Curvature bounds on length-minimizing discs

Abstract

We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.

1 Introduction

We extend the results of [24, 27] to the case of curvature bounds different from 0 and provide full proofs of some consequences of the main theorems, which are mentioned in [27] in the case of $\textsc {CAT}(0)$ spaces.

We need some notation, in order to state the main result. Let $f:{\mathbb {D}} \rightarrow Y$ be a continuous map from the closed unit disc ${\mathbb {D}}$ into a $\textsc {CAT}(\kappa )$ space Y. For $x,z\in {\mathbb {D}}$ we define the length pseudodistance $\langle x-z\rangle _f \in [0, \infty ]$ induced by f as

$$\begin{aligned} \langle x -z \rangle _f:= \inf _{\gamma } \ell _Y (f\circ \gamma ) \in [0,\infty ]\;, \end{aligned}$$

(1)

where $\ell _Y(\eta )$ denotes the length of a curve $\eta$ in the metric space Y, and the infimum in (1) is taken over all curves $\gamma$ in ${\mathbb {D}}$ connecting x and z.

We say that f is length-connected if the value $\langle x -z \rangle _f$ is finite, for all $x,z\in {\mathbb {D}}$. In this case, the length pseudodistance defines a metric space by identifying points with length pseudodistance 0. This space is denoted by $\langle {\mathbb {D}} \rangle _f$ and is called the length metric space induced by f.

We say that f is length-continuous if the canonical projection $\hat{\pi } _f:{\mathbb {D}}\rightarrow \langle {\mathbb {D}} \rangle _f$ is continuous. Length-continuity always holds if f is a composition of a homeomorphism $\Phi : {\mathbb {D}}\rightarrow {\mathbb {D}}$ and a Lipschitz map $g:{\mathbb {D}}\rightarrow Y$.

A continuous map $f:{\mathbb {D}} \rightarrow Y$ is length-minimizing if for all continuous $g:{\mathbb {D}} \rightarrow Y$ with $g|_{{\mathbb {S}}^1} =f|_{{\mathbb {S}}^1}$ and

$$\begin{aligned} \ell _Y(g\circ \gamma ) \le \ell _Y (f\circ \gamma )\,, \end{aligned}$$

(2)

for all curves $\gamma$ in ${\mathbb {D}}$, equality must hold in (2), for all curves $\gamma$.

Now we can state our main result:

Theorem 1.1

Let $f:{\mathbb {D}}\rightarrow Y$ be a length-continuous and length-minimizing map from the disc ${\mathbb {D}}$ to a $\textsc {CAT}(\kappa )$ space Y. Then ${\mathbb {D}}$ with the length metric induced by f is a $\textsc {CAT}(\kappa )$ space.

We use this result in order to find upper curvature bounds on discs satisfying more common minimality assumptions. For the first consequence, we assume some familiarity with the notion of harmonic maps due to Korevaar–Schoen [11].

Corollary 1.2

Let Y be a CAT$(\kappa )$ space and let the continuous map $f:{\mathbb {D}}\rightarrow Y$ be harmonic. If the boundary curve $f:{\mathbb {S}}^1 \rightarrow X$ has finite length then ${\mathbb {D}}$ with the length metric induced by f is a CAT$(\kappa )$ space.

If f is conformal, the conclusion of Corollary 1.2 appears in [17] and a closely related statement appears in [19]. For non-conformal harmonic discs it seems barely possible to prove Corollary 1.2 by purely analytic means.

Another application concerns ruled discs:

Corollary 1.3

Let $\eta _0, \eta _1:[0,1]\rightarrow Y$ be rectifiable curves in a CAT$(\kappa )$ space Y. If $\kappa >0$, we assume that the distance between $\eta _0(a)$ and $\eta _1(a)$ is less than $\frac{\pi }{\sqrt{\kappa }}$, for all $a\in [0,1]$. For any $a>0$, consider the geodesic $\gamma _a:[0,1] \rightarrow Y$ between $\eta _0(a)$ and $\eta _1(a)$ parametrized proportionally to arclength. Then $[0,1]\times [0,1]$ with the length metric induced by the map $f(a,t):=\gamma _a(t)$ is CAT$(\kappa )$.

The statement about the inheritance of upper curvature bounds by ruled discs appeared in somewhat different generality with a sketchy proof in [2], the paper that gave birth to the theory of CAT$(\kappa )$ spaces. Missing details in Alexandrovs proof were recently provided by Nagano–Shioya–Yamaguchi in [23].

It seems possible to extend our proof to the case of non-rectifiably boundary curves and to dispense of the assumption on the parametrization of the geodesics. However, it would require rather technical considerations. The generality we have chosen is sufficient for most applications and follows directly from Theorem 1.1.

We finish the introduction with several comments.

• Our notion of length-minimality corresponds to the notion called metric-minimality in [24]. In [27] a new, stronger and less natural notion of metric-minimality was introduced. With this new stronger notion, a version of our main theorem for CAT(0) spaces was proved only under the assumption of length-connectedness of f, instead of the stronger length-continuity. We have only been able to verify the validity of this stronger metric-minimality assumption in cases covered by our Theorem 1.1. In order to find a clearer and better comprehensible way to Corollaries 1.2, 1.3, we have decided to work with the original, more natural notion of metric-minimality used in [24]. In order to distinguish it from metric-minimality used in [27], we have given it a different name.

• In the last section we formulate several questions concerning generalizations of our main results.

• The length metrics arising in Theorem 1.1 and, therefore, in Corollaries 1.2, 1.3 are homeomorphic to deformation retracts of ${\mathbb {D}}$.

• A general existence result in Sect. 5 shows that length-minimizing discs are abundant, beyond harmonic and ruled discs.

• The main argument closely follows [27]. A central technical step in the proof is the reduction of the theorem to the case, where the map f has totally disconnected fibers. In order to achieve this reduction, another metric induced by f on the disc is investigated, the so-called connecting pseudometric. It has better topological and metric properties than the more natural induced length metric. Roughly speaking, while the induced length metric collapses curves sent by f to a point, the connecting pseudometric also collapses pseudocurves sent by f to a point. In general, the precise relation between the two metrics seems to be rather complicated. However, as verified in [27], the length-minimality together with the length-connectedness, imply that the induced length metric is just the intrinsic metric induced by the connecting pseudometric.

• The heart of the proof is an approximation of the induced length metric by polyhedral metrics, whose restriction to the 1-skeleton is length-minimizing. This idea presented in [24] works in our setting with minor modifications. The case of $\kappa >0$ requires some additional considerations, due to the absence of the theorem of Cartan–Hadamard.

2 Preliminaries

2.1 Topology

A curve in a topological space X is a continuous map $\gamma : I\rightarrow X$, where $I\subset {\mathbb {R}}$ is an interval. A Jordan curve in X is a subset homeomorphic to ${\mathbb {S}}^1$.

A topological space X is a Peano continuum if X is metrizable, compact, connected and locally connected. In this case X is arcwise connected.

A disc retract is a compact space homeomorphic to a subset $X'$ of the closed unit disc ${\mathbb {D}}$ such that $X'$ is a homotopy retract of ${\mathbb {D}}$. A Peano continuum is a disc retract if and only if it is homeomorphic to a non-separating subset of the plane, [21, p. 27]. We will only need the following statement about disc retracts, a variant of a classical theorem of Moore, [20, 27, Proposition 3.3]:

If $f:{\mathbb {D}}\rightarrow X$ is a surjective, continuous map, such that all fibers are connected and non-separating subsets of ${\mathbb {R}}^2$, then X is a disc retract.

Remark 2.1

The following properties will not be used below. The reader might find them helpful, since disc retracts are central objects of this paper.

A space X is a disc retract if and only if there exists a closed curve $\gamma :{\mathbb {S}}^1\rightarrow X$, such that the mapping cylinder $([0,1]\times {\mathbb {S}}^1) \cup _{\gamma } X$ is homeomorphic to ${\mathbb {D}}$, [8, 27]. The image of $\gamma$ is the boundary of X, defined as the set of points at which X is not a 2-manifold.

Any maximal subset of a disc retract X which is not a point and does not contain cut points is a closed disc [21, p.27], [27]. The number of these cyclic components of X is at most countable.

By approximation, it follows that any disc retract admits a CAT$(-1)$ metric. The space of (isometry classes of) disc retracts with a geodesic metric, uniformly bounded ${\mathcal {H}}^2$-measure and ${\mathcal {H}}^1$-measure of the boundary and a quadratic isoperimetric inequality is compact in the Gromov–Hausdorff topology, [8]. In particular, the space of isometry classes of CAT$(\kappa )$ disc retracts with boundary of ${\mathcal {H}}^1$-measure at most $r< \frac{2\pi }{\sqrt{\kappa }}$ is compact in the Gromov–Hausdorff topology.

2.2 Metric geometry

We stick to the notations and conventions used in [27], and refer to [4] and [26] for introductions to metric geometry.

We denote the distance in a metric space Y by $|*-*|_Y=\langle *-*\rangle =\langle *-*\rangle _Y$.

The length of a curve $\gamma$ in Y will be denoted by $\ell _Y(\gamma )=\ell (\gamma )$. A geodesic is a curve $\gamma$ connecting points $y_1,y_2$ in Y, such that the length of $\gamma$ equals $\langle y_1-y_2 \rangle _Y$. We make no a priori assumption on the parametrization of a geodesic.

The space Y is a length space if the distance between any pair of its points equals the infimum of lengths of curves connecting these points. The space is a geodesic space if any pair of points is connected by a geodesic.

A pseudometric space is a metric space in which the distance can also assume values 0 and $\infty$. If the value $\infty$ is not assumed by a pseudometric on Y, we identify subsets of points of Y with pseudo-distance equal to 0 and obtain the induced metric space [26, Sect. 1.C].

