††footnotetext: A. Petrunin was partially supported by NSF grant DMS 1309340.
S. Stadler was supported by DFG grants STA 1511/1-1 and SPP 2026.

# Metric minimizing surfaces revisited

Anton Petrunin and Stephan Stadler

###### Abstract

A surface that does not admit a length nonincreasing deformation is called
_metric minimizing_. We show that metric minimizing surfaces in $\mathop{\rm
CAT}\nolimits(0)$ spaces are locally $\mathop{\rm CAT}\nolimits(0)$ with
respect to their length metrics.

##  1 Introduction

Main result. Assume $s$ is a Lipschitz embedding of the disc $\mathbb{D}$ into
the Euclidean space $\mathbb{R}^{3}$. We say $s$ is _metric minimizing_ if its
intrinsic metric is minimal. Explicitly, this means that if a map
$s^{\prime}\colon\mathbb{D}\to\mathbb{R}^{3}$ agrees with $s$ on
$\partial\mathbb{D}=\SS^{1}$ and fulfills

$\mathop{\rm length}\nolimits s\circ\gamma\geqslant\mathop{\rm
length}\nolimits s^{\prime}\circ\gamma$

for all curves $\gamma$ in $\mathbb{D}$, then equality holds for all curves
$\gamma$.

As it follows from the main theorem below, the induced metric on the disc for
a metric minimizing embedding is $\mathop{\rm CAT}\nolimits(0)$.

To formulate the theorem in full generality we need to extend the definition
of metric minimizing maps in two ways. First, we will only assume continuity
of $s$, so the map $s$ might not be an embedding and not necessary Lipschitz;
this part is tricky — a straightforward generalization produces a too weak
condition. Secondly, we also have to define it with arbitrary metric spaces as
targets; this part is straightforward.

Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ be a continuous map.
Let $\gamma\colon\SS^{1}\to Y$ be a closed rectifiable curve. We say that
$s\colon\mathbb{D}\to Y$ _spans_ $\gamma$, if $s$ is an extension of $\gamma$;
that is, $s|_{\SS^{1}}=\gamma$.

Consider the induced length-pseudometric on $\mathbb{D}$ defined as

$\langle x-y\rangle_{s}=\inf_{\alpha}\\{\mathop{\rm length}\nolimits
s\circ\alpha\\},$

where the greatest lower bound is taken over all paths $\alpha$ from $x$ to
$y$ in $\mathbb{D}$. The distance $\langle x-y\rangle_{s}$ might take infinite
values. We denote by $\langle\mathbb{D}\rangle_{s}$ the corresponding metric
space; see the next section for a precise definition.

The space $\langle\mathbb{D}\rangle_{s}$ comes with the projections
$\mathbb{D}\xrightarrow{\hat{\pi}_{s}}\langle\mathbb{D}\rangle_{s}\xrightarrow{\hat{s}}Y$;
the restriction $\hat{\pi}_{s}|_{\SS^{1}}$ will be denoted by $\delta_{s}$.

Assume that $s$ and $s^{\prime}$ are two maps spanning the same curve. We
write $s\succcurlyeq s^{\prime}$ if there is a _majorization_
$\mu\colon\langle\mathbb{D}\rangle_{s}\to\langle\mathbb{D}\rangle_{s^{\prime}}$;
meaning that $\mu$ is a _short_ map and
$\delta_{s^{\prime}}=\mu\circ\delta_{s}$; here and further _short_ stands for
_distance non-incerasing_.

(Note that if $s$ and $s^{\prime}$ are Lipschitz embeddings, then any
majorization
$\langle\mathbb{D}\rangle_{s}\to\langle\mathbb{D}\rangle_{s^{\prime}}$ admits
a continuous lifting $\mathbb{D}\to\mathbb{D}$; in general such a lifting may
not exist.)

A map $s\colon\mathbb{D}\to Y$ will be called _metric minimizing disc_ if
$s\succcurlyeq s^{\prime}$ implies that the corresponding majorization $\mu$
is an isometry.

A topological space $W$ together with a choice of a closed curve
$\delta\colon\SS^{1}\to\nobreak W$ is called _disc retract_ if the mapping
cylinder of $\delta$

$W_{\delta}=W\bigsqcup_{\delta(u)\sim(u,0)}\SS^{1}\times[0,1]$

is homeomorphic to the disc $\mathbb{D}$. The curve $\delta$ will be called
_boundary curve_ of the disc retract $W$.

An example of a disc retract that is not a disc is shown on the picture; note
that it has essentially different boundary curves, in the sense that one is
not a reparametrization of the other. The definition is motivated by the
following observation: if the boundary curve of a disc retract $W$ is a simple
closed curve, then $W$ is homeomorphic to $\mathbb{D}$.

1.1. Main theorem. Assume $Y$ is a $\mathop{\rm CAT}\nolimits(0)$ space and
$s\colon\mathbb{D}\to Y$ is a metric minimizing disc. Then
$\langle\mathbb{D}\rangle_{s}$ is a $\mathop{\rm CAT}\nolimits(0)$ disc
retract with boundary curve $\delta_{s}$.

In particular, if $s$ spans a simple closed curve, then
$\langle\mathbb{D}\rangle_{s}$ is a $\mathop{\rm CAT}\nolimits(0)$ disc.

Note that apart from continuity we did not make any regularity assumptions on
the map $s$; in particular, before hand, the space
$\langle\mathbb{D}\rangle_{s}$ might have wild topology.

If we remove the condition that the boundary curve is rectifiable, then the
space $\langle\mathbb{D}\rangle_{s}$ might have points at infinite distance
from each other. However, our proof shows that all triangles with finite sides
in $\langle\mathbb{D}\rangle_{s}$ are still thin; in particular each metric
component of $\langle\mathbb{D}\rangle_{s}$ is a $\mathop{\rm
CAT}\nolimits(0)$ space. So in this case, one could consider
$\langle\mathbb{D}\rangle_{s}$ as a $\mathop{\rm CAT}\nolimits(0)$ space where
infinite distances between points are legal.

The class of metric minimizing discs is huge — if the ambient space $Y$ is
$\mathop{\rm CAT}\nolimits(0)$, then one can find a metric minimizing disc
$\succcurlyeq$-_below_ for any map $s\colon\mathbb{D}\to Y$ such that the
metric $(x,y)\mapsto\langle x,y\rangle_{s}$ is continuous (this can be proves
by applying ultralimit+projection construction described below). Metric
minimizing discs include many well-studied maps from $\mathbb{D}$ to metric
spaces. The ruled discs considered by Alexandrov [1] are evidently metric
minimizing since one cannot shorten a geodesic. In addition, harmonic maps in
the sense of Korevaar–Schoen [12] from the disc to $\mathop{\rm
CAT}\nolimits(0)$ spaces are metric minimizing. Indeed, harmonic discs are
solutions to the Dirichlet problem; that is energy minimizing fillings of
given loops. Since the Korevaar–Schoen energy is convex, such solutions are
unique. But decreasing the intrinsic metric will only decrease the energy of
the corresponding map. As a consequence, minimal discs in the sense of
Lytchak–Wenger [15] are metric minimizing; these are Douglas–Rado solutions to
the Plateau problem and therefore harmonic.

As intended, the main theorem subsumes and generalizes several previously
known results, including Alexandr Alexandrov’s theorem about ruled surfaces in
[1] and the main theorem by the first author announced in [20], assuming it is
formulated correctly; see Section 9. It is closely related to the classical
Gauss formula and results on saddle surfaces by Samuel Shefel in [23] and
[24]. It also generalizes the result on minimal surfaces by Alexander Lytchak
and Stefan Wenger [17, Theorem 1.2] and an earlier result of Chikako Mese
[18]; see also [3, Chapter 4] and [22]. Despite that some special cases of
Theorem 1 were known, the result is new even for harmonic discs.

Let us list a few applications of metric minimizing surfaces. They were used
by Alexander Lytchak [13] to study sets of positive reach (ruled surfaces). In
[2], Stephanie Alexander and Richard Bishop used them to generalize the Gauss
equation to non-smooth spaces (ruled surfaces). In [8], Mikhael Gromov used
them to bound the complexity of smooth maps (general metric minimizing
surfaces). In [14], Alexander Lytchak and the second author used them to
deform general $\mathop{\rm CAT}\nolimits(0)$ spaces (minimal discs). In [25],
the second author used them in the proof a $\mathop{\rm CAT}\nolimits(0)$
version of the Fary–Milnor theorem to control the mapping behavior of minimal
surfaces (minimal discs).

In general it is hard to check whether a given surface is metric minimizing.
Note that any smooth metric minimizing disc in a Euclidean space is saddle.
The converse fails in general (see Section 10), but the following statement
gives a local converse in dimension three. We expect that in the four-
dimensional case even the local converse does not hold.

1.2. Proposition. Any smooth strictly saddle surface in $\mathbb{R}^{3}$ is
locally metric minimizing.

Structure of the paper. In Section 2 we define a couple of metrics induced by
a given continuous map. This naturally leads to the monotone-light
factorization theorem in a metric context.

In Section 3 we obtain topological control on the length space associated to a
metric minimizing disc. This part follows from Moore’s quotient theorem on
cell-like maps.

Section 4 establishes compactness of a certain class of nonpositively curved
surfaces. This will be used later, when we approximate metric minimizing maps.

In Section 6 we prove the key lemma which says that the restriction to a
finite set of a metric minimizing disc factorizes as a composition of short
maps over a nonpositively curved surface. The proof uses the properties of
_metric minimizing graphs_ discussed in Section 5.

In Section 7 we prove an extension result which is of independent interest and
logically detached from the rest of the paper. It gives a criterion that
allows to extend maps from subsets to the whole space, once extensions on
finite subsets are guaranteed.

In Section 8 we assemble the proof of the main theorem. We show that a metric
minimizing disc factorizes as a composition of short maps over a nonpositively
curved surface. The proof is finished by showing that the first map of this
factorization induces an isometry.

In Section 9, we give corrected forulations of the theorem in the old paper
and show that they follow from the main theorem.

