}
Let $M^n$ be a closed Riemannian manifold of dimension $n$.  Let
$\gamma(t)$ denote a closed piecewise differentiable curve.  Then
the homological filling function $FH=FH_1: {\bf R_+} \longrightarrow {\bf R_+}$
will be defined as follows: $FH(x)=\max_{\{\gamma(t)\vert
length (\gamma(t)) \leq x\}}
\min_{\{\Sigma_2\vert \partial \Sigma_2 = \gamma(t)\}} Area (\Sigma_2)$, 
where $\Sigma_2$ is a singular $2$-chain, and $Area(\Sigma_2)$ denotes
its area.

\end{Def}

\begin{Def}[Homological filling function in dimension $k>1$.]
\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}




































