We assume some familiarity with the Gromov–Hausdorff convergence and, for readers interested in the non-locally compact targets Y in our main results, with ultralimits. We refer to [26, Sects. 5,6] and [5, Sect. I.5].

2.3 CAT$(\kappa )$ spaces

We assume familiarity with properties of CAT$(\kappa )$ spaces, the reader is refered to [1, 4, 5] and [3]. We will stick to the convention that CAT$(\kappa )$ spaces are complete and geodesic.

By $M_{\kappa }$ we denote the model surface of constant curvature $\kappa$ and by $R_{\kappa }$ its diameter. Thus, $R_{\kappa }=\frac{\pi }{\sqrt{\kappa }}$ if $\kappa >0$ and $R_{\kappa }=\infty$ if $\kappa \le 0$.

3 Metrics induced by maps

We recall some notions and notations introduced explicitly or implicitly in [27, Sect. 2]. All statements in this section are proved there.

3.1 Length metric induced by a map

Let X be a topological space and Y a metric space. A continuous map $f:X\rightarrow Y$ induces a pseudometric on X by

$$\begin{aligned} \langle x-z\rangle _f =\inf \{ \ell _Y(f\circ \gamma ): \gamma \;\; \text {is a curve in}\ X\ \text {joining}\ x\ \text {to}\ z\}. \end{aligned}$$

This pseudometric is called the length pseudometric on X induced by f.

As in the introduction, we call a continuous map $f:X\rightarrow Y$ length-connected if the length pseudometric induced by f is finite for all pairs of points. Thus, f is length-connected if any $x,z \in X$ are connected in X by a curve $\gamma$, whose image $f\circ \gamma$ has finite length in Y.

If $f:X\rightarrow Y$ is length-connected then the metric space arising from the pseudometric $\langle x-z\rangle _f$ by identifying points at pseudistance 0 will be called the length metric space induced by f. We denote it by $\langle X\rangle _f$.

The canonical surjective projection from X onto $\langle X\rangle _f$ will be denoted by

$$\begin{aligned} \hat{\pi }_f:X\rightarrow \langle X\rangle _f. \end{aligned}$$

We will say that a continuous map $f:X\rightarrow Y$ is length-continuous if f is length-connected and the induced projection $\hat{\pi }_f:X\rightarrow \langle X\rangle _f$ is continuous.

For a compact metric space X, a map $f:X \rightarrow Y$ is length-continuous if and only if for any $\varepsilon >0$ there exists $\delta >0$, such that any pair of points $x,z\in X$ with $\langle x -z \rangle _X<\delta$ is connected by some curve $\gamma$ in X with $\ell _Y(f\circ \gamma ) <\varepsilon$.

If X is a length metric space and $f:X\rightarrow Y$ is locally Lipschitz continuous then f is length-continuous.

For any length-connected $f:X\rightarrow Y$, there exists a unique 1-Lipschitz map

$$\begin{aligned} \hat{f}: \langle X\rangle _f \rightarrow Y, \end{aligned}$$

such that $\hat{f} \circ \hat{\pi }_f =f$. The following is stated between the lines in [27].

Lemma 3.1

Let $f:X\rightarrow Y$ be length-connected. Let $\gamma$ be a curve in X. The curve $f\circ \gamma$ has finite length in Y if and only if $\hat{\pi } _f \circ \gamma$ is a curve of finite length in $\langle X\rangle _f$. In this case

$$\begin{aligned} \ell _Y (f\circ \gamma )= \ell _{\langle X\rangle _f} (\hat{\pi } _f \circ \gamma ). \end{aligned}$$

Proof

If $\hat{\pi } _f \circ \gamma$ is a curve of finite length, then $\ell _Y (f\circ \gamma ) \le \ell _{\langle X\rangle _f} (\hat{\pi } _f \circ \gamma )$, since $\hat{f}$ is 1-Lipschitz.

Assume now that $f\circ \gamma$ is of finite length. Then, for any t, s in the interval I of definition of $\gamma$, we have

$$\begin{aligned} \langle \gamma (t)-\gamma (s)\rangle _f=\langle (\hat{\pi } _f \circ \gamma ) (t) -(\hat{\pi } _f \circ \gamma (s) \rangle _{\langle X\rangle _f} \le \ell _Y (f\circ \gamma |_{[t,s]}). \end{aligned}$$

Since the length of $f\circ \gamma$ on small intervals around a fixed point t goes to 0 with the length of the interval, $\hat{\pi } _f \circ \gamma$ is continuous. Applying the above inequality to arbitrary partitions of I, we deduce $\ell _{\langle X\rangle _f} (\hat{\pi } _f \circ \gamma ) \le \ell _Y (f\circ \gamma )$. $\square$

3.2 Connecting pseudometric

Another pseudometric on X associated with a continuous map $f:X\rightarrow Y$ is the connecting pseudometric $|*-*|_f$ defined as

$$\begin{aligned} | x-z|_f =\inf \{\text {diam}\; f(C): C\subset X \; \text {connected and}\; x,z\in C \}. \end{aligned}$$

Whenever the connecting pseudometric assumes only finite values, we consider the associated metric spaces and denote it by $\big | X\big |_f$.

In this case we have the canonical projection map, denoted by

$$\begin{aligned} \bar{\pi }_f: X\rightarrow \big | X\big |_f. \end{aligned}$$

Moreover, there exist a uniquely defined 1-Lipschitz map

$$\begin{aligned} \bar{f} : \big |X\big |_f \rightarrow Y, \end{aligned}$$

such that $\bar{f} \circ {\bar{\pi }}_f =f.$

If $f:X\rightarrow Y$ is length-connected then the connecting pseudometric assumes only finite values and there exists a uniquely defined, surjective 1-Lipschitz map

$$\begin{aligned} \tau _f:\langle X\rangle _f \rightarrow \big | X\big |_f, \end{aligned}$$

such that

$$\begin{aligned} \bar{\pi } _f = \tau _f \circ \hat{\pi } _f. \end{aligned}$$

3.3 Basic properties

Recall that a map $f:X\rightarrow Y$ between topological spaces is called monotone (respectively, light) if any fiber of f is connected (respectively, totally disconnected). We recall from [27, Sect. 2]:

Lemma 3.2

Let X be a Peano continuum and Y a metric space. Let $f:X\rightarrow Y$ be continuous. Then

• (a) The map $\bar{\pi }_f:X\rightarrow \big | X\big |_f$ is continuous. Hence, $\big | X\big |_f$ is a Peano continuum.

• (b) $\ell _{|X|_f} (\gamma )=\ell _Y(\bar{f}\circ \gamma )$, for every curve $\gamma$ in $\big | X \big |_f$.

• (c) The map $\bar{\pi }_f:X\rightarrow \big | X\big |_f$ is monotone.

• (d) The map $\bar{f}:\big |X\big |_f\rightarrow Y$ is light.

Lemma 3.2(b) implies that $\big | X\big |_f$ with the induced length metric is isometric to $\big | X\big |_f$ with the length metric induced by $\bar{f}$.

From Lemmas 3.1 and 3.2 we obtain:

Lemma 3.3

Let X be a Peano continuum, Y a metric space and $f:X\rightarrow Y$ be length-connected. Then, for all curves $\gamma$ in X,

$$\begin{aligned} \ell _Y(f\circ \gamma )=\ell _{|X|_f} (\bar{\pi } _f \circ \gamma )=\ell _{|X|_f} (\tau _f \circ \hat{\pi } _f \circ \gamma ). \end{aligned}$$

If, in addition, $\ell _Y(f\circ \gamma )$ is finite or if f is length-continuous then $\hat{\pi } _f \circ \gamma$ is continuous and

$$\begin{aligned} \ell _Y(f\circ \gamma )= \ell _{\langle X\rangle _f} (\hat{\pi } _f \circ \gamma ). \end{aligned}$$

For a Peano continuum X and a length-continuous map $f:X\rightarrow Y$, the natural 1-Lipschitz map $\tau _f:\langle X\rangle _f \rightarrow \big | X\big |_f$ may be non-injective [25, Example 4.2]. Since every connected subset of a finite graph is arcwise connected, this pathology cannot occur if X is a finite, connected graph. The following result proven in [27, Lemma 3.3, Proposition 9.3] is much less trivial:

Lemma 3.4

Let $f:{\mathbb {D}}\rightarrow Y$ be length-connected and let any fiber $f^{-1} (y)$ be a non-separating subset in ${\mathbb {R}}^2$. Then $|{\mathbb {D}}|_f$ is a disc retract. If, in addition, f is length-continuous then the map $\tau _f:\langle {\mathbb {D}}\rangle _f \rightarrow | {\mathbb {D}}|_f$ is a homeomorphism, which preserves the length of all curves in $\langle {\mathbb {D}}\rangle _f$.

4 Length-minimizing maps

4.1 Definition and first properties

Let X be a topological space, Z a closed subset of X. Let $f:X\rightarrow Y$ be a continuous map into a metric space. For another map $g:X\rightarrow Y$ we will write $f\unrhd g$ (rel Z) if f and g coincide on Z and

$$\begin{aligned} \ell _Y(f\circ \gamma )\geqslant \ell _Y(g\circ \gamma ) \,, \; \; \text {for every curve} \; \; \gamma \;\; \text {in} \;\; X. \end{aligned}$$

(3)

We will call f length-minimizing relative to Z if for every map $f\unrhd g$ (rel Z) equality holds in (3), for every curve $\gamma$.

If X is arcwise connected and Z is empty then length-minimizing maps relative to Z are constant. From now on, we will always assume that Z is not empty.

Lemma 4.1

Let X be a topological space, Y be a metric space and $Z\subset X$ be closed. Let $f:X\rightarrow Y$ be length-connected. A continuous map $g:X\rightarrow Y$ satisfies $f\unrhd g$ (rel. Z) if and only if g is length-connected, coincides with f on Z and

$$\begin{aligned} \langle x_1 -x_2\rangle _f \ge \langle x_1-x_2 \rangle _g, \end{aligned}$$

for all $x_1,x_2\in X$.