In the last section, we discuss the relation between saddle surfaces and
metric minimizing discs.

Acknowledgement. We want to thank Sergei Ivanov, Nina Lebedeva, Carlo
Sinestrari, Peter Topping and Burkhard Wilking for help. We also would like to
thank Alexander Lytchak for explaining his recent work with Stefan Wenger and
for several helpful discussions.

##  2 Definitions

Metrics and pseudometrics. Let $X$ be a set. A _pseudometric_ on $X$ is a
function $X\times X\to[0,\infty]$ denoted as $(x,y)\mapsto|x-y|$ such that

  * $\diamond$

$|x-x|=0$, for any $x\in X$;

  * $\diamond$

$|x-y|=|y-x|$, for any $x,y\in X$;

  * $\diamond$

$|x-y|+|y-z|\geqslant|x-z|$ for any $x,y,z\in X$.

If in addition $|x-y|=0$ implies $x=y$, then the pseudometric $|{*}-{*}|$ is
called a _metric_ ; some authors prefer to call it _$\infty$ -metric_ to
emphasize that the distance between points might be infinite. The value
$|x-y|$ will also be called _distance_ form $x$ to $y$.

A (pseudo)metric space $X$ is the underlying set which is also denoted by $X$
equipped with a (pseudo)metric which often will be denoted by $|{*}-{*}|_{X}$.
We will use $X$ as an index if we want to emphasize that we are working in the
space $X$; for example the ball of radius $R$ centered at $z$ in $X$ can be
denoted as

$B(z,R)_{X}=\\{\,{x\in X}\,:\,{|z-x|_{X}<R}\,\\}.$

For any pseudometric on a set there is an equivalence relation “$\sim$” such
that

$x\sim y\quad\iff\quad|x-y|=0.$

The pseudometric induced on the set of equivalence classes

$[x]=\\{\,{x^{\prime}\in X}\,:\,{x^{\prime}\sim x}\,\\}$

becomes a metric. The obtained metric space will be denoted as $[X]$; it comes
with the projection map $X\to[X]$ defined as $x\mapsto[x]$.

For a metric space we can consider the equivalence relation “$\approx$”
defined as

$x\approx y\quad\iff\quad|x-y|<\infty.$

Its equivalence classes are called _metric components_. Note that by
definition each metric component is a _genuine metric space_ , meaning that
distances between points are finite. Consequently, any metric space is a
disjoint union of genuine metric spaces.

Pseudometrics induced by a map. Assume $X$ is a topological space and $Y$ is a
metric space. Let $f\colon X\to Y$ be a continuous map.

Let us define the _length pseudometric_ on $X$ induced by $f$ as

$\langle x-y\rangle_{f}=\inf\\{\,{\mathop{\rm
length}\nolimits(f\circ\gamma)_{Y}}\,:\,{\gamma\ \text{a path in}\ X\
\text{from}\ x\ \text{to}\ y}\,\\}.$

Denote by $\langle X\rangle_{f}$ the corresponding metric space; that is,

$\langle X\rangle_{f}=[(X,\langle{*}-{*}\rangle_{f})].$

Similarly, define a _connecting pseudometric_ $|{*}-{*}|_{f}$ on $X$ in the
following way

$|x-y|_{f}=\inf\\{\mathop{\rm diam}\nolimits f(K)\\},$

where the greatest lower bound is taken over all connected sets $K\subset X$
that contain $x$ and $y$; if there is no such set we set $|x-y|_{f}=\infty$.
The associated metric space will be denoted as $|X|_{f}$; that is,

$|X|_{f}=[(X,|{*}-{*}|_{f})].$

For the projections

$\bar{\pi}_{f}\colon X\to|X|_{f}\quad\text{and}\quad\hat{\pi}_{f}\colon
X\to\langle X\rangle_{f}$

we will also use the shortcut notations

$\bar{x}=\bar{\pi}_{f}(x)\quad\text{and}\quad\hat{x}=\hat{\pi}_{f}(x).$

2.1. Lemma. Let $X$ be a locally connected topological space and $Y$ a metric
space. Assume that $f\colon X\to Y$ is a continuous map. Then
$\bar{\pi}_{f}\colon X\to|X|_{f}$ is continuous.

In particular, if in addition $X$ is compact, then so is $|X|_{f}$.

Proof. For a point $x\in X$ and $\varepsilon>0$ we denote by $U$ the connected
component of $f^{-1}[B(f(x),\varepsilon)_{Y}]$ that contains $x$. Since $X$ is
locally connected, the set $U$ is open.

Note that $\bar{\pi}_{f}(U)\subset B(\bar{x},2{\hskip 0.5pt\cdot\hskip
0.5pt}\varepsilon)_{|X|_{f}}$; hence the result. ∎

$|X|_{f}$$\langle
X\rangle_{f}$$Y$$X$$\tau_{f}$$f$$\hat{\pi}_{f}$$\hat{f}$$\bar{\pi}_{f}$$\bar{f}$

Note that $\tau_{f}\colon\hat{x}\to\bar{x}$ defines a map
$\tau_{f}\colon\langle X\rangle_{f}\to\nobreak|X|_{f}$ and by construction it
preserves the lengths of all curves coming from $X$. Since $\langle
X\rangle_{f}$ is a length space, the latter implies that $\tau_{f}$ is short.
The map $\tau_{f}$ might not induce an isometry

$\langle X\rangle_{f}\to\langle|X|_{f}\rangle_{\bar{f}}.$

Moreover, $\tau_{f}$ does not have to be injective, an example is given in
[21, 4.2]. However, for metric minimizing discs $f\colon\mathbb{D}\to Y$ both
statements hold true; see Proposition 3.

The space $\langle|X|_{f}\rangle_{\bar{f}}$ is the intrinsic metric on
$|X|_{f}$, which is to $\langle|X|_{f}\rangle$ in the notation defined below.
We will denote it briefly by $\langle|X|\rangle_{f}$. The corresponding
pseudometric will be called _intrinsic pseudometric_ on $X$ induced by $f$; it
will be denoted by $\langle|{*}-{*}|\rangle_{f}$. This is a more natural way
to pullback intrinsic metric to $X$. If $X$ is compact, then
$\langle|{*}-{*}|\rangle_{f}$ coincides with the pseudometric
$\mathrm{pull}_{f}$ defined in [21]. It will show up in Section 9.

The maps $\bar{f}\colon|X|_{f}\to Y$ and $\hat{f}\colon\langle X\rangle_{f}\to
Y$ are uniquely defined by the identity

$f(x)=\bar{f}(\bar{x})=\hat{f}(\hat{x})$

for any $x\in X$. By construction, the diagram commutes.

Moreover, if $X$ is compact, then $\bar{\pi}_{f}$ has connected fibers, see
Lemma 2.

2.2. Lemma. Let $X$ be a compact metric space. Then $\bar{f}\colon|X|_{f}\to
Y$ preserves the length of every curve.

Proof. Since $\bar{f}$ is short, we have $\mathop{\rm
length}\nolimits(\bar{f}\circ\gamma)\leq\mathop{\rm length}\nolimits(\gamma)$
for every curve $\gamma$ in $|X|_{f}$. Let a rectifiable curve $\gamma$ be
given and let $\varepsilon>0$. Choose points $x_{i}$ on $\gamma$ such that
$\mathop{\rm
length}\nolimits(\gamma)\leq\sum_{i}|x_{i},x_{i+1}|_{|X|_{f}}+\varepsilon$.
Denote by $\gamma_{i}$ the piece of $\gamma$ between $x_{i}$ and $x_{i+1}$.
Since $\bar{\pi}_{f}$ has connected fibers, each set
$\bar{\pi}_{f}^{-1}(\gamma_{i})$ is connected. Hence

$\displaystyle\mathop{\rm length}\nolimits(\gamma)$
$\displaystyle\leq\sum_{i}\mathop{\rm diam}\nolimits
f(\bar{\pi}_{f}^{-1}(\gamma_{i}))+\varepsilon=$
$\displaystyle=\sum_{i}\mathop{\rm
diam}\nolimits(\bar{f}(\gamma_{i}))+\varepsilon\leq$
$\displaystyle\leq\mathop{\rm
length}\nolimits(\bar{f}\circ\gamma)+\varepsilon.$

The claim follows since $\varepsilon$ was arbitrary. ∎

Metrics induced by metrics. If $X$ is a metric space, the two constructions
above can be applied to the identity map $\mathrm{id}\colon X\to X$. In this
case the obtained spaces $\langle X\rangle_{\mathrm{id}}$ and
$|X|_{\mathrm{id}}$ will be denoted by $\langle X\rangle$ and $|X|$
correspondingly. The space $\langle X\rangle$ is $X$ equipped with induced
length metric. All three spaces $\langle X\rangle$, $|X|$ and $X$ have the
same underlying set; in other words they can be considered as a single space
with different metrics and tautological maps between them. Both tautological
maps

$\langle X\rangle\to|X|\to X$

are short and length-preserving.

As a consequence of Lemma 2, the tautological map
$\langle|X|_{f}\rangle_{\bar{f}}\to\nobreak\langle|X|_{f}\rangle$ is an
isometry; that is, for any continuous map $f\colon X\to Y$ the induced length
metric on $|X|_{f}$ coincides with the length metric induced by
$\bar{f}\colon|X|_{f}\to Y$.

Recall that a geodesic in a metric space is a curve whose length coincides
with the distance between its endpoints. A metric space is called _geodesic_
if any two points at finite distance can be joined by a geodesic.

2.3. Lemma. Let $X$ be a compact metric space. Then $\langle X\rangle$ is a
complete geodesic space.

The second statement is classical (see for example [9, II-§8 Thm. 3]) but we
were not able to find the first one in the literature.

Proof. Assume that $\langle X\rangle$ is not complete. Fix a Cauchy sequence
$(x_{n})$ in $\langle X\rangle$ that is not converging in $\langle X\rangle$.
After passing to a subsequence, we can assume that the points of the sequence
appear on a rectifiable curve $\hat{\gamma}\colon[0,1)\to\nobreak\langle
X\rangle$ in the same order.