Proof

Assume $f\unrhd g$ (rel. Z). By definition, f and g coincide on Z. For any $x_1,x_2\in X$, we find a curve $\gamma$ in X connecting $x_1$ and $x_2$, such that $f\circ \gamma$ has finite length arbitrarily close to $\langle x_1 -x_2\rangle _f$. Since $\ell _Y( f\circ \gamma ) \ge \ell _Y(g\circ \gamma )$ we deduce that $\langle x_1 -x_2\rangle _g$ is finite and not larger than $\langle x_1 -x_2\rangle _f$.

Assume on the other hand, that g satisfies the conditions in the statement of the Lemma. In order to prove $f\unrhd g$ (rel. Z), consider an arbitrary curve $\gamma$ in X. If $f\circ \gamma$ has infinite length, then $\ell _Y (f\circ \gamma ) \ge \ell _Y (g\circ \gamma )$.

If $f\circ \gamma$ has finite length, then $\ell _Y (f\circ \gamma )= \ell _{\langle X \rangle _f} (\hat{\pi } _f \circ \gamma )$ by Lemma 3.1. By assumption, the canonical map $\mu :\langle X \rangle _f \rightarrow \langle X \rangle _g$ is 1-Lipschitz and commutes with the projections $\hat{\pi } _f$ and $\hat{\pi } _g$. Thus, $\hat{\pi } _g \circ \gamma$ is continuous and has length at most $\ell _{\langle X \rangle _f} (\hat{\pi } _f \circ \gamma )$. Applying Lemma 3.1 twice, we deduce $\ell _Y (f\circ \gamma ) \ge \ell _Y (g\circ \gamma )$. $\square$

The property of being length-minimizing is inherited by restrictions:

Lemma 4.2

Let X be a topological space, let $Z, S \subset X$ be closed. Let Y be a metric space and $f: X\rightarrow Y$ be length-minimizing relative to Z. Then the restriction $f:S\rightarrow Y$ of f is length-minimizing relative to $Z_S:=\partial S \cup (Z\cap S)$.

Proof

Assume the contrary. Then there exists a map $h: S \rightarrow Y$ such that $f\big |_{S}\unrhd h$ (rel $Z_S$) together with a curve $\gamma _0: I_0\rightarrow S$ such that $\ell _Y(f\circ \gamma _0 )> \ell _Y(h\circ \gamma _0 )$.

Define $g: X\rightarrow Y$ by setting $g=h$ on S and $g=f$ on $X\setminus S$. The maps h and f agree on $\partial S$, hence g is continuous. By construction, $g=f$ on Z. Moreover,

$$\begin{aligned} \ell _Y(h\circ \gamma _0)=\ell _Y(g\circ \gamma _0 ) < \ell _Y (f\circ \gamma _0 ). \end{aligned}$$

Let now $\gamma :I\rightarrow X$ an arbitrary curve. Then $f\circ \gamma$ and $g\circ \gamma$ agree on the closed set $C= \gamma ^{-1} (\overline{X{\setminus } S})$. The complement $I{\setminus } C$ is a countable union of open intervals $I_j$. For any j, the length of the restriction of $g\circ \gamma$ to $I_j$ does not exceed the length of $f\circ \gamma \big |_{I_j}$, since $f\big |_{S}\unrhd h$. Computing the length via the 1-dimensional Hausdorff measure ${\mathcal {H}}^1$, see [4, Exercise 2.6.4], we deduce

$$\begin{aligned} \ell (g\circ \gamma ) \le \ell (f\circ \gamma ). \end{aligned}$$

This contradicts the length-minimality of f. $\square$

As a consequence we obtain the following non-bubbling property:

Corollary 4.3

Let X be a Peano space and let $f:X\rightarrow Y$ be length-minimizing relative to a closed subset $Z\subset X$. Then, for any $y\in Y$ and any connected component U of $X\setminus f^{-1} (y)$ the intersection $U\cap Z$ is not empty.

Proof

Assume $U\cap Z= \emptyset$ and let S be the closure of U in X. Since connected components are always closed in the ambient space, the boundary $\partial S$ of S in X, satisfies $\partial S \subset f^{-1} (y)$. Moreover, $S\cap Z = \partial S \cap Z$.

By Lemma 4.2, $f:S \rightarrow Y$ is length-minimizing relative to $\partial S$. The constant map $g:S \rightarrow Y$, which sends every point to y satisfies $f\unrhd g$. The length-minimality of f implies that the image of every curve in U has length 0. Therefore, f is constant on U, hence on S. Thus, U is contained in $f^{-1} (y)$, which is impossible. $\square$

Lemma 4.4

Let X be a topological space, $Z\subset X$ be closed. Let Y be a CAT$(\kappa )$ space. Let $K\subset Y$ be closed, convex and contained in a ball of radius $<\frac{R_{\kappa }}{2}$ in Y.

Let $f:X\rightarrow Y$ be a length-connected, length-minimizing map relative to Z. If f(Z) is contained in K then f(X) is contained in K.

Proof

Assume that there exists a point $x\in X$ with $f(x) \notin K$. We find a curve $\gamma _0$ in X between a point $z\in Z$ and x such that $f\circ \gamma _0$ has finite length.

Due to [12, Theorem 1.1], there exists a 1-Lipschitz retraction $\Psi : Y\rightarrow K$, which decreases the length of any rectifiable curve of positive length not completely contained in K. Hence, the composition $g:=\Psi \circ f$ satisfies $f\unrhd g$ (rel. Z) and $\ell _Y(g\circ \gamma _0) < \ell _Y(f\circ \gamma _0)$. This contradicts the length-minimality of f. $\square$

Lemma 4.5

Under the assumptions of Lemma 4.4, let $g:X\rightarrow Y$ satisfy $f\unrhd g$ (rel. Z). Then $f=g$.

Proof

By definition, g is length-minimizing relative to Z as well. By Lemma 4.4, the images of f and of g are contained in K.

Denote by $\Delta \subset K\times K$ the diagonal. We apply [12, Corollary 1.2] and obtain a 1-Lipschitz retraction $\Pi :K\times K \rightarrow \Delta$, such that $\Pi$ decreases the length of any curve of finite positive length not completely contained in $\Delta$.

We identify $\Delta$ with K, rescaled by $\sqrt{2}$ and consider the map $h:X\rightarrow Y$

$$\begin{aligned} h(x):= \Pi (f(x),g(x)). \end{aligned}$$

Then h coincides with f and with g on Z. For any curve $\gamma$ in X, we have

$$\begin{aligned} \ell _Y(h\circ \gamma ) \le \ell _Y(f\circ \gamma )=\ell _Y(g\circ \gamma ). \end{aligned}$$

Whenever $\ell _Y(f\circ \gamma )$ is finite and positive, and $f\circ \gamma$ does not coincide with $g\circ \gamma$, the above inequality is strict.

If f does not coincide with g on X, we find a curve $\gamma _0$ in X, such that $f\circ \gamma _0$ and $g\circ \gamma _0$ are different curves of finite length, since f is length-connected. We infer $f\unrhd h$ (rel. Z) and that f is not metric minimizing. Contradiction. $\square$

We will apply Lemmas 4.5, 4.4 only for balls $K= B_r(y)\subset Y$ with $r<\frac{R_{\kappa }}{2}$. However, these lemmas might be useful in other cases as well, cf. [28].

4.2 Length-minimizing graphs

The results in this subsection are contained explicitly or between the lines [27, Sect. 5].

By a finite graph we mean a connected, finite 1-dimensional CW-complex. Thus, we allow multiple edges and edges connecting a vertex with itself.

Proposition 4.6

Let Y be a geodesic metric space, $\Gamma$ a finite graph and A a set of its vertices. If $f:\Gamma \rightarrow Y$ is length-minimizing relative to A then the restriction of f to any edge is a geodesic.

Proof

By Lemma 4.2, the restriction $f_0:E\rightarrow Y$ of f to any edge $E\subset \Gamma$ is length-minimizing relative to its endpoints. We can identify E with the interval [0, a], such that a is the distance $a=\langle f(0)-f(a) \rangle _Y$ between the endpoints. Consider a geodesic $\gamma :[0,a]\rightarrow Y$ with the same endpoints as f, parametrized by arclength. Then $C:=\gamma ([0,a])$ is an isometric embedding of the interval [0, a]. Since intervals are injective metric spaces, cf. [26, Lecture 3], there exists a 1-Lipschitz retraction $\Phi :Y\rightarrow C$. Hence, for $g:=\Phi \circ f_0$ we have $f \unrhd g$ (rel. $\{0,a\}$).

Hence, also the map $g:[0,a]\rightarrow C$ is length-minimizing relative to $\{0,a\}$. Due to the non-bubbling result, Corollary 4.3, the map g provides a monotone parametrization of the simple arc C. Hence, the length of the curve g is exactly the length of the geodesic $\gamma$. Since $f_0$ is length-minimizing, the length of $f_0$ cannot be larger than the length of g. Hence, $f_0:E\rightarrow C$ is a geodesic. Since the edge E was arbitrary, this finishes the proof. $\square$

It follows that any length-minimizing graph $f:\Gamma \rightarrow Y$ as in Proposition 4.6 is length-continuous. Any connected subset of $\Gamma$ is arcwise connected. Therefore, the canonical map $\tau _f:\langle \Gamma \rangle _f \rightarrow |\Gamma |_f$ is a homeomorphism. Moreover, the space $\langle \Gamma \rangle _f$ is a finite union of geodesics intersecting only at their endpoints. Thus, $\langle \Gamma \rangle _f$ is finite topological graph with a geodesic metric. A vertex in $\langle \Gamma \rangle _f$ corresponds to connected subgraphs of $\Gamma$ sent by f to a single point. Any edge $\hat{E}$ of $\langle \Gamma \rangle _f$ is the image of an edge E in $\Gamma$ and the restricton $\hat{\pi } _f: E\rightarrow \hat{E}$ is monotone. The induced map $\hat{f}:\langle \Gamma \rangle _f \rightarrow Y$ sends any edge isometric onto a geodesic in Y. Finally, any curve in $\langle \Gamma \rangle _f$ lifts to a curve in $\Gamma$. By definition, the last statement implies:

Corollary 4.7

Let $f:\Gamma \rightarrow Y$ be length-minimizing relative to A, as in Proposition 4.6. Then $\hat{f}: \langle \Gamma \rangle _f \rightarrow Y$ is length-minimizing relative to $\hat{\pi } _f (A)$.

The following result is proved in detail in [27, Proposition 5.2] for $\kappa =0$. Since the proof only requires the first variational inequality for distances [1, Inequality 6.7] and Reshetnyak’s majorization theorem [1, Theorem 9.56] in spaces of directions, the proof applies literally in CAT$(\kappa )$ spaces:

Lemma 4.8

Let Y be $\textsc {CAT}(\kappa )$. Let $\Gamma$ be a finite geodesic graph and A be a subset of its vertices. Let $f: \Gamma \rightarrow Y$ be length-minimizing relative to A and assume that the restriction of f to any edge is an isometry.

Let $p\in \Gamma \setminus A$ be a vertex. Let $\gamma _1,..., \gamma _n$ be the images in Y of the edges in $\Gamma$ starting in p and enumerated in an arbitrary order. Then the sum of the n consecutive angles satisfies

$$\begin{aligned} \measuredangle _p(\gamma _1, \gamma _2)+\ldots +\measuredangle _p (\gamma _{n-1}, \gamma _n)+\measuredangle _p (\gamma _n,\gamma _1) \geqslant 2\pi . \end{aligned}$$

In particular, any vertex $p\in \Gamma \setminus A$ is contained in at least 2 edges of $\Gamma$.

4.3 Length-minimizing discs

A length-minimizing disc will denote a map $f:{\mathbb {D}} \rightarrow Y$ length-minimizing relative to ${\mathbb {S}}^1$.

Due to Corollary 4.3 and Lemma 3.4, for any length-connected and length-minimizing disc $f:{\mathbb {D}} \rightarrow Y$, the spaces $|{\mathbb {D}} |_f$ is a disc retract. If, in addition, f is length-continuous, then the map $\tau _f:\langle {\mathbb {D}} \rangle _f\rightarrow | {\mathbb {D}} |_f$ is a homeomorphism preserving the length of all curves.

Lemma 4.9

Let $f:{\mathbb {D}}\rightarrow Y$ be length-connected and length-minimizing disc in a $\textsc {CAT}(\kappa )$ space Y. Let G be a Jordan curve of length $l < 2 \cdot {R_{\kappa }}$ in the disc retract $|{\mathbb {D}}|_f$. Denote by J the closed disc bounded by G in $|{\mathbb {D}}|_f$. Then

• (1) The restriction $\bar{f}:J\rightarrow Y$ is a length-connected and length-minimizing disc.

• (2) The image $\bar{f}(J)$ is contained in a ball of radius $r <\frac{R_{\kappa }}{2}$ in Y.

• (3) If f is length-continuous then so is $\bar{f}:J\rightarrow Y$.

Proof

The map $\bar{f}$ is 1-Lipschitz, hence length of the closed curve $\bar{f} (G)$ is less than $2 \cdot {R_{\kappa }}$. Reshetnyak’s majorization theorem [1, Theorem 9.56] implies that $\bar{f}(G)$ is contained in a ball B of radius $r <\frac{R_{\kappa }}{2}$ in Y.

Denote by $\tilde{G}$ and $\tilde{J}$ the preimages of G and J in ${\mathbb {D}}$, respectively.

For any point $p\in J$ consider any preimage $\tilde{p}\in \tilde{J}$ of p. We find a curve $\gamma$ in ${\mathbb {D}}$, which connects $\tilde{p}$ with a point in $\tilde{G}$ and such that $f\circ \gamma$ has finite length. By cutting the curve, if needed, we may assume that $\gamma$ is contained in $\tilde{J}$. Then the projection $\bar{\gamma } =\bar{\pi } _f\circ \gamma$ is a curve connecting p to a point on G, such that $\bar{f}\circ \bar{\gamma }$ has finite length.

Since G has finite length and $\bar{f}$ preserves all lengths, $\langle p-q \rangle _{\bar{f}}$ is finite, for any $p\in J$ and any $q\in G$. By the triangle inequality, $\bar{f}$ is length-connected.

Assume now that $g:J\rightarrow Y$ satisfies $\bar{f}\unrhd g$ (rel. G). Set

$$\begin{aligned} \tilde{g}:=g\circ \bar{\pi } _f:\tilde{J}\rightarrow Y. \end{aligned}$$

Then $\tilde{g}$ coincides with f on $\tilde{G}$ and $\tilde{J}\setminus \tilde{G}$ does not intersect the boundary ${\mathbb {S}}^1$. Due to Lemma 4.2, the restriction $f:\tilde{J}\rightarrow Y$ is length-minimizing relative to $\tilde{G}$.

The assumption $\bar{f}\unrhd g$ (rel. G) implies $f|_{\tilde{J}} \unrhd \tilde{g}$. Due to Lemma 4.4, $f(\tilde{J})=\bar{f}(J)$ is contained in the ball B, proving (2).

Due to Lemma 4.5, f equals $\tilde{g}$ on $\tilde{J}$. Hence, $\bar{f}= g$. This proves (1).

Let, finally, f be length-continuous. By Lemma 3.4, $\langle X \rangle _f$ is a geodesic metric space and $\tau _f: \langle X \rangle _f \rightarrow |X |_f$ is a homeomorphism. Consider the disc $\hat{J}=\tau _f ^{-1}(J) \subset \langle X \rangle _f$. Its boundary $\hat{G}= \tau _f^{-1} (G)$ has finite length, therefore, the length metric of the subset $\hat{J}$ of $\langle X \rangle _f$ induces the subset topology of $\hat{J}$, cf. [18, Lemma 2.1]. The restriction $\hat{f}:\hat{J}\rightarrow Y$ is 1-Lipschitz, if $\hat{J}$ is equipped with its length metric. Hence, $\hat{f}:\hat{J}\rightarrow Y$, and therefore also $\bar{f}:J\rightarrow Y$, is length-continuous. $\square$

5 An existence result

The following result is essentially [27, Proposition 5.1].

Lemma 5.1

Let X be a Peano continuum, Z a closed subset of X and Y a CAT$(\kappa )$ space. Let the map $f:X\rightarrow Y$ be length-continuous. If f(Z) is a contained in a closed ball $B \subset Y$ of radius $\le \frac{R_{\kappa }}{2}$ then there exists a map $g:X\rightarrow Y$, length-minimizing relative to Z, with $f\unrhd g$ (rel. Z).

Proof

We consider the set ${\mathcal {F}}$ of all maps $h:X\rightarrow Y$ which coincide with f on Z and satisfy $f\unrhd h$ (rel. Z). We are looking for a minimal element in the set ${\mathcal {F}}$ with the partial ordering given by $\unrhd$ (rel. Z).

By [12, Theorem 1.1] there exists a 1-Lipschitz retraction $\Phi :Y\rightarrow B$. Hence, for any $h\in {\mathcal {F}}$, we have $\Phi \circ h \in {\mathcal {F}}$ and $h\unrhd \Phi \circ h$.

We replace ${\mathcal {F}}$ by its subset ${\mathcal {F}}_0$ of all maps $h\in {\mathcal {F}}$ whose image is contained in the ball B. The existence of the retraction $\Phi$ implies that an element g in ${\mathcal {F}}_0$ minimal with respect to the partial ordering $\unrhd$ (rel. Z) will be length-minimizing.

In order to find such a minimal element in ${\mathcal {F}}_0$ we apply Zorn’s lemma. It thus suffices to show that for any totally ordered subset ${\mathcal {T}}$ of ${\mathcal {F}}_0$ there exists an element $g\in {\mathcal {F}}_0$ with $h\unrhd g$ (rel. Z) for all $h\in {\mathcal {T}}$.

For any $h\in {\mathcal {T}}$, the identity $id:Y\rightarrow Y$ induces a 1-Lipschitz map $\mu :\langle Y \rangle _f \rightarrow \langle Y \rangle _h$. The map h then factorizes as $h= \hat{h} \circ \mu \circ \hat{\pi } _f$. Since the maps $\hat{h}$ and $\mu$ are 1-Lipschitz and $\hat{\pi } _f$ is continuous by assumption, the family ${\mathcal {T}}$ is equicontinuous.