The corresponding curve $\gamma\colon[0,1)\to X$ has the same length. Since
$X$ is compact we can extend it to a path $\gamma_{+}\colon[0,1]\to X$. The
curve

$\hat{\gamma}_{+}=\hat{\pi}\circ\gamma_{+}\colon[0,1]\to\langle X\rangle$

has the same length. Therefore $\hat{\gamma}_{+}(1)$ is the limit of
$(x_{n})$, a contradiction.

It remains to show that $\langle X\rangle$ is geodesic. Assume $\gamma_{n}$ is
a sequence of constant speed paths from $x$ to $y$ in $X$ such that
$\mathop{\rm length}\nolimits(\hat{\gamma}_{n})\to\langle x-y\rangle$ as
$n\to\infty$. Since $X$ is compact, we can pass to a partial limit $\gamma$ of
$\gamma_{n}$. The corresponding curve $\hat{\gamma}=\hat{\pi}\circ\gamma$ is
the needed geodesic from $\hat{x}$ to $\hat{y}$ in $\langle X\rangle$. ∎

Monotone-light factorization. Let $f\colon X\to Y$ be a map between
topological spaces. Recall that

  * $\diamond$

$f$ is called _monotone_ if the inverse image of each point is connected,

  * $\diamond$

$f$ is called _light_ if the inverse image of any point is totally
disconnected.

Since a connected set is nonempty by definition, any monotone map is onto.

2.4. Lemma. Assume $X$ is a locally connected compact metric space and $Y$ is
a metric space. Let $f\colon X\to Y$ be a continuous map. Then the map
$\bar{\pi}_{f}$ is monotone and $\bar{f}$ is light. In particular

$f=\bar{f}\circ\bar{\pi}_{f}$

is a monotone-light factorization.

Proof. First we prove the monotonicity of $\bar{\pi}_{f}$.

Assume the contrary; that is, for some $x\in X$ the equivalence class

$K=\bar{\pi}_{f}^{-1}(\bar{x})=\\{\,{x^{\prime}\in
X}\,:\,{|x-x^{\prime}|_{f}=0}\,\\}$

is not connected. Since $X$ is normal, we can cover $K$ by disjoint open sets
$U,V\subset X$ such that both intersections $K\cap U$ and $K\cap V$ are
nonempty.

By Lemma 2, $K$ is closed.

Suppose that $x\in U$ and pick $x^{\prime}\in K\cap V$. Then there is a
sequence of connected sets $K_{n}\ni x,x^{\prime}$ such that $\mathop{\rm
diam}\nolimits f(K_{n})<\tfrac{1}{n}$. For each $n$ we choose a point
$k_{n}\in K_{n}\backslash(U\cup V)$. Let $k$ be a partial limit of the
sequence $(k_{n})$. It follows that $k\in K\backslash(U\cup V)$, a
contradiction.

Assume $\bar{f}$ is not light; that is, the inverse image of some $y\in Y$
contains a closed connected set $C\subset|X|_{f}$ with more than one point.
Note that the inverse image $Z:=\bar{\pi}_{f}^{-1}C$ is connected due to
monotonicity of $\bar{\pi}_{f}$. It follows that $|z-z^{\prime}|_{f}=0$ for
any two points $z,z^{\prime}\in Z$, a contradiction. ∎

##  3 Disc retracts

A disc retract as defined above is nothing but the image of strong deformation
retraction of $\mathbb{D}$; the restriction of retraction to the boundary can
be taken as the corresponding boundary curve. We will not need this statement,
but it follows from Moore’s quotient theorem quoted below.

3.1. Proposition. Let $Y$ be a metric space and $s\colon\mathbb{D}\to\ Y$ be a
metric minimizing disc. Then $|\mathbb{D}|_{s}$ is a disc retract with
boundary curve $\delta_{s}=\bar{\pi}_{s}|_{\SS^{1}}$. Moreover, the map
$\tau_{s}\colon\langle\mathbb{D}\rangle_{s}\to|\mathbb{D}|_{s}$ is injective
and defines an isometry
$\langle\mathbb{D}\rangle_{s}\to\nobreak\langle|\mathbb{D}|\rangle_{s}$; that
is, $\tau_{s}$ is an isometry from $\langle\mathbb{D}\rangle_{s}$ to
$|\mathbb{D}|_{s}$ equipped with induced length metric.

We need a little preparation before giving the proof.

Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ be a continuous map. We
say that $s$ has _no bubbles_ if for any point $p\in Y$ every connected
component of the complement $\mathbb{D}\backslash s^{-1}\\{p\\}$ contains a
point from $\partial\mathbb{D}$.

3.2. Lemma. Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ be a metric
minimizing disc. Then $s$ has no bubbles.

Proof. Assume the contrary; that is, there is $y\in Y$ such that the
complement $\mathbb{D}\backslash s^{-1}(y)$ contains a connected component
$\Omega$ with $\partial\mathbb{D}\cap\Omega=\varnothing$.

Let us define a new map $s^{\prime}\colon\mathbb{D}\to\ Y$ by setting
$s^{\prime}(z)=y$ for any $x\in\Omega$ and $s^{\prime}(x)=s(x)$ for any
$x\notin\Omega$.

By construction, $s^{\prime}$ and $s$ agree on $\partial\mathbb{D}$. Moreover,
$s\succcurlyeq s^{\prime}$ because of the majorization
$\mu\colon\hat{\pi}_{s}(x)\mapsto\nobreak\hat{\pi}_{s^{\prime}}(x)$.

Note that

$\langle x-x^{\prime}\rangle_{s}>0=\langle x-x^{\prime}\rangle_{s^{\prime}}$

for a pair of distinct points $x,x^{\prime}\in\Omega$. In particular, $\mu$ is
not an isometry, a contradiction. ∎

3.3. Lemma. Let $Y$ be a metric space and assume that a map
$f\colon\mathbb{D}\to Y$ has no bubbles. Then $|\mathbb{D}|_{f}$ is
homeomorphic to a disc retract with boundary curve $\bar{\pi}_{f}|_{\SS^{1}}$.

This lemma is nearly identical to [16, Corollary 7.12]; it could be considered
a disc version of Moore’s quotient theorem [19], [7] which states that if a
continuous map $f$ from the sphere $\SS^{2}$ to a Hausdorff space $X$ has
acyclic fibers, then $f$ can be approximated by a homeomorphism; in particular
$X$ is homeomorphic to $\SS^{2}$.

Proof. From Lemma 2 we know that $\bar{\pi}_{f}$ is continuous and hence
$|\mathbb{D}|_{f}$ is a compact metric space.

The mapping cone over $\mathbb{D}$ along its boundary is homeomorphic to the
sphere $\SS^{2}$; denote by $\Sigma$ the mapping cone over $|\mathbb{D}|_{f}$
with respect to $\bar{\pi}_{s}|_{\SS^{1}}$. Let us extend the map
$\bar{\pi}_{f}$ to a map between the mapping cones $\SS^{2}\to\Sigma$. Note
that this map satisfies Moore’s quotient theorem, hence the statement follows.
∎

Proof of Proposition 3. The first two statements follow from Lemma 3 and Lemma
3.

Since $|\mathbb{D}|_{s}$ is a disc retract, the mapping cylinder over the
boundary curve of $|\mathbb{D}|_{s}$ is homeomorphic to $\mathbb{D}$. Denote
by $r\colon\mathbb{D}\to|\mathbb{D}|_{s}$ the corresponding retraction.

Note that $\langle\mathbb{D}\rangle_{\bar{s}\circ r}$ is isometric to
$|\mathbb{D}|_{s}$ equipped with the induced length metric. Recall that the
map $\tau_{s}\colon\langle\mathbb{D}\rangle_{s}\to|\mathbb{D}|_{s}$ is short
and the induced map
$\mu\colon\langle\mathbb{D}\rangle_{s}\to\nobreak\langle\mathbb{D}\rangle_{\bar{s}\circ
r}$ is a majorization. Since $s$ is metric minimizing, $\mu$ is an isometry.
Hence the statement follows. ∎

Assume $W$ is a disc retract with a boundary curve $\delta$. Recall that a
point $p$ in a connected space $W$ is a _cut point_ if the complement
$W\backslash\\{p\\}$ is not connected.

3.4. Lemma. Suppose that $W$ is a disc retract. Let $\Delta\subset W$ be a
maximal connected subset that contains no cut points. Assume $\Delta$ has at
least two points. Then closure of $\Delta$ is homeomorphic to $\mathbb{D}$.

Proof. Let $\delta$ be a boundary curve of $W$, so the mapping cylinder
$W_{\delta}$ is homeomorphic to $\mathbb{D}$.

Note that $p$ is a cut point of $W$ if and only if $\delta^{-1}\\{p\\}$ has at
least two connected components. In particular, any cut point of $W$ lies on
its boundary curve.

Denote by $\bar{\Delta}$ the closure of $\Delta$. Note that for any
$x\notin\bar{\Delta}$ there is a cut point $p\in\bar{\Delta}$ that cuts $x$
from $\Delta$. Moreover, the map $\sigma\colon x\mapsto p$ is uniquely defined
on $W\backslash\bar{\Delta}$; extend this map to whole $W$ by identity in
$\bar{\Delta}$. By Moore’s theorem, $\bar{\Delta}$ is a disc retract with a
boundary curve $\sigma\circ\delta$. Namely we apply Moore’s theorem to
$\SS^{2}=W_{\delta}/(\SS^{1}\times 1)$ and the quotient map
$\SS^{2}\to\SS^{2}/\sim$ for the minimal equivalence relation such that $x\sim
y$ if $\sigma(x)=\sigma(y)$.

The space $\bar{\Delta}$ has no cut points, in other words $\sigma\circ\delta$
is monotonic. It follows that $\sigma\circ\delta$ can be reparameterized into
a simple closed curve. By Jordan–Schoenflies theorem, the statement follows. ∎

The following lemma will be used in the final step in the proof of the main
theorem, Section 8.