Fix a dense sequence $(x_n)$ in the compact space X. For any pair (n, m), set

$$\begin{aligned} d_{m,n}:=\inf \{ \langle x_n-x_m \rangle _h \; \; | \; \; h \in {\mathcal {T}}\}. \end{aligned}$$

For any triple of natural numbers (n, m, k) we find some $h=h_{k,n,m}$ in ${\mathcal {T}}$ such that

$$\begin{aligned} d_{m,n}+ \frac{1}{k} > \langle x_n-x_m \rangle _h. \end{aligned}$$

We enumerate the set of all triples and use a diagonal sequence argument in order to find a sequence $h_1 \unrhd h_2 \unrhd h_3 \unrhd ...$ of elements in ${\mathcal {T}}$ such that for any $x_n,x_m$

$$\begin{aligned} \lim _{i\rightarrow \infty } \langle x_n-x_m \rangle _{h _i} =d_{m,n}. \end{aligned}$$

If Y is proper, hence B compact, we take a convergent subsequence of the equi-continuous sequence $h_i$ and obtain a uniform limit $g:=\lim _i h_i:X\rightarrow B.$

If Y is non-proper, we take the ultralimit

$$\begin{aligned} h_{\omega }=\lim _{\omega } h_i: X\rightarrow B^{\omega } \end{aligned}$$

from Y to the ultracompletion $B^{\omega }$ of B. Then we apply [12, Theorem 1.1] again and find a 1-Lipschitz retraction $\Psi : B^{\omega } \rightarrow B$. Set now

$$\begin{aligned} g:=\Psi \circ h_{\omega }. \end{aligned}$$

Since all $h_i$ agree with f on Z, so does g. Since length of curves is semi-continuous under Gromov–Hausdorff convergence and under ultraconvergence, we have

$$\begin{aligned} \lim \ell _Y(h_i \circ \gamma ) \ge \ell _{B^\omega } (h_{\omega } \circ \gamma ) \ge \ell _B(g\circ \gamma ) , \end{aligned}$$

for any curve $\gamma$ in X. In particular, g is contained in ${\mathcal {F}}_0$. Moreover, for any n, m, i

$$\begin{aligned} d_{m,n}\le \langle x_n-x_m \rangle _g \le \langle x_n - x_m \rangle _{h_i} . \end{aligned}$$

Thus, equality holds on the left. By the assumption on $h_i$, for any $h\in {\mathcal {T}}$,

$$\begin{aligned} \langle x_n -x_m \rangle _h \ge \langle x_n -x_m \rangle _g \end{aligned}$$

Since the sequence $x_m$ is dense in X and, for all $h\in {\mathcal {T}}$, the distance $\langle *-*\rangle _h$ is continuous on X, we deduce

$$\begin{aligned} \langle x -z \rangle _h \ge \langle x -z \rangle _g, \end{aligned}$$

for any pair of points $x,z\in X$ and any $h\in {\mathcal {T}}$.

Thus, $h\unrhd g$ (rel. Z), for all $h\in {\mathcal {T}}$. This finishes the proof. $\square$

The CAT$(\kappa$)-assumption and the assumption that the image is contained in a small ball is only used for the existence of a 1-Lipschitz retraction $\Pi :Y^{\omega } \rightarrow Y$. Thus, in the proper setting the proof provides the following result, which we state for the sake of completeness:

Proposition 5.2

Let X be a Peano space, $Z\subset X$ closed. Let $f:X\rightarrow Y$ be a length-continuous map into a proper metric space Y. Then there exists a length-continuous map $g:X\rightarrow Y$, which is length-minimizing relative to Z and satisfies $f\unrhd g$ (rel Z).

The first part of the proof of Proposition 4.6, shows that for any finite graph $\Gamma$ and any continuous map $f:\Gamma \rightarrow Y$ into a geodesic space Y there exists a map $g:\Gamma \rightarrow Y$ such that $f\unrhd g$ and such that the restriction of g to any edge is a geodesic in Y. Since any such g is length-continuous, we deduce from Proposition 5.1

Corollary 5.3

Let $\Gamma$ be a finite graph and $A\subset \Gamma$ a set of vertices. Let B be a ball of radius $\le \frac{R_{\kappa }}{2}$ in a CAT$(\kappa )$ space Y. For any continuous map $f:\Gamma \rightarrow B$, there exists a length-minimizing map $h:\Gamma \rightarrow B$ relative to A such that $f\unrhd h$.

6 Main result

The following proposition is the technical heart of all results in this paper. This is a slight generalization of [27, Key-Lemma 6.2]. The proof follows [27] filling some details and some additional arguments in positive curvature.

Proposition 6.1

Let $\Gamma$ be a finite graph, which is the 1-skeleton of a triangulation of the disc ${\mathbb {D}}$. Let ${\mathcal {V}}$ be the set of vertices of $\Gamma$ and $A:={\mathcal {V}}\cap {\mathbb {S}}^1$. Let B be a ball of radius $<\frac{R_{\kappa }}{2}$ in some $\textsc {CAT}(\kappa )$ space Y. Let $f:\Gamma \rightarrow B$ be continuous. Let $0<\varepsilon < \frac{R_{\kappa }}{2}$ be such that, for the vertices $x_1,x_2,x_3$ of any triangle T in the triangulation of ${\mathbb {D}}$, the distances satisfy $\langle f(x_i)- f(x_j) \rangle _Y \le \varepsilon$.

Then there exists some $\textsc {CAT}(\kappa )$ disc retract W, a 1-Lipschitz map $q:W\rightarrow Y$ and a map $p:{\mathbb {D}}\rightarrow W$ with the following properties

• (1) All fibers of p are contractible and $p({\mathcal {V}})$ is $\varepsilon$-dense in W.

• (2) $q\circ p|_{A} =f|_A$.

• (3) For every curve $\gamma$ in $\Gamma$, we have $\ell _Y (f\circ \gamma )\ge \ell _W (p\circ \gamma ) = \ell _Y (q\circ p \circ \gamma )$.

Proof

We may replace Y by B and assume $Y=B$. Due to Corollary 5.3, we find a map $h:\Gamma \rightarrow Y$ length-minimizing relative to A such that $f\unrhd h$ (rel. A).

Consider the induced length-metric space $\langle \Gamma \rangle _h$, which is a geodesic graph. Due to Corollary 4.7, the map $\hat{h}:\langle \Gamma \rangle _h \rightarrow B$ is length-minimizing relative to $\hat{\pi } _h (A)$. By Proposition 4.6, the restriction of $\hat{h}$ to any edge of $\langle \Gamma \rangle _h$ is an isometry and the projection $\hat{\pi } _h: \Gamma \rightarrow \langle \Gamma \rangle _h$ is monotone. Thus, the conclusions of Lemma 4.8 are valid for the map $\hat{h}:\langle \Gamma \rangle _h \rightarrow Y$.

For any triangle $T \subset {\mathbb {D}}$ bounded by 3 edges of $\Gamma$ with vertices $x_1,x_2,x_3$ we consider the unique, possibly degenerated triangle $T^{\kappa }$ in the surface $M^2_{\kappa }$ of constant curvature $\kappa$, such that the sides of $T^{\kappa }$ have distances equal to

$$\begin{aligned} \langle x_i-x_j \rangle _{{\langle \Gamma \rangle } _h} = \langle h(x_i)-h(x_j) \rangle _Y \le \varepsilon . \end{aligned}$$

We glue all these triangles $T^{\kappa }$ (possibly degenerated to a vertex or to an edge) together in the same way as the corresponding triangles T are glued in ${\mathbb {D}}$. We denote the obtained metric space by W.

The metric space W is glued from triangles of curvature $\kappa$ and, possibly, some edges, cf. [1, definition 12.1]. The gluing maps are simplicial. A priori, edges or vertices of a single triangle T may be identified in W.

We have a tautological length-preserving map $\iota : \langle \Gamma \rangle _h \rightarrow W$. Some caution is in order: two non-degenerate edges in a triangle whose third side degenerates to a point are sent by $\iota$ to the same curve in W, hence $\iota$ may be non-injective. However, all maps defined below respect the identifications on $\langle \Gamma \rangle _h$ given by $\iota$. Thus, we will identify $\langle \Gamma \rangle _h$ with $\iota (\langle \Gamma \rangle _h)$ below.

For any triangle $T \subset {\mathbb {D}}$, the restriction of $\hat{\pi } _h:\Gamma \rightarrow \langle \Gamma \rangle _h$ to $\partial T$ defines a map $f_T:\partial T\rightarrow \partial T^{\kappa }$. We claim that $f_T$ admits an extension to a map $\tilde{f}_T:T\rightarrow T^{\kappa }$, such that any fiber of $\tilde{f}_T$ is topologically a point, an interval or a disc.

If $T^{\kappa }$ degenerates to a point $w\in W$ we choose the constant map $\tilde{f}_T\rightarrow \{w\}$.

If $T^{\kappa }$ degenerates to an edge E, the map $f_T$ sends one side of $\partial T$ to a point and the two other edges monotonically to E. Identifying T with a Euclidean triangle, we then obtain a unique extension $\tilde{f}_T:T\rightarrow E$ of $f_T$ with all fibers convex subsets. This finishes the construction in the degenerated case.

If none of the three sides of $T^{\kappa }$ degenerates to a point, $T^{\kappa }$ is a topological disc with boundary $\partial T^{\kappa }$. Identifying the two triangles T and $T^{\kappa }$ with the disc and coning the map $f_T$, we obtain an extension map $\tilde{f}_T:T\rightarrow T^{\kappa }$ of $f_T$. The preimages of points under the map $\tilde{f}_T$ are (possibly degenerated) compact intervals.

All the maps $\tilde{f}_T$ constructed above coincide on common edges and points of different intersecting triangles T. Thus, all $\tilde{f}_T$ glue together to a continuous map $p:{\mathbb {D}}\rightarrow W$. By construction, the preimage $P:=p^{-1} (w)$ of any point $w\in W$ is a Peano continuum and the intersection of P with any triangle T is contractible.

We claim that any such preimage $P=p^{-1} (w)$ is contractible. Otherwise, we would find a non-contractible circle S in P. Denote by O the open Jordan domain of S in ${\mathbb {D}}$. Since the intersection of P with each triangle T is empty or contractible, S is not contained in any triangle. Hence, O has a non-empty intersection with $\Gamma$. The open disc O does not contain points in A. By Corollary 4.3, applied to the length-minimizing map h, the subset $O\cap \Gamma$ must be contained in $h^{-1} (w)$, hence in P. Since any triangle intersects P in an empty or in a contractible set, we deduce $T\cap O\subset P$, for all triangles T, with $T\cap O\ne \emptyset$. Therefore, $O\subset P$. This contradicts the non-contractibility of S.

Hence, p has contractible fibers. Therefore, W is a disc-retract [27, Lemma 3.3]. In particular, W is contractible.

The diameter of any triangle $T^{\kappa }$ equals to the maximum of the distances between its vertices. Hence, this diameter is at most $\varepsilon$, by assumption. Therefore, the vertices of $T^{\kappa }$ are $\varepsilon$-dense in $T^{\kappa }$ and $p({\mathcal {V}})$ is $\varepsilon$-dense in W.