3.5. Lemma. Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ be a metric
minimizing map. Assume that there is a $\mathop{\rm CAT}\nolimits(0)$ disc
retract $W$ with boundary curve $\delta$ and a short map
$f\colon\langle\mathbb{D}\rangle_{s}\to W$ such that
$f\circ\delta_{s}=\delta$. If there exists a short map $q\colon W\to Y$ with
$q\circ\delta=s|_{\partial\mathbb{D}}$, then the map $f$ is an isometry.

Proof. Let $r\colon\mathbb{D}\to W$ be the projection from the mapping
cylinder $\mathbb{D}=W_{\delta}$. Note that $r$ is a retraction,
$r|_{\partial\mathbb{D}}=\delta$ and the composition
$\mathbb{D}\xrightarrow{r}W\xrightarrow{q}Y$ fulfills

$q\circ r|_{\partial\mathbb{D}}=s|_{\partial\mathbb{D}}.$

Note that $\langle W\rangle_{q}=\langle\mathbb{D}\rangle_{q\circ r}$ and the
natural projection $\rho\colon W\to\langle W\rangle_{q}$ is short. It follows
that $\rho\circ
f\colon\langle\mathbb{D}\rangle_{s}\to\langle\mathbb{D}\rangle_{q\circ r}$ is
a majorization. Since $s$ is metric minimizing, $\rho\circ f$ is an isometry.

Therefore $f$ is an isometric embedding that contains $\delta$ in its image.
By Lemma 2 and Proposition 3, $\langle\mathbb{D}\rangle_{s}$ is a complete
geodesic space. So $f$ has to be surjective and therefore an isometry. ∎

##  4 Compactness lemma

A sequence of pairs $(X_{n},\gamma_{n})$, where $X_{n}$ is a metric space and
$\gamma_{n}\colon\SS^{1}\to X_{n}$ is a closed curve is said to _converge_ to
$(X_{\infty},\gamma_{\infty})$ if there is a convergence of $X_{n}$ to
$X_{\infty}$ in the sense of Gromov–Hausdorff for which $\gamma_{n}$ converges
to $\gamma_{\infty}$ pointwise.

More precisely, we ask the following.

  1. (1)

There is a metric $\rho$ on the disjoint union

$\bm{X}=X_{\infty}\sqcup X_{1}\sqcup X_{2}\dots$

that restricts to the given metric on each
$X_{\alpha},\alpha\in\\{1,2,\dots,\infty\\}$, and such that $X_{n}$ converge
to $X_{\infty}$ in the sense of Hausdorff as subsets in $(\bm{X},\rho)$.

  2. (2)

The sequence of compositions $\gamma_{n}\colon\SS^{1}\to
X_{n}\hookrightarrow\bm{X}$ converges to
$\gamma_{\infty}\colon\SS^{1}\to\nobreak X_{\infty}\hookrightarrow\bm{X}$
pointwise.

Consider the class $\mathcal{K}_{\ell}$ of $\mathop{\rm CAT}\nolimits(0)$ disc
retracts whose marked boundary curves have Lipschitz constant $\ell$.

4.1. Compactness lemma. $\mathcal{K}_{\ell}$ is compact in the topology
described above.

The lemma follows from the two lemmas below.

4.2. Lemma. $\mathcal{K}_{\ell}$ is precompact in the topology described
above.

Proof. Let $K$ be a metric space with the isometry class in
$\mathcal{K}_{\ell}$.

Denote by $\mathop{\rm area}\nolimits A$ the two-dimensional Hausdorff measure
of $A\subset K$. By the Euclidean isoperimetric inequality we have

$\mathop{\rm area}\nolimits K\leqslant\pi{\hskip 0.5pt\cdot\hskip
0.5pt}\ell^{2}.$

Fix $\varepsilon>0$. Set $m=\lceil 10{\hskip 0.5pt\cdot\hskip
0.5pt}\tfrac{\ell}{\varepsilon}\rceil$. Choose $m$ points $y_{1},\dots,y_{m}$
on $\partial K$ that divide $\partial K$ into arcs of equal length.

Consider the maximal set of points $\\{x_{1},\dots,x_{n}\\}$ such that
$d(x_{i},x_{j})>\varepsilon$ and $d(x_{i},y_{j})>\varepsilon$.

Note that the set $\\{x_{1},\dots,x_{n},y_{1},\dots,y_{m}\\}$ is an
$\varepsilon$-net in $(K,d)$. Further note that the balls
$B_{i}=B_{\varepsilon/2}(x_{i})$ do not overlap.

By comparison,

$\mathop{\rm area}\nolimits B_{i}\geqslant\tfrac{\pi{\hskip 0.5pt\cdot\hskip
0.5pt}\varepsilon^{2}}{4}.$

It follows that $n\leqslant 4{\hskip 0.5pt\cdot\hskip
0.5pt}\left(\tfrac{\ell}{\varepsilon}\right)^{2}$. In particular, there is a
integer valued function $N(\varepsilon)$, such that any $K$ as above contains
an $\varepsilon$-net with at most $N(\varepsilon)$ points.

The latter means that the class $\mathcal{K}_{\ell}$ is uniformly totally
bounded. By the selection theorem [6, 7.4.15], the class of metrics with this
property is precompact in the Gromov–Hausdorff topology.

Since the set of $\ell$-Lipschitz maps defined on $\SS^{1}$ with compact
target is compact with respect to pointwise convergence, we conclude that
$\mathcal{K}_{\ell}$ is precompact in the topology defined above. ∎

4.3. Lemma. $\mathcal{K}_{\ell}$ is closed in the topology described above.

Proof. Let $(X_{n},\gamma_{n})$ be a sequence in $\mathcal{K}_{\ell}$. Assume
$X_{n}\to X$ and $\gamma_{n}\to\gamma$. Choose a point $o_{n}\in\partial
X_{n}$ and define $f_{n}:\mathbb{D}\to X_{n}$ by sending the geodesic
$[0,\theta]$ for $\theta\in\partial\mathbb{D}=\SS^{1}$ to the geodesic path
$[o_{n},\gamma_{n}(\theta)]$ with constant speed.

By comparison, $f_{n}$ is a $(5{\hskip 0.5pt\cdot\hskip 0.5pt}\ell)$-Lipschitz
continuous ruled disc. The limit map $f\colon\mathbb{D}\to X$ is also a
$(5{\hskip 0.5pt\cdot\hskip 0.5pt}\ell)$-Lipschitz continuous ruled disc. In
particular, $f$ is metric minimizing. By Lemma 3, $f$ has no bubbles.
Therefore, by Lemma 3, $X$ is a disc retract with boundary curve $\gamma$. ∎

##  5 Metric minimizing graphs

_Metric minimizing graphs_ are defined analogously to metric minimizing discs.

Namely, let $Y$ be a metric space, $\Gamma$ be a finite graph and $A$ be a
subset of its vertexes. Given two maps $f,f^{\prime}\colon\Gamma\to Y$, we
write $f\succcurlyeq f^{\prime}\ ({\rm rel.}\ {A})$ if $f$ and $f^{\prime}$
agree on $A$ and there is a majorization
$\mu\colon\langle\Gamma\rangle_{f}\to\langle\Gamma\rangle_{f^{\prime}}$ such
that $f(a)=\nobreak f^{\prime}(\mu(a))$ for any $a\in A$.

A map $f\colon\Gamma\to Y$ is called _metric minimizing relative to $A$_ if
$f\succcurlyeq f^{\prime}\ ({\rm rel.}\ {A})$ implies that the majorization
$\mu$ is an isometry.

5.1. Proposition. Let $Y$ be a $\mathop{\rm CAT}\nolimits(0)$ space, $\Gamma$
be a finite graph and $A$ be a subset of its vertexes.

Given a continuous map $f\colon\Gamma\to Y$ there is a map $h\colon\Gamma\to
Y$ that is metric minimizing relative to $A$ and $f\succcurlyeq h\ ({\rm
rel.}\ {A})$.

Proof. Let us parametrize each edge of $\Gamma$ by $[0,1]$. A map
$h\colon\Gamma\to Y$ will be called _straight_ if it sends each edge of
$\Gamma$ to a constant-speed geodesic path in $Y$.

If $h\colon\Gamma\to Y$ is straight, then $f\succcurlyeq h\ ({\rm rel.}\ {A})$
if and only if

$|f(v)-f(w)|_{Y}\geqslant|h(v)-h(w)|_{Y}$

for any two adjacent vertexes $v$ and $w$ in $\Gamma$. In particular, we can
assume that the given map $f$ is straight.

By finiteness of the number of vertexes and Zorn’s lemma, it is sufficient to
prove that for any ordered sequence of straight maps $f_{1}\succcurlyeq
f_{2}\succcurlyeq\dots$ there exists a map $f\preccurlyeq f_{n}$ for all $n$.

Assume contrary; let us apply the _ultralimit+projection construction_.

Namely, fix an ultrafilter $\omega$; denote by $Y^{\omega}$ the ultrapower of
$Y$. Then $Y^{\omega}$ is a $\mathop{\rm CAT}\nolimits(0)$ space that contains
$Y$ as a closed convex subset. The $\omega$-limit $f_{\omega}\colon\Gamma\to
Y^{\omega}$ is well defined, since all $f_{n}$ are Lipschitz continuous.
Denote $f^{\prime}$ the composition of $f_{\omega}$ with the nearest point
projection $Y^{\omega}\to Y$. The nearest point projection to a closed convex
set in $\mathop{\rm CAT}\nolimits(0)$ space is short. Therefore
$f_{n}\succcurlyeq f^{\prime}$ for any $n$. Let $f^{\prime\prime}$ denote the
straightening of $f^{\prime}$. Then $f^{\prime}\succcurlyeq f^{\prime\prime}$
and the claim follows. ∎

5.2. Proposition. Let $Y$ be a $\mathop{\rm CAT}\nolimits(0)$ space, $\Gamma$
be a finite graph and $A$ be a subset of its vertexes. Assume that the
assignment $v\mapsto v^{\prime}$ is a metric minimizing map $\Gamma\to Y$
relative to $A$. Then

  1. (a)

each edge of $\Gamma$ maps to a geodesic;

  2. (b)

for any vertex $v\notin A$ and any $x\neq v^{\prime}$ there is an edge $[v,w]$
in $\Gamma$ such that
$\measuredangle[v^{\prime}\,{}^{w^{\prime}}_{x}]\geqslant\tfrac{\pi}{2}$;

  3. (c)

for any vertex $v\notin A$ and any cyclic order $w_{1},\dots,w_{n}$ of
adjacent vertexes we have

$\measuredangle[v^{\prime}\,{}^{w^{\prime}_{1}}_{w^{\prime}_{2}}]+\dots+\measuredangle[v^{\prime}\,{}^{w^{\prime}_{n-1}}_{w^{\prime}_{n}}]+\measuredangle[v^{\prime}\,{}^{w^{\prime}_{n}}_{w^{\prime}_{1}}]\geqslant
2{\hskip 0.5pt\cdot\hskip 0.5pt}\pi.$

Remark. The conditions in the proposition do not guarantee that the map $f$ is
metric minimizing. An example can be guessed from the diagram, where the solid
points form the set $A$.

Proof. The first condition is evident.

Assume the second condition does not hold at a vertex $v\notin\nobreak A$;
that is, there is a point $x\in Y$ such that
$\measuredangle[v^{\prime}\,{}^{w^{\prime}}_{x}]<\tfrac{\pi}{2}$ for any
adjacent vertex $w$. In this case moving $v^{\prime}$ toward $x$ along
$[v^{\prime},x]$ decreases the lengths of all edges adjacent to $v$, a
contradiction.

Assume the third condition does not hold; that is, the sum of the angles
around a fixed interior vertex $v^{\prime}$ is less than $2{\hskip
0.5pt\cdot\hskip 0.5pt}\pi$.

Recall that the space of directions $\Sigma_{v^{\prime}}$ is a $\mathop{\rm
CAT}\nolimits(1)$ space. Denote by $\xi_{1},\dots,\xi_{n}$ the directions of
$[v^{\prime},w^{\prime}_{1}],\dots,[v^{\prime},w^{\prime}_{n}]$ in
$\Sigma_{v^{\prime}}$. By assumption, we have

$|\xi_{1}-\xi_{2}|_{\Sigma_{v^{\prime}}}+\dots+|\xi_{n}-\xi_{1}|_{\Sigma_{v^{\prime}}}<2{\hskip
0.5pt\cdot\hskip 0.5pt}\pi.$

By Reshetnyak’s majorization theorem, the closed broken line
$[\xi_{1},\dots,\xi_{k}]$ is majorized by a convex spherical polygon $P$.

Note that $P$ lies in an open hemisphere with pole at some point in $P$.
Choose $x\in Y$ so that the direction from $v^{\prime}$ to $x$ coincides with
the image of the pole in $\Sigma_{f(v)}$. This choice of $x$ contradicts
(_b_). ∎

##  6 Key lemma

6.1. Lemma. Let $Y$ be a $\mathop{\rm CAT}\nolimits(0)$ space and
$s\colon\mathbb{D}\to Y$ be a metric minimizing disc. Assume
$F\subset\mathbb{D}$ is a finite set such that $\hat{\pi}_{s}(F)$ has finite
diameter in $\langle\mathbb{D}\rangle_{s}$. Then there exists a finite
piecewise geodesic graph $\Gamma$ embedded in $\langle\mathbb{D}\rangle_{s}$
that contains a geodesic between any pair of points in $\hat{\pi}_{s}(F)$.

Proof. For any pair $x,y\in F$, connect $\hat{x}$ to $\hat{y}$ by a minimizing
geodesic in $\langle\mathbb{D}\rangle_{s}$. We can assume that the constructed
geodesics are either disjoint or their intersection is formed by finite
collections of arcs and points.

Indeed, if some number of geodesics $\gamma_{1},\dots,\gamma_{n}$ already has
this property and we are given points $x$ and $y$, then we choose a minimizing
geodesic $\gamma_{n+1}$ from $x$ to $y$ that maximizes the time it spends in
$\gamma_{1},\dots,\gamma_{n}$ in the order of importance. Namely,

  * $\diamond$

among all minimizing geodesics connecting $x$ to $y$ choose one that spends
maximal time in $\gamma_{1}$ — in this case $\gamma_{n+1}$ intersects
$\gamma_{1}$ along the empty set, a one-point set or a closed arc.

  * $\diamond$

among all minimizing geodesics as above choose one that spends maximal time in
$\gamma_{2}$ — in this case $\gamma_{n+1}$ intersects $\gamma_{2}$ along at
most two arcs and points.

  * $\diamond$

and so on.

It follows that together the constructed geodesics form a finite graph
$\Gamma$ as required. ∎

6.2. Key lemma. Let $Y$ be a $\mathop{\rm CAT}\nolimits(0)$ space and
$s\colon\mathbb{D}\to Y$ be a metric minimizing disc. Given a finite set
$F\subset\mathbb{D}$ there is

  1. (1)

a $\mathop{\rm CAT}\nolimits(0)$ disc retract $W$ with boundary curve
$\delta$;

  2. (2)

a map $p\colon F\to W$ such that

$|p(x)-p(y)|_{W}\leqslant\langle x-y\rangle_{s}$

for $x,y\in F$ and $p(x)=\delta(x)$ for $x\in F\cap\partial\mathbb{D}$;

  3. (3)

a short map $q\colon W\to Y$ such that

$s(x)=q\circ p(x)$

for any $x\in\partial\mathbb{D}\cap F$.

Proof. If $\partial\mathbb{D}\cap F=\varnothing$, then one can take a one-
point space as $W$ and arbitrary maps $p\colon F\to W$ and $q\colon W\to Y$.
So suppose $\partial\mathbb{D}\cap F\neq\varnothing$.

Without loss of generality we may assume that the distance $\langle
x-y\rangle_{s}$ between any pair of points $x,y\in F$ is finite. Indeed, since
the boundary curve $s|_{\partial\mathbb{D}}$ is rectifiable, this always holds
for pairs of points in $\partial\mathbb{D}\cap F$. Consider the subset
$F^{\prime}\subset F$ that lies at finite $\langle{*}-{*}\rangle_{s}$-distance
from one (and therefore any) point in $\partial\mathbb{D}\cap F$. Suppose
$p^{\prime}\colon F^{\prime}\to W$ and $q\colon W\to Y$ are maps satisfying
the proposition for $F^{\prime}$. Extend $p^{\prime}$ to $F$ by sending
$F\backslash F^{\prime}$ to one point in $W$. The resulting map $p$ together
with $q$ will then satisfy the proposition for $F$.

By Lemma 6, there exists a finite piecewise geodesic graph $\Gamma$ embedded
in $\langle\mathbb{D}\rangle_{s}$ that contains $F$ as a subset of its
vertexes. According to Proposition 3, $\tau_{s}$ embeds $\Gamma$ in
$|\mathbb{D}|_{s}$. By Proposition 3, $|\mathbb{D}|_{s}$ is a disc retract.
Therefore $\Gamma$ can be (and will be) considered as a graph embedded into
the plane.

By Proposition 5, there is a map $u\colon\Gamma\to Y$ metric minimizing
relative to $A=F\cap\partial\mathbb{D}$ such that

$s|_{\Gamma}\succcurlyeq u\ ({\rm rel.}\ {A}).$ ➊

Fix an open disc $\Delta$ cut by $\Gamma$ from $|\mathbb{D}|_{s}$. By
Reshetnyak’s theorem, the closed curve $u|_{\partial\Delta}$ is majorized by a
convex plane polygon, possibly degenerate to a point or a line segment. Note
that the angle of the majorizing polygon cannot be smaller than the angle
between the corresponding edges in $u(\Gamma)\subset Y$.

Let us glue the majorizing polygons into $\langle\Gamma\rangle_{u}$; denote by
$W$ the resulting space. According to Proposition 5(c), the angle around each
inner vertex has to be at least $2{\hskip 0.5pt\cdot\hskip 0.5pt}\pi$. Clearly
$W$ is a disc retract; in particular, it is simply connected. It follows that
$W$ is a $\mathop{\rm CAT}\nolimits(0)$ space.

The short map $q\colon W\to Y$ is constructed by gluing together the maps
provided by Reshetnyak’s majorization theorem. The space $W$ comes with a
natural short map $\langle\Gamma\rangle_{u}\to W$.

Define $p(x)$ for $x\in F$ as the image of the corresponding vertex of
$\Gamma$ in $W$. By ➊ ‣ 6,

$|p(x)-p(y)|_{W}\leqslant\langle x-y\rangle_{s}$

for any $x,y\in F$.

By construction the pair of maps $p,q$ meet all conditions. ∎

The following establishes a connection between the key lemma (6) and the
extension lemma (7).

6.3. Lemma. Let $Y$ be a $\mathop{\rm CAT}\nolimits(0)$ space and
$s\colon\mathbb{D}\to Y$ be a metric minimizing disc. Let $W$ be a
$\mathop{\rm CAT}\nolimits(0)$ disc retract with boundary curve $\delta$. For
a given finite set $F\subset\mathbb{D}$ we define $\mathfrak{S}_{F}$ to be the
family of maps $p\colon F\to W$ such that

$|p(x)-p(y)|_{W}\leqslant\langle x-y\rangle_{s}$

for $x,y\in F$ with $p(x)=\delta(x)$ for $x\in F\cap\partial\mathbb{D}$ and
such that there exists a short map $q\colon W\to Y$ with

$s(x)=q\circ p(x)$

for any $x\in\partial\mathbb{D}\cap F$. Then $\mathfrak{S}_{F}$ is closed
under pointwise convergence.

The proof is an application of the ultralimit+projection construction; we used
it once before and will use it again later.