For any edge E in $\langle \Gamma \rangle _h$, the map $\hat{h}: E\rightarrow Y$ is an isometric embedding. For any degenerated triangle $T^{\kappa }\subset W$, this gives an isometric embedding $\tilde{h}_T:=\hat{h}:T^{\kappa } \rightarrow Y$.

If $T^{\kappa }$ is non-degenerated, then the boundary $\partial T^{\kappa }$ is the comparison triangle of the triangle $\hat{h} (\partial T ^{\kappa })$ in Y. Reshetnyak’s majorization theorem, [1, Theorem 9.56, Proposition 9.54] implies that $\hat{h}$ extends to a 1-Lipschitz map $\tilde{h}_T:T^{\kappa } \rightarrow Y$.

The maps $\tilde{h}_T$ coincide on common edges and vertices, where they are given by $\hat{h}$. Hence, these maps glue together to a 1-Lipschitz map $q:W\rightarrow Y$, whose restriction to $\langle \Gamma \rangle _h \subset W$ is $\hat{h}$.

By construction, $q\circ p|_A=h|_A=f|_A$. For every curve $\gamma$ in $\Gamma$, we have

$$\begin{aligned} \ell _Y(f\circ \gamma ) \ge \ell _Y (h\circ \gamma ) = \ell _W (p\circ \gamma ) . \end{aligned}$$

It remains to show that the space W is CAT$(\kappa )$. By construction, W is obtained by gluing together (simplically) some number of triangles of constant curvature $\kappa$. Since the space W is a disc retract, the space of directions $\Sigma _w W$ at any point $w\in W$ topologically embeds into a circle. Hence, $\Sigma _w W$ is topologically either a circle or a disjoint union of (possibly degenerated) intervals.

If W is not locally CAT$(\kappa )$, we find a point $w\in W$ and a closed local geodesic of length less than $2\pi$ in $\Sigma _wW$ [1, Theorem 12.2]. Then the whole space $\Sigma _wW$ must be a circle of length less than $2\pi$. Hence, w must be a vertex of $\langle \Gamma \rangle _h$ and the cyclic sum of angles $\alpha _j$ (in triangles $T_j ^{\kappa }$) adjacent to w is less than $2\pi$.

By construction and angle comparison, the angles in $T^j_{\kappa }$ are not smaller than the corresponding angles in $\hat{h} (\partial T_i ^{\kappa } )\subset Y$.

Thus, the cyclic sum of all angles $\angle _{\hat{h} (w)} (\hat{h} (e_i),\hat{h} (e_{i+1}))$ in Y is less than $2\pi$, where $e_i$ run over all edges in $\langle \Gamma \rangle _h$ adjacent to w in $\langle \Gamma \rangle _h$. This contradicts the length-minimality of $\hat{h}$ and Lemma 4.8.

Therefore, the space W is locally CAT($\kappa$). Since W is simply connected, we may apply the Cartan–Hadamard theorem, [1, Theorem 9.6], and deduce that W is CAT$(\kappa )$, if $\kappa <0$. The case $\kappa >0$ requires an additional argument.

Thus, we assume now $\kappa >0$.

The fact that the total angle at any interior point is at least $2\pi$ implies that no two sides of a non-degenerate triangle $T^{\kappa }$ are identified in W. Therefore, the triangles $T^{\kappa }$ define a triangulation of W with 1-skeleton $\iota (\langle \Gamma \rangle _h)$.

If W is not CAT($\kappa$), we find an isometrically embedded circle S of length $<2R_{\kappa }$ in W [6, Theorem 2.2.11]. Moreover, we find and fix such a circle with the smallest possible length. Upon rescaling we may assume that S has length $2\pi$. Then $\kappa <1$. Since there are no closed geodesics of length $<2\pi$, the space W is CAT(1) and locally CAT($\kappa$).

Consider the subdisc $W_0$ of W bounded by S, which is a convex subset of W. We are going to construct a 1-Lipschitz map $\tilde{q}:W_0 \rightarrow Y$, which equals q on S and such that $\tilde{q}$ shortens the length of some curve $\gamma \subset \langle \Gamma \rangle _h \cap W_0$. Once $\tilde{q}$ is constructed, the map

$$\begin{aligned} j: \langle \Gamma \rangle _h \rightarrow Y \end{aligned}$$

given by q on $\langle \Gamma \rangle _h \setminus W_0$ and by $\tilde{q}$ on $W_0$ would satisfy $\hat{h} \unrhd j$ (rel. A) and contradict the length-minimality of $\hat{h}$.

It remains to construct the 1-Lipschitz $\tilde{q}:W_0 \rightarrow Y$, which shortens the length of some curve in $\langle \Gamma \rangle _h \cap W_0$.

Denote by H the hemisphere of curvature 1 with pole o. Glue H to $W_0$ identifying S with the boundary of the hemisphere H. The arising space

$$\begin{aligned} W_1:=W_0\cup _S H \end{aligned}$$

is CAT(1) by Reshetnyak’s gluing theorem. There is a canonical 1-Lipschtitz retraction $\Pi :W_1\rightarrow H$, defined as follows (compare [10, Lemma 1.3] [12]):

• $\Pi (x)=x$ if $|o-x|_{W_1} \le \frac{\pi }{2}$.

• $\Pi (x)=o$ is $|o-x|_{W_1} \ge \pi$.

• $\Pi (x)=\eta _x (\pi - |o-x|_{W_1})$ if $\pi> |o-x|_{W_1} > \frac{\pi }{2}$, where $\eta _x$ is the geodesic from o to x in $W_1$ parametrized by arclength.

The hemisphere H has constant curvature $1 > \kappa$. We apply the Kirzsbraun–Lang–Schroeder extension theorem, [13, 1, Theorem 10.14], and find a 1-Lipschitz map $g:H\rightarrow Y$ which extends the 1-Lipschitz map $q:S\rightarrow Y$.

We define the 1-Lipschitz map $\tilde{q}:W_0\rightarrow Y$ as the composition $\tilde{q}= g\circ \Pi$. Then $\tilde{q}$ coincides with q on S. It remains to find a curve in $G:=\langle \Gamma \rangle _h \subset W_0$, whose length is strictly contracted by $\tilde{q}$.

Assume on the contrary, that $\tilde{q}$ preserves the length of all curves in G. The closed geodesic S cannot be contained in a single triangle $T^{\kappa }$ nor can it intersect twice the same side of the same triangle $T^{\kappa }$. It follows that G together with S constitute the 1-skeleton of a triangulation of $W_0$.

By assumption, $\tilde{q}$ is 1-Lipschitz on G. On the other hand $\hat{h}$ is length-preserving and length-minimizing on G relative to $G\cap S$. Therefore, $\tilde{q}$ is length-minimizing on G relative to $G\cap S$ as well. Hence, $\tilde{q}$ is an isometry on every edge of the graph G. Since g is 1-Lipschitz, we deduce that the 1-Lipschitz map $\Pi :W_0 \rightarrow H$ restricts to an isometry to every edge of G.

We find a triangle $T_0 =xpz$ in the triangulation of $W_0$ defined by $G\cup S$, such that one side xz of $T_0$ lies on S. The definition of the map $\Pi$ and the equality

$$\begin{aligned} {|}p-x|_{W_0} = |p-x|_{W_1} = |\Pi (p)- \Pi (x)| _H \end{aligned}$$

imply that $|p-x|_{W_0} \le \frac{\pi }{2}$ and that the triangle opx is of constant curvature 1. Since $W_0$ is locally CAT($\kappa$), it implies that the triangle opx is degenerated. Hence px meets S at x orthogonally. The same is true for the geodesic pz.

Since $W_0$ is CAT(1), we deduce that p has distance $\frac{\pi }{2}$ to x and to z and that the triangle pxz has constant curvature 1. Since $W_0$ is locally CAT($\kappa$), this implies that the triangle $T_0$ is degenerated, which is impossible.

This contradiction shows that W is CAT($\kappa$) and finishes the proof. $\square$

We are now in position to provide

Proof of Theorem 1.1

Thus, let Y be CAT$(\kappa )$ and $f:{\mathbb {D}} \rightarrow Y$ be length-continuous and length-minimizing relative to ${\mathbb {S}}^1$. We need to prove that $\langle {\mathbb {D}}\rangle _f$ is CAT$(\kappa )$.

Due to Lemma 3.4, $\langle {\mathbb {D}}\rangle _f$ is a disc retract. It is sufficient to prove that any Jordan triangle G (thus a Jordan curve, built by 3 geodesics) of length $<2R_{\kappa }$ in $\langle {\mathbb {D}}\rangle _f$ is not thicker than its comparison triangle in $M^2_{\kappa }$. We fix G and denote by J the closed disc bounded by G in the disc retract $\langle {\mathbb {D}}\rangle _f$.

It is sufficient to prove that J with its intrinsic metric is CAT$(\kappa )$. Since the boundary G of J has finite length, J is a geodesic metric space in its intrinsic metric.

The restriction $\hat{f}:J\rightarrow Y$ is length-minimizing relative to G and its image is contained in a ball $B\subset Y$ of radius $r<\frac{R_{\kappa }}{2}$, by Lemma 4.9. Moreover, $\hat{f}:J\rightarrow Y$ is a light map, which preserves the length of all curves in J, by Lemma 3.4 and Lemma 3.2.

We thus have a geodesic metric space J homeomorphic to a closed disc, whose boundary G is a geodesic triangle of perimeter less than $2R_{\kappa }$. Upon renaming B into Y, we have a light map $\hat{f}:J\rightarrow Y$ into a CAT$(\kappa )$ space Y which is contained in a ball of radius less than $\frac{R_{\kappa }}{2}$ around some of its points. The map $\hat{f}$ preserves the length of all curves in J and $\hat{f}$ is length-minimizing relative to G. We need to verify that J is CAT($\kappa$).