Proof. Consider a converging sequence $p_{n}\in\mathfrak{S}_{F}$; denote by
$p_{\infty}$ its limit. For each $p_{n}$ there is a short map $q_{n}\colon
W\to Y$ satisfying the condition above. Pass to its ultralimit
$q_{\omega}\colon W\to Y^{\omega}$. Recall that $Y$ is a closed convex set in
$Y^{\omega}$. In particular the nearest point projection $\nu\colon
Y^{\omega}\to Y$ is well defined and short. Therefore, the composition
$q=\nu\circ q_{\omega}$ is short. Finally note that the maps $p_{\infty}\colon
F\to W$ and $q\colon W\to Y$ satisfy the condition above. ∎

##  7 Extension lemma

7.1. Extension lemma. Suppose that $X$ is a set and $Y$ is a compact
topological space. Assume that for any finite set $F\subset X$ a nonempty set
$\mathfrak{S}_{F}$ of maps $F\to Y$ is given, such that

  * $\diamond$

$\mathfrak{S}_{F}$ is closed under pointwise convergence;

  * $\diamond$

for any subset $F^{\prime}\subset F$ and any map $h\in\mathfrak{S}_{F}$ the
restriction $h|_{F^{\prime}}$ belongs to $\mathfrak{S}_{F^{\prime}}$.

Then there is a map $h\colon X\to Y$ such that $h|_{F}\in\mathfrak{S}_{F}$ for
any finite set $F\subset X$.

Proof. Consider the space $Y^{X}$ of all maps $X\to Y$ equipped with the
product topology.

Denote by $\bar{\mathfrak{S}}_{F}$ the set of maps $h\in Y^{X}$ such that its
restriction $h|_{F}$ belongs to $\mathfrak{S}_{F}$. By assumption, the sets
$\bar{\mathfrak{S}}_{F}\subset Y^{X}$ are closed and any finite interection of
these sets is nonempty.

According to Tikhonov’s theorem, $Y^{X}$ is compact. By the finite
intersection propery, the intersection $\bigcap_{F}\bar{\mathfrak{S}}_{F}$ for
all finite sets $F\subset X$ is nonempty. Hence the stattement follows. ∎

Note that if $X$ and $Y$ are metric spaces and $A$ is a subset in $X$ then one
can take as $\mathfrak{S}_{F}$ the short maps $F\to Y$ that coincide with a
given short map $A\to Y$ on $A\cap F$. This way we obtain the folloing
corollary; it is closely related to [10, Proposition 5.2]. In a similar
fashion, we will use the lemma in the proof of our main theorem.

7.2. Corollary. Let $X$ and $Y$ be metric spaces, $A\subset X$ and $f\colon
A\to Y$ a short map. Assume $Y$ is compact and for any finite set $F\subset X$
there is a short map $F\to Y$ that agrees with $f$ in $F\cap A$. Then there is
a short map $X\to Y$ that agrees with $f$ in $A$.

##  8 Proof assembling

Proof of the main theorem. Given a finite set $F\subset\mathbb{D}$, denote by
$\mathcal{W}_{F}$ the set of isometry classes of spaces $W$ that meet the
conditions of the key lemma 6 for $F$; according to the key lemma,
$\mathcal{W}_{F}\neq\varnothing$. Note that for two finite sets $F\subset
F^{\prime}$ in $\mathbb{D}$, we have
$\mathcal{W}_{F}\supset\mathcal{W}_{F^{\prime}}$.

According to the compactness Lemma (4) $\mathcal{W}_{F}$ is compact. Therefore

$\mathcal{W}=\bigcap_{F}\mathcal{W}_{F}\neq\varnothing$

where the intersection is taken over all finite subsets $F$ in $\mathbb{D}$.

Fix a space $W$ from $\mathcal{W}$; the space $W$ is a $\mathop{\rm
CAT}\nolimits(0)$ disc retract, such that given a finite set
$F\subset\mathbb{D}$ there is a map $h_{F}\colon F\to W$ that is short with
respect to $\langle{*}-{*}\rangle_{s}$ and a short map $q_{F}\colon W\to Y$
such that $q_{F}\circ h_{F}$ agrees with $s$ on $\partial\mathbb{D}\cap F$.

Given a finite set $F\subset\mathbb{D}$, denote by $\mathfrak{S}_{F}$ the set
of all maps $h_{F}\colon F\to W$ described above.

By Lemma 6, $\mathfrak{S}_{F}$ is closed. The condition on the restriction of
$h_{F}\in\mathfrak{S}_{F}$ in the extension lemma (7) is evident. Applying the
lemma, we get a map $h\colon\mathbb{D}\to W$ such that
$h|_{F}\in\mathfrak{S}_{F}$ for any finite set $F\subset\mathbb{D}$.

Our next aim is to show that there is a single map $q$ such that for all
finite sets $F$ the composition $q\circ h|_{F}$ agrees with $s$ on
$\partial\mathbb{D}\cap F$. This is done by applying the ultralimit+projection
construction:

Choose a sequence of finite sets $F_{n}$ such that $F_{n}$ get denser and
denser in $\mathbb{D}$ and the intersections $F_{n}\cap\partial\mathbb{D}$ get
denser and denser in $\partial\mathbb{D}$; denote by $q_{n}$ the corresponding
maps. Let $q_{\omega}\colon W\to Y^{\omega}$ be the ultralimit of $q_{n}$ and
set $q=\nu\circ q_{\omega}$, where $\nu\colon Y^{\omega}\to Y$ is the nearest
point projection. By construction $q\colon W\to Y$ is short and $q\circ h$
agrees with $s$ on $\partial\mathbb{D}$. Note that we cannot conclude
$s\succcurlyeq q\circ h$ because $h$ might not be continuous.

By construction, the map $h$ induces a short map
$\hat{h}\colon\langle\mathbb{D}\rangle_{s}\to W$ such that
$\hat{h}\circ\delta_{s}$ is the boundary curve of $W$. By Lemma 3, $\hat{h}$
is an isometry and the statement follows. ∎

##  9 About the old theorem

In this section we formulate and prove two versions of the main theorem in
[20] with corrections.

Note that the meaning of term _metric minimizing_ in the present paper differs
from its meaning in [20] — the old paper used a weaker definition and the
theorem requires an additional assumption.

In Section 2, we introduced three ways to pull back the metric along a map
$f\colon X\to Y$ from a topological space $X$ to a metric space $Y$: the
induced length metric $\langle x-y\rangle_{f}$, the induced intrinsic metric
$\langle|x-y|\rangle_{f}$ and the induced connecting metric $|x-y|_{f}$. From
the definitions we have that

$\langle x-y\rangle_{f}\geqslant\langle|x-y|\rangle_{f}\geqslant|x-y|_{f}$

for any $x,y\in X$. The second inequality is strict for generic maps, an
example of a map $f$ with strict first inequality is given in [21, 4.2]. (If
$f$ is an embedding then equality holds [11, 4.5].)

$\langle|X|\rangle_{f}$$|X|_{f}$$\langle
X\rangle_{f}$$Y$$X$$\hat{\tau}_{f}$$\bar{\tau}_{f}$$\hat{\bar{\pi}}_{f}$$\hat{\bar{f}}$$f$$\hat{\pi}_{f}$$\hat{f}$$\bar{\pi}_{f}$$\bar{f}$

The diagram is an extension of the diagram on page 2 that includes
$\langle|X|\rangle_{f}$. From above, both maps $\hat{\tau}_{f}$ and
$\bar{\tau}_{f}$ are short and $\tau_{f}=\bar{\tau}_{f}\circ\hat{\tau}_{f}$.
Note that $\tau_{f}$ might be not injective while $\bar{\tau}_{f}$ is always
injective.

Our first formulation uses $\langle|{*}-{*}|\rangle$-metric instead of
$\langle{*}-{*}\rangle$ which was used in the old formulation.

9.1. Old theorem for $\bm{\langle|{*}-{*}|\rangle}$. Let $Y$ be a $\mathop{\rm
CAT}\nolimits(0)$ space and $s\colon\mathbb{D}\to Y$ a continuous map that
satisfies the following property: if a continuous map
$s^{\prime}\colon\mathbb{D}\to Y$ agrees with $s$ on $\partial\mathbb{D}$ and

$\langle|x-y|\rangle_{s^{\prime}}\leqslant\langle|x-y|\rangle_{s}$

for any $x,y\in\mathbb{D}$ then the equality holds for all pairs of $x$ and
$y$. Assume that the space $\langle|\mathbb{D}|\rangle_{s}$ is compact. Then
$\langle|\mathbb{D}|\rangle_{s}$ is $\mathop{\rm CAT}\nolimits(0)$.

Proof. Note that $s$ has no bubbles; it can be proved the same way as Lemma 3.
By Lemma 3, $|\mathbb{D}|_{s}$ is a disc retract.

Note that the natural map
$\bar{\tau}_{s}\colon\langle|\mathbb{D}|\rangle_{s}\to|\mathbb{D}|_{s}$ is
injective and continuous. Since $\langle|\mathbb{D}|\rangle_{s}$ is compact
and $|\mathbb{D}|_{s}$ is Hausdorff, the map $\bar{\tau}_{s}$ is a
homeomorphism. Since $\bar{\pi}_{s}$ is continuous, so is
$\hat{\bar{\pi}}_{s}$.

In particular $\langle|\mathbb{D}|\rangle_{s}$ is a disc retract as well as
$|\mathbb{D}|_{s}$. Therefore the mapping cylinder of the boundary curve in
$|\mathbb{D}|_{s}$ is homeomorphic to $\mathbb{D}$. Let us identify
$\mathbb{D}$ with the mapping cylinder of the boundary curve in
$|\mathbb{D}|_{s}$. Denote by $h\colon\mathbb{D}\to|\mathbb{D}|_{s}$ the
natural projection; it maps the cylinder to the boundary curve and does not
move the points in $|\mathbb{D}|_{s}$. (If $|\mathbb{D}|_{s}$ is a disc, we
can assume instead that $h$ is a homeomorphism.) Note that
$\langle\mathbb{D}\rangle_{\bar{s}\circ h}$ is isometric to
$\langle|\mathbb{D}|\rangle_{s}$.

If $s$ is not metric minimizing, then there is an other map
$h^{\prime}\colon\mathbb{D}\to Y$ such that $\bar{s}\circ h\succcurlyeq
h^{\prime}$ with nonisometric majorization
$\mu\colon\langle|\mathbb{D}|\rangle_{\bar{s}\circ
h}=\langle|\mathbb{D}|\rangle_{s}\to\langle\mathbb{D}\rangle_{h^{\prime}}$.
For the composition $s^{\prime}=\hat{h}\circ\mu\circ\hat{\bar{\pi}}_{s}$, we
have that

$\langle x-y\rangle_{s^{\prime}}\leqslant\langle|x-y|\rangle_{s}$

and therefore

$\langle|x-y|\rangle_{s^{\prime}}\leqslant\langle|x-y|\rangle_{s}$ ➋

for any $x,y\in\mathbb{D}$.

By the assumption, equality holds in ➋ ‣ 9 for any $x$ and $y$. Since
$\langle|\mathbb{D}|\rangle_{s}$ is compact, applying ultralimit+projection
construction, we can assume that $s^{\prime}$ is metric minimizing. By the
main theorem $\langle|\mathbb{D}|\rangle_{s^{\prime}}$ is $\mathop{\rm
CAT}\nolimits(0)$ disc retract.

Note that we can assume that
$\mu\colon\langle|\mathbb{D}|\rangle_{s}\to\langle\mathbb{D}\rangle_{h^{\prime}}$
is saddle; otherwise it can be shorten which would lead to a strict inequality
in ➋ ‣ 9 for some $x$ and $y$. From the main theorem in [22] it follows that
$\mu$ is monotonic (we need to apply the theorem for each disc provided by
Lemma 3 in the disc retract $\langle\mathbb{D}\rangle_{h^{\prime}}$). Since
equality holds in ➋ ‣ 9, $\mu$ has to be an isometry — a contradiction. ∎

In the second formulation use the metric $\langle{*}-{*}\rangle$ as in the
original formulation.

9.2. Old theorem for $\bm{\langle{*}-{*}\rangle}$. Let $Y$ be a $\mathop{\rm
CAT}\nolimits(0)$ space and $s\colon\mathbb{D}\to Y$ a continuous map that
satisfies the following property: if a continuous map
$s^{\prime}\colon\mathbb{D}\to Y$ agrees with $s$ on $\partial\mathbb{D}$ and

$\langle x-y\rangle_{s^{\prime}}\leqslant\langle x-y\rangle_{s}$

for any $x,y\in\mathbb{D}$, then the equality holds for all pairs of $x$ and
$y$. Assume that the function $(x,y)\mapsto\langle x-y\rangle_{s}$ is
continuous. Then $\langle\mathbb{D}\rangle_{s}$ is $\mathop{\rm
CAT}\nolimits(0)$.

Note that continuity of the function $(x,y)\mapsto\langle x-y\rangle_{s}$
implies that $\langle\mathbb{D}\rangle_{s}$ is compact. Therefore the former
condition is stronger than the latter. The sketch of proof given in [20]
implicitly used that the metric $(x,y)\mapsto\nobreak\langle x-y\rangle_{s}$
is continuous. (We do not know if compactness of
$\langle\mathbb{D}\rangle_{s}$ alone is sufficient.)

Theorem 9 follows from the following proposition and Theorem 9.

9.3. Proposition. Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ is a
continuous map without bubbles. Assume that the function $(x,y)\mapsto\langle
x-y\rangle_{s}$ is continuous. Then

$\langle x-y\rangle_{s}=\langle|x-y|\rangle_{s}$

for any $x,y\in\mathbb{D}$.

Before going into the proof, let us give an example showing that the
proposition is not trivial.

Consider a pseudo-arc $P\subset\mathbb{D}$ and let $s$ be the quotient map
$\mathbb{D}\to\mathbb{D}/P$. Evidently

$\langle|x-y|\rangle_{s}=0$

for any $x,y\in P$. However since $P$ contains no curves it is not at all
evident that

$\langle x-y\rangle_{s}=0$

for any $x,y\in P$.

We present an argument of Taras Banakh [4]; it works for two-dimensional disc
and we do not know a generalization of the proposition to higher dimensions.

9.4. Lemma. Let $Y$ be a metric space and $s\colon\mathbb{D}\to Y$ a
continuous map without bubbles. Assume that the function $(x,y)\mapsto\langle
x-y\rangle_{s}$ is continuous. Then there is a finite collection of curves in
$\mathbb{D}$ with finite total $\langle{*}-{*}\rangle$-length that divide
$\mathbb{D}$ into subsets with arbitrary small
$\langle{*}-{*}\rangle$-diameter.

Proof. Note that given $\varepsilon>0$ there is $\delta>0$ such that a set of
diameter $\delta$ in $\langle\mathbb{D}\rangle_{s}$ can not separate a set of
diameter at least $\varepsilon$ from the boundary curve. If this is not the
case, then sets of arbitrary small diameter can separate a set of diameter at
least $\varepsilon$ from the boundary. Passing to a limit we get a one point
set that separates a set from the boundary curve; that is, $s$ has a bubble —
a contradiction.

Let us subdivide $\mathbb{D}$ into small pieces by curves, say by vertical and
horizontal lines. Since the function $(x,y)\mapsto\langle x-y\rangle_{s}$ is
continuous, we can assume that all pieces have small
$\langle{*}-{*}\rangle_{s}$-diameter; that is, given $\varepsilon>0$ we can
assume that $\langle x-y\rangle_{s}<\varepsilon$ for any two points $x$ and
$y$ in one piece.

It remains to modify the decomposition to make the boundary curves
$\langle{*}-{*}\rangle_{s}$-rectifiable.

Subdivide the curves into arcs with $\langle{*}-{*}\rangle_{s}$-diameter
smaller $\tfrac{\delta}{5}$ and exchange this piece by a curve of
$\langle{*}-{*}\rangle_{s}$-length smaller than $\tfrac{\delta}{5}$. The new
arc together with the old one form a set of
$\langle{*}-{*}\rangle_{s}$-diameter at most $\delta$.111Note that this arc
might travel far in the Euclidean metric on $\mathbb{D}$. Therefore we might
add to a piece a subset of diameter at most $\varepsilon$ and the total
$\langle{*}-{*}\rangle_{s}$-diameter of each piece remains below $3{\hskip
0.5pt\cdot\hskip 0.5pt}\varepsilon$. The curves might cut more pieces from
$\mathbb{D}$, but by the same argument each of these pieces will have
$\langle{*}-{*}\rangle_{s}$-diameter below $3{\hskip 0.5pt\cdot\hskip
0.5pt}\varepsilon$. ∎

Proof of 9. Fix points $x,y\in\mathbb{D}$. It is sufficient to construct a
path $\alpha$ from $x$ to $y$ in $\mathbb{D}$ such that the length of
$s\circ\alpha$ is arbitrary close to $\langle|x-y|\rangle_{s}$.

Since $(x,y)\mapsto\langle x-y\rangle_{s}$ is continuous,
$\langle\mathbb{D}\rangle_{s}$ and therefore $\langle|\mathbb{D}|\rangle_{s}$
are compact. In particular, there is a minimizing geodesic $\gamma$ from
$\hat{\bar{x}}=\hat{\bar{\pi}}(x)$ to $\hat{\bar{y}}=\hat{\bar{\pi}}(y)$ in
$\langle|\mathbb{D}|\rangle_{s}$. Denote by $\Gamma$ the inverse image of
$\gamma$ in $\mathbb{D}$; this is a connected compact set which does not have
to be path connected.

To construct the needed path $\alpha$, it is sufficient to prove the following
claim:

  * $\bigstar$

Given $\varepsilon>0$ there is a set $\Gamma^{\prime}\subset\Gamma$ and a
collection of paths $\alpha_{0},\dots,\alpha_{n}$ such that (1) the total
length of $s(\alpha_{i}\backslash\Gamma)$ is at most $\varepsilon$, (2) The
set $\Gamma^{\prime}$ is a union of finite collection of closed connected set
$\Gamma_{0},\dots,\Gamma_{n}$, (3) diameter of each $\Gamma_{i}$ is at most
$\varepsilon$, (4) $x\in\alpha_{0}$, $y\in\alpha_{n}$ and (5) the union of
$\Gamma^{\prime}\cup\alpha_{0}\cup\dots\cup\alpha_{n}$ is connected.

Indeed, once the claim $\bigstar$ is proved, one can apply it recursively for
a sequence of $\varepsilon_{n}$ that converge to zero very fast. Namely we can
apply the claim to each of the subsets $\Gamma_{n}$ and take as
$\Gamma^{\prime\prime}$ the union of all closed subsets provided by the claim.
This way we obtain a nested sequence of closed sets
$\Gamma\supset\Gamma^{\prime}\supset\Gamma^{\prime\prime}\supset\dots$ which
break into finite union of closed connected subsets of arbitrary small
diameter and a countable collection of arcs with total length at most
$\varepsilon_{1}+\varepsilon_{2}+\dots$ outside of $\Gamma$. Set

$\Phi=\Gamma\cap\Gamma^{\prime}\cap\Gamma^{\prime\prime}\cap\dots$

Note that there is a simple curve from $x$ to $y$ that runs in the constructed
arcs and $\Phi$. The part of the curve in $\Gamma$ contributes at most
$\langle|x-y|\rangle_{s}$ to its $\langle{*}-{*}\rangle_{s}$-length. Therefore
the total length of the curve can not exceed

$\langle|x-y|\rangle_{s}+\varepsilon_{1}+\varepsilon_{2}+\dots;$

hence the result will follow.

It remains to prove $\bigstar$.

Fix a subdivision $\Upsilon_{1},\dots,\Upsilon_{k}$ of $\mathbb{D}$ provided
by Lemma 9 for the given $\varepsilon$. Denote by $\Delta$ the union of all
the cutting curves.