For any natural n we find a finite, connected, piecewise geodesic graph $\tilde{\Gamma }_n$ in J, which contains G and satisfies the following conditons, see [7, Theorem 1.2] and [22, Proposition 5.2]:

The boundary of any connected component $T_0$ of $J{\setminus } \tilde{\Gamma } _n$ is a Jordan triangle and the closure T of $T_0$ has diameter less than $\frac{1}{n}$. The embedding of $\tilde{\Gamma } _n$ with its intrinsic metric into J is a $\frac{1}{n}$-isometry, thus,

$$\begin{aligned} |x-z|_{ \tilde{\Gamma } _n } \le |x-z|_J +\frac{1}{n} , \end{aligned}$$

for all $x,z\in \tilde{\Gamma } _n$.

Note that the graph $\tilde{\Gamma } _n$ provided by [22, Proposition 5.2] does not need to be the 1-skeleton of a triangulation of J: A vertex of one triangle may lie on the side of another triangle.

We refine $\tilde{\Gamma } _n$ by adding one vertex $o_T$ inside of each triangle T defined by $\tilde{\Gamma } _n$ and by connecting $o_T$ by pairwise disjoint curves inside T to all vertices of $\tilde{\Gamma } _n$ on the sides of T. Denote the arising graph by $\Gamma _n$.

By construction, $\Gamma _n$ defines a triangulation of J and each triangle of the triangulation has diameter at most $\frac{1}{n}$.

We now apply Proposition 6.1 and obtain a CAT$(\kappa )$ disc retract $W_n$, maps $p_n:J\rightarrow W_n$ and $q_n:W_n\rightarrow Y$ with the properties (1), (2), (3) stated there, for $\varepsilon =\frac{1}{n}$ and $f=\hat{f}$.

Hence, $p_n$ is surjective and has contractible fibers, and $q_n$ is 1-Lipschitz.

Denote by ${\mathcal {V}}_n$ the vertices of the subgraph $\tilde{\Gamma } _n$ of $\Gamma _n$. Any vertex of $\Gamma _n$ lies at distance at most $\frac{1}{n}$ to some vertex of $\tilde{\Gamma } _n$. Since $\hat{f}$ is length-preserving and therefore 1-Lipschitz, this distance estimate holds for the images of the vertices in Y and, therefore for vertices of the triangulation of $W_n$. Together with the property (2), this implies that $p_n({\mathcal {V}}_n)$ is $\frac{2}{n}$-dense in $W_n$.

For every pair of points $x,z\in {\mathcal {V}}_n$, we choose a curve $\gamma \subset \Gamma _n$ realizing the distance between x and z in $\Gamma _n$. Then

$$\begin{aligned} |x-z|_J + \frac{1}{n} \ge \ell _J (\gamma ) \ge \ell _Y (\hat{f}\circ \gamma ) \ge \ell _{W_n} (p_n\circ \gamma ) \ge |p_n(x)-p_n (z)|_{W_n} . \end{aligned}$$

After choosing a subsequence we obtain a Gromov–Hausdorff converging sequence of images $p_n(\Gamma _n)$ to a compact metric space W. Moreover, the spaces $W_n$ converge to the space W as well and the maps $p_n$ converge to a surjective 1-Lipschitz map $p:J\rightarrow W$. Since the spaces $W_n$ are CAT$(\kappa )$, the limit space W is a CAT$(\kappa )$ space as well.

Set $A_n:={\mathcal {V}}_n \cap G$. Then $A_n$ is $\frac{1}{n}$-dense in G and, by Proposition 6.1, the restriction of $q_n\circ p_n$ to $A_n$ coincides with $\hat{f}|_{A_n}$. In the limit we obtain

$$\begin{aligned} \hat{f}|_{G} = q\circ p |_{G}. \end{aligned}$$

By assumption, $\hat{f}$ preserves the length of all curves. Since $q\circ p$ is 1-Lipschitz, we deduce $\hat{f}\unrhd q\circ p$ (rel. G). By assumption, $\hat{f}$ is length-minimizing relative to G, hence $\hat{f}=q\circ p$, by Lemma 4.5. Since p and q are 1-Lipschitz, the map p must preserve the length of any curve.

Since $\hat{f}$ is a light map, also p must be a light map. The fibers of $p_n$ converge to subsets of the corresponding fibers of p. The fibers of $p_n$ are connected, hence any limit set is connected as well. Since all fibers of p are totally disconnected, we deduce that any limit of fibers $p_n^{-1} (w_n)$ must be a singleton. Denoting by $\varepsilon _n$ the maximal diameter of fibers of $p_n$, we deduce $\lim \varepsilon _n =0$.

We claim that p is injective. Otherwise, we find some $x\ne z \in p^{-1} (w)$. Then $|p_n(x)-p_n(z)|_{W_n}$ converge to 0. Consider the geodesic $e_n$ between $p_n(x)$ and $p_n(z)$ in $W_n$ and set $E_n:=p^{-1}_n (e_n)$. Since $p_n$ has connected fibers, $E_n$ is a connected subset of J.

Taking a subsequence, we obtain in the limit a connected subset E of J, which contains x and z and is sent by p to the point w. Since p is a light map, this is a contradiction.

Therefore, $p:J\rightarrow W$ is injective, hence bijective. Since J is compact, p is a homeomorphism. Since p preserves the length of any curve, p is an isometry and $J=W$ is a CAT$(\kappa )$ space. $\square$

7 Ruled discs

We are going to prove Corollary 1.3 in this section. Thus, let $\eta _0, \eta _1:[0,1]\rightarrow Y$ be rectifiable curves in a CAT$(\kappa )$ space Y. For $a\in [0,1]$, let $\gamma _a (t):[0,1]\rightarrow Y$ be geodesics of length $<R_{\kappa }$ parametrized proportionally to arclength and connecting $\eta _{0} (a)$ with $\eta _1(a)$.

We consider the square ${\mathcal {Q}}:=[0,1]\times [0,1]$ as a topological disc. We define $f:{\mathcal {Q}}\rightarrow Y$ by $f(a,t):=\gamma _a(t)$. We need to verify that $\langle {\mathcal {Q}} \rangle _f$ is CAT$(\kappa )$.

Since geodesics of length $<R_{\kappa }$ depend continuously on the endpoints, we deduce that f is continuous. By continuity, we find some $\delta >0$, such that all geodesics $\gamma _a$ have length at most $R_{\kappa } -\delta$. By the quadrangle comparison, we find some $L, \rho >0$ with the following property:

Whenever $|\eta _i (a) -\eta _i (b)| _Y \le \rho$, for $i=0,1$, then, for all $t\in [0,1]$:

$$\begin{aligned} |\gamma _a(t) -\gamma _b(t)|_Y \le L \cdot \big (|\eta _0(a) -\eta _0(b)| _Y + |\eta _1(a) -\eta _1(b)|_Y\big ). \end{aligned}$$

Therefore, for any $s\in [0,1]$, the curve $\eta _s (t):=f(t,s)$ is of finite length, bounded from above by $L\cdot (\ell _Y(\eta _0) +\ell _Y(\eta _1))$. Moreover, once $\eta _{0}, \eta _1$ have length at most r on an interval $[a,b] \subset [0,1]$, then any of the horizontal curves $\eta _t, t\in [0,1]$ has length less than $2\cdot L\cdot r$.

Hence, any point $(a,t)\in {\mathcal {Q}}$ sufficiently close to a given point $(a_0,t_0)$ can be connected to $(a_0,t_0)$ by a concatenation of a vertical and horizontal segments in ${\mathcal {Q}}$, which is mapped to a curve of a small length. Therefore, f is length-continuous.

Therefore, Corollary 1.3 is a direct consequence of Theorem 1.1 and the following:

Lemma 7.1

The map $f:{\mathcal {Q}} \rightarrow Y$ is length-minimizing relative to $\partial {\mathcal {Q}}$.

Proof

Let a continuous map $g:{\mathcal {Q}}\rightarrow Y$ satisfy $f\unrhd g$ (rel. $\partial {\mathcal {Q}}$). For the vertical segment $t\rightarrow (a,t) \in {\mathcal {Q}}$, the image curve $t\rightarrow g(a,t)$ has length not larger than the geodesic $\gamma _a$ and connects the same boundary points.

Hence $t\rightarrow g(a,t)$ coincides with $\gamma _a$ up to parametrization. Since also

$$\begin{aligned} \ell _Y(f\circ \gamma ) \ge \ell _Y(g\circ \gamma ) \end{aligned}$$

for all subsegments $\gamma$ of the segment $t\rightarrow (a,t)$, parametrizations must coincide, hence $f(a,t)=g(a,t)$, for all a and t.

Thus, $f\unrhd g$ (rel. $\partial {\mathcal {Q}}$) implies $f=g$. Hence f is length-minimizing. $\square$

8 Harmonic disks

8.1 Basics on Sobolev discs

In this section, we assume some knowledge on Sobolev and harmonic maps with values in metric spaces. We refer the reader to [9, 11, 15, 29], for introductions to this subject.

Throughout the section let Y be a CAT$(\kappa )$ space and $\Omega$ be a bounded, open subset of ${\mathbb {R}}^2$. Any Sobolev map $f\in W^{1,2} (\Omega , Y)$ has a (Korevaar–Schoen) energy $E(f)\in [0, \infty )$ [11, 15, Proposition 4.6].

If $\Omega$ is a Lipschitz domain, hence the boundary $\partial \Omega$ is a union of Lipschitz curves, then for any Sobolev map $f\in W^{1,2} (\Omega , Y)$ there is a trace of f, $tr(f)\in W^{1,2} (\partial \Omega , Y)$, [11, Theorem 1.12.2]. If $f: \bar{\Omega } \rightarrow Y$ is continuous, then the trace is just the restriction of f to the boundary $\partial \Omega$.