By the regularity of $\langle{*}-{*}\rangle_{s}$-length, we may cover
$\Delta\cap\Gamma$ by a finite collection of arcs with total
$\langle{*}-{*}\rangle_{s}$-length arbitrary close to
$\langle{*}-{*}\rangle_{s}$-length of $\Delta\cap\Gamma$. Denote these arcs by
$\alpha_{0},\dots,\alpha_{n}$. Without loss of generality we may assume that
$x\in\alpha_{0}$ and $y\in\alpha_{n}$.

Consider a finite graph with the vertexes labeled by
$\alpha_{0},\dots,\alpha_{n}$; two vertexes $\alpha_{i}$ and $\alpha_{j}$ are
connected by an edge if the there is a connected set
$\Theta\subset\Gamma\cap\Upsilon_{k}$ for some $k$ such that $\Theta$
intersects $\alpha_{i}$ and $\alpha_{j}$. Note that the graph is connected
therefore we may choose a path from $\alpha_{0}$ to $\alpha_{n}$ in the graph.

The path corresponds to a sequence of arcs $\alpha_{i}$ and a sequence of
$\Theta$-sets. The $\Theta$-sets that correspond to the edges in the path can
be taken as $\Gamma_{i}$ in the claim. Hence the claim and therefore the
proposition follow. ∎

##  10 Saddle surfaces

In this section we will discuss the relation between metric minimizing discs
and saddle discs.

Recall that a map $s\colon\mathbb{D}\to\mathbb{R}^{m}$ is called _saddle_ if
for any hyperplane $\Pi\subset\mathbb{R}^{m}$ each of the connected components
of $\mathbb{D}\backslash s^{-1}\Pi$ meets the boundary.

If $s$ is a smooth embedding in $\mathbb{R}^{3}$, then it is saddle if and
only if the obtained surface has nonpositive Gauss curvature. An old
conjecture of Samuel Shefel states that any saddle disc in a $\mathbb{R}^{3}$
is $\mathop{\rm CAT}\nolimits(0)$ with respect to its length metric, see [24].

It is evident that any metric minimizing disc $s$ in a Euclidean space is
saddle.

Three-dimensional case. In general a saddle disc may not be globally metric
minimizing. An example is shown in the picture. It is a saddle polyhedral disc
made from 10 triangles with a hexagon boundary. The boundary curve goes along
the Y-shape marked with bold lines; each segment of the Y-shape is traveled
twice back and forth.

The picture is rotationally symmetric by the angle $\tfrac{2}{3}{\hskip
0.5pt\cdot\hskip 0.5pt}\pi$. A shortening deformation can be obtained by
rotating the central triangle slightly counterclockwise and extending the map
on the remaing 9 triangles linerarly.

By smoothing this example one can produce a smooth saddle disc that is not
metric minimizing.

Proposition 1 states that there are no local examples of that type that are
smooth and _strictly saddle_ meaning that the principal curvatures at each
interior point have opposite signs. To prove this proposition we need to
introduce a certain energy functional.

Let $s\colon\mathbb{D}\to\mathbb{R}^{3}$ be a smooth map.

Fix an array of vector fields $\bm{v}=(v_{1},\dots,v_{k})$ on $\mathbb{D}$.
Assume that each integral curve of vector fields $v_{i}$ goes from boundary to
boundary yielding sweep outs of the whole disc $\mathbb{D}$.

Consider the energy functional

$E_{\bm{v}}s:=\sum_{i}\int\limits_{\mathbb{D}}|v_{i}s|^{2},$

where $vs=ds(v)$ denotes the derivative of $s$ in the direction of the field
$v$. Set

$\Delta_{\bm{v}}s=\sum_{i}v_{i}(v_{i}s).$

It is convenient to think of the operator $s\mapsto\Delta_{\bm{v}}s$ as an
analog of the Laplacian.

Note that

  1. (i)

$E_{\bm{v}}$ is well defined for any Lipschitz map $s$.

  2. (ii)

$E_{\bm{v}}$ is convex; that is,

$E_{\bm{v}}s_{t}\leqslant(1-t){\hskip 0.5pt\cdot\hskip
0.5pt}E_{\bm{v}}s_{0}+t{\hskip 0.5pt\cdot\hskip 0.5pt}E_{\bm{v}}s_{1},$

where $s_{t}=(1-t){\hskip 0.5pt\cdot\hskip 0.5pt}s_{0}+t{\hskip
0.5pt\cdot\hskip 0.5pt}s_{1}$ and $0\leqslant t\leqslant 1$. Moreover the
equality holds for any $t$ if and only if for any $i$ we have
$v_{i}s_{0}=v_{i}s_{1}$ almost everywhere.

  3. (iii)

If $s_{0}$ is a smooth $E_{\bm{v}}$-minimizing map in the class of Lipschitz
maps with given boundary data, then $s_{0}$ is metric minimizing.
Indeed, if $s_{1}\preccurlyeq s_{0}$, then $s_{1}$ has to be Lipschitz. It
follows that $E_{\bm{v}}s_{1}\leqslant E_{\bm{v}}s_{0}$ and from convexity
$E_{\bm{v}}s_{t}\leqslant E_{\bm{v}}s_{0}$ if $0\leqslant t\leqslant 1$. Since
$s_{0}$ is $E_{\bm{v}}$-minimizing, $E_{\bm{v}}s_{t}=E_{\bm{v}}s_{0}$ for any
$t$. Hence $v_{i}s_{0}=v_{i}s_{1}$ almost everywhere. Since $\mathbb{D}$ is
sweeped out by arcs of integral curves of $v_{i}$, the latter implies
$s_{0}=s_{1}$.

  4. (iv)

A smooth map $s\colon\mathbb{D}\to\mathbb{R}^{3}$ is a $E_{\bm{v}}$-minimizing
map among the class of Lipschitz maps with given boundary if and only if

$\Delta_{\bm{v}}s=0.$

The discussion above reduces Proposition 1 to the following.

10.1. Claim. Assume $s\colon\mathbb{D}\to\mathbb{R}^{3}$ is a smooth strictly
saddle surface. Then for any interior point $p\in\mathbb{D}$ there is an array
of 4 vector fields $\bm{v}=(v_{1},v_{2},v_{3},v_{4})$ such that the equation

$\Delta_{\bm{v}}s=0$ ➌

holds in an open neighborhood of $p$.

Proof. Denote by $\kappa_{1},\kappa_{2}$ the principal curvatures, and by
$e_{1},e_{2}$ the corresponding unit principal vectors. Further, denote by
$a_{1},a_{2}$ a pair of asymptotic vectors; that is, the normal curvatures in
these directions vanish. We can assume that $a_{1},a_{2}$ form coordinate
vector fields in a neighborhood of $p$.

Set $v_{1}=\tfrac{1}{\sqrt{|\kappa_{1}|}}{\hskip 0.5pt\cdot\hskip 0.5pt}e_{1}$
and $v_{2}=\tfrac{1}{\sqrt{|\kappa_{2}|}}{\hskip 0.5pt\cdot\hskip
0.5pt}e_{2}$. It remains to show that one can choose smooth functions
$\lambda_{1}$ and $\lambda_{2}$ so that ➌ ‣ 10 holds in a neighborhood of $p$
for $v_{3}=\lambda_{1}{\hskip 0.5pt\cdot\hskip 0.5pt}a_{1}$ and
$v_{4}=\lambda_{1}{\hskip 0.5pt\cdot\hskip 0.5pt}a_{1}$.

Note that the sum $v_{1}(v_{1}s)+v_{2}(v_{2}s)$ has vanishing normal part.
That is

$v_{1}(v_{1}s)+v_{2}(v_{2}s)$

is a tangent vector to the surface.

Since the $a_{i}$ are asymptotic, the vectors $a_{1}(a_{1}s)$ and
$a_{2}(a_{2}s)$ have vanishing normal part. Therefore, for any choice of
$\lambda_{i}$, the following two vectors are also tangent

$\displaystyle v_{3}(v_{3}s)$ $\displaystyle=\lambda_{1}^{2}{\hskip
0.5pt\cdot\hskip 0.5pt}a_{1}(a_{1}s)+\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip
0.5pt}a_{1}\lambda_{1}^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}a_{1}s$
$\displaystyle v_{4}(v_{4}s)$ $\displaystyle=\lambda_{2}^{2}{\hskip
0.5pt\cdot\hskip 0.5pt}a_{2}(a_{2}s)+\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip
0.5pt}a_{2}\lambda_{2}^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}a_{2}s.$

Set $w=(\lambda_{1}^{2},\lambda_{2}^{2})$. Note that the system ➌ ‣ 10 can be
rewritten as

$\left(\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\right)w_{x}+\left(\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right)w_{y}=h(x,y,w),$

where $h\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ is a smooth function.

Change coordinates by setting $x=t+z$ and $y=t-z$. Then the system takes the
form

$w_{t}+\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)w_{z}=h(t+z,t-z,w),$

which is a semilinear hyperbolic system. According to [5, Theorem 3.6], it can
be solved locally for smooth initial data at $t=0$.

It remains to choose $v_{3}$ and $v_{4}$ for solution so that
$\lambda_{1},\lambda_{2}>0$ in a small neighborhood of $p$. ∎

Four-dimensional case. Except for constructing an energy as we did above, we
do not see any way to show that a given smooth surface is metric minimizing.
Locally, the appropriate energy functional can be described by three functions
defined on the disc. These three functions are subject to certain differential
equations. Straightforward computations show that on generic smooth saddle
surfaces in $\mathbb{R}^{4}$ there is no solution even locally.

By that reason we expect that generic smooth saddle surfaces in
$\mathbb{R}^{4}$ are not locally metric minimizing. That is, arbitrary small
neighborhoods of any point admit deformations that shrink the length metric
and keep the boundary fixed. On the other hand we do not have an example of a
saddle surface for which this condition would hold at a single point.

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Anton Petrunin,

Department of Mathematics, PSU, University Park, PA 16802, USA

Email<EMAIL_ADDRESS>
Stephan Stadler,

Mathematisches Institut der Universität München, Theresienstr. 39, D-80333
München, Germany

Email<EMAIL_ADDRESS>