If $\partial \Omega$ is not Lipschitz we say that $f,g\in W^{1,2} (\Omega , Y)$ have equal traces, if $\langle f-g \rangle _Y$ is contained in $W^{1,2}_0 (\Omega , {\mathbb {R}})$, see [29, Sect. 1.4], [16, Sect. 4.1].

A map $f\in W^{1,2} (\Omega , Y)$ is called harmonic if f has the smallest energy among all maps $g\in W^{1,2} (\Omega , Y)$ with the same trace as f.

For any $f\in W^{1,2}(\Omega , B)$, where B is a ball in Y of radius $<\frac{R_{\kappa }}{2}$, there exists a unique harmonic map $g\in W^{1,2} (\Omega , Y)$ with the same trace as f, [29, Theorem 1.16]. Moreover, the image of this harmonic g is contained in B as well and g has a (unique) locally Lipschitz representative [29, Theorem 3.1]. We will always use this representative below. Finally, the restriction of a harmonic map to any subdomain $\tilde{\Omega } \subset \Omega$ is harmonic as well, [16, Lemma 4.2].

Assume that $f,g:\bar{\Omega } \rightarrow Y$ are continuous and that $f\in W^{1,2} (\Omega , Y)$. Assume further that $f\unrhd g$. Then $\ell _Y(f\circ \gamma )\ge \ell _Y(g\circ \gamma )$, for any curve $\gamma$ in $\Omega$. Using the upper-gradient definition of Sobolev maps [9] this implies that $g \in W^{1,2} (\Omega , Y)$ as well. Moreover, at almost all points of $\Omega$ the approximate metric differential of f is not less than the corresponding approximate metric differential of g, [15, Proposition 4.10]. Then also the energies satisfy $E(f)\ge E(g)$, [15, Proposition 4.6]. In particular, if f is harmonic then g is harmonic as well.

If $f\in W^{1,2} ({\mathbb {D}}, Y)$ is harmonic and $tr(f):{\mathbb {S}}^1\rightarrow Y$ is continuous, then f is continuous on the closed disc ${\mathbb {D}}$, [14, Proposition 4.4]. The following Lemma is essentially contained in the proof of [14, Proposition 4.4].

Lemma 8.1

Let $f:{\mathbb {D}}\rightarrow Y$ be a harmonic map such the trace $f:{\mathbb {S}}^1 \rightarrow Y$ is a curve of finite length. Then f is length-continuous.

Proof

The map f is locally Lipschitz on ${\mathbb {D}}_0:={\mathbb {D}}\setminus {\mathbb {S}}^1$. Hence, $\hat{\pi } _f:{\mathbb {D}}\rightarrow \langle {\mathbb {D}} \rangle _f$ is continuous on ${\mathbb {D}}_0$.

Let $z\in {\mathbb {S}}^1$ and $\varepsilon >0$ be arbitrary. We find an arbitrary small ball $B_r(z)$ such that the lengths of the two curves $f(\partial B_r(z) \cap {\mathbb {D}})$ and $f({\mathbb {S}}^1\cap B_r(z))$ are smaller than $\varepsilon$. Moreover, if r is small enough, the energy of the restriction of f to $B_0:=B_r(z) \cap {\mathbb {D}}$ is also smaller than $\varepsilon$.

Hence $f:B_0 \rightarrow Y$ is a continuous, harmonic and of energy smaller than $\varepsilon$. We claim that there exists some $\delta (\varepsilon )>0$ which goes to 0 with $\varepsilon$, such that the following holds true: Any point $p\in B_0$ can be connected by a curve $\gamma$ with a point on $\partial B_0$ such that $\ell _Y(f\circ \gamma )$ is at most $\delta$.

Once the claim is verified, we can use the fact that the restriction of f to $\partial B_0$ has length $<\varepsilon$ to deduce, for every $p\in B_0$,

$$\begin{aligned} \langle z -p \rangle _f \le \delta +\varepsilon . \end{aligned}$$

The right hand side goes to 0 as r converges to 0. Hence the map $\hat{\pi } _f$ would be continuous at z. Thus, $\hat{\pi } _f$ would continuous on all of ${\mathbb {D}}$.

It remains to verify the claim above. Since precomposing with a conformal diffeomorphism between domains in ${\mathbb {C}}$ does not change the energy (and the induced intrinsic distances), we use the Riemann mapping theorem and may replace $B_0$ by ${\mathbb {D}}$ and p by the origin $0\in {\mathbb {D}}$.

We denote by $\eta _{\theta } (t)=t\cdot \theta$ the radial segment in the direction $\theta \in {\mathbb {S}}^1$, which connects 0 with the point $\theta \in {\mathbb {S}}^1=\partial {\mathbb {D}}$. Denote by $\eta _{\theta }^+$ the first half of the segment connecting 0 with $\frac{1}{2} \theta$ and by $\eta _{\theta }^-$ the second half of $\eta _{\theta }$.

The restriction of f to the smaller ball $B_{\frac{1}{2}} (0)$ is $L\cdot \varepsilon$-Lipschitz, where L is some uniform constant, see [29, Theorem 2.2]. Hence, for any segment $\eta _{\theta } ^+$, the length of $f\circ \eta _{\theta } ^+$ is at most $\frac{1}{2} \cdot L\cdot \cdot \varepsilon$.

On the other hand, integrating the energy in polar coordinates, we find that the length of $f\circ \eta _{\theta }^-$ is at most $\sqrt{\varepsilon }$, for at least one $\theta \in {\mathbb {S}}^1$. Indeed, denoting the absolute gradient of u by $\rho \in L^2({\mathbb {D}})$, we obtain:

$$\begin{aligned} \varepsilon =E(u) \ge \frac{1}{2} \int _{{\mathbb {D}}} \rho ^2 \ge \frac{1}{4} \int _{\theta \in {\mathbb {S}}^1} \left( \int _{\eta _{\theta } ^-} \rho ^2 \right) \, d\theta \ge \frac{1}{4} \int _{{\mathbb {S}}^1} \ell _Y ^2 (f\circ \eta _{\theta } ^-) \, d\theta . \end{aligned}$$

Thus, for this $\theta$, the length of $f\circ \eta _{\theta }$ is at most $\frac{1}{2} \cdot L \cdot \varepsilon + \sqrt{\varepsilon }$. This finishes the proof of the claim and of the lemma. $\square$

It is possible to prove (but requires some rather technical considerations in the case of positive $\kappa$) that any harmonic disc $f:{\mathbb {D}}\rightarrow Y$ is length-minimizing. We arrive at the proof of Corollary 1.2 faster restricting the domain of definition:

Proof of Corollary 1.2

Thus, let $f:{\mathbb {D}}\rightarrow Y$ be continuous and harmonic and such that $f({\mathbb {S}}^1)$ has finite length. If, for some $y\in Y$, the set ${\mathbb {D}} \setminus f^{-1} (y)$ has a connected component O not intersecting ${\mathbb {S}}^1$, then the restriction of f to this component O could not be harmonic. Hence, this is impossible. Therefore, the geodesic space $\langle {\mathbb {D}} \rangle _f$ is a disc retract by Lemma 8.1 and Lemma 3.4.

Let G be a Jordan triangle in $\langle {\mathbb {D}} \rangle _f$ of length less than $2R_{\kappa }$ and let J be the closed disc bounded by G in the disc retract $\langle {\mathbb {D}} \rangle _f$. As in the proof of Theorem 1.1, it suffices to show that $\langle J\rangle _{\hat{f}}$ with the induced length metric is CAT($\kappa$). Due to Theorem 1.1 it suffices to prove that $\hat{f}:J\rightarrow Y$ is length-minimizing relative to G.

Let $\tilde{G}$ be the preimage of G in ${\mathbb {D}}$ and let $O\subset {\mathbb {D}}$ be the preimage of $J\setminus G$. Then $f(\tilde{G})=\hat{f}(G)$ is a curve of length less than $2R_{\kappa }$, hence it is contained in a ball B of radius less than $\frac{R_{\kappa }}{2}$ in Y. Applying a strictly 1-Lipschitz retraction to B, [12], we deduce that the image of the harmonic map $f:O\rightarrow Y$ is contained in B.

Assume now that $\hat{f}$ is not length-minimizing relative to G and let $g:J\rightarrow Y$ with $\hat{f}\unrhd g$ (rel. G) be given. Then the map $\tilde{g}:=g\circ \hat{\pi } _f:\bar{O} \rightarrow Y$ satisfies $f \unrhd \tilde{g}$ (rel. $\tilde{G}$). As seen above, $\tilde{g}$ is harmonic on O. By the uniqueness of harmonic maps with values in B, we deduce $\tilde{g}=f$. Hence $\hat{f}=g$. This finishes the proof of the fact that $\hat{f}$ is length-minimizing and of the Corollary. $\square$

9 Questions

Here is the promised list of (mostly technical) questions.

Question 9.1

What does it mean in differential-geometric terms for a smooth map $f:{\mathbb {D}}\rightarrow M$ into a Riemannian manifold M to be length-minimizing?

See [27, Sect. 10] for partial answers to this question.

Question 9.2

Does our main theorem hold true for length-minimizing discs which are length-connected but not length-continuous?

This question was the essential motivation for the definition of metric-minimality created in [27].

Question 9.3

Does the conclusion of Lemma 3.4 hold true without the non-bubbling assumption? For which topological spaces X does the conclusion of Lemma 3.4 hold true, for all length-continuous maps $f:X\rightarrow Y$? In particular, does Lemma 3.4 hold true for Euclidean balls X of dimension larger than 2.

The last part of this question appears in the arXiv-version of [25].

Question 9.4

Let $f:{\mathbb {D}}\rightarrow Y$ be length-connected. Can f be not length-continuous if $\langle {\mathbb {D}} \rangle _f$ is compact? What about more general Peano spaces X instead of ${\mathbb {D}}$?