Proof. Follows immediately from [Reference RanganathanRan17, Theorem A].

Proof. Since no edge is contracted, the lemma reduces to checking local realizability at each vertex. Fix a vertex $v$ of the domain $\Gamma$ . To $v$ , we associate $\mathbb{P}^1$ marked at a finite collection of distinct points $S \subset \mathbb{P}^1$ in bijection with the tangent directions (flags) of edges based at $v$ . At each point $p$ we have a slope $a_p$ coming from $F$ . The divisor $\sum _{p \in S} a_p [p]$ has degree $0$ and is therefore principal. Any trivializing section of this line bundle gives a local realization of $F$ at $v$ .

Proof. It suffices to show that $F'$ is H-modifiable when a single additional ray is added to $\Gamma$ . Denote this ray by $e$ ; its image under the collapsing map is a point $v$ in $\Gamma$ . Hence, $F'$ maps $e$ to $F(v)$ .

By hypothesis, there exists an H-modification $\hat {F}\colon \hat {\Gamma } \to \hat {\Delta }$ of the original map $F\colon \Gamma \to \mathbb{R}$ . We first glue the ray $e$ to $v \in \hat {\Gamma }$ in the canonical way; it has no target in the modification yet. Hence, we add a ray $r$ to $F(v) \in \hat {\Delta }$ ; this fixes the target of the putative H-modification of $F'$ , while we only modify the domain later on.

In order to resolve local realizability at $v$ , we further add $d_v(\hat {F})-1$ new rays to $v$ ; along with $e$ , each is mapped to $r$ with slope $1$ . Now, the only obstructions to local realizability are the vertices $u \in \hat {\Gamma }$ different from $v$ such that $\hat {F}(u)$ equals $F(v)$ . Resolving a case like this is straightforward: we add $d_u(\hat {F})$ new rays to $u$ and map each to $r$ with slope $1$ . We are done.

Proof. We argue by induction on $n$ ; the case $n=0$ is trivial. Assume the statement holds for $n-1$ and set

\begin{align*} \Sigma _{n-1}=\Sigma \cup \bigcup _{i=1}^{n-1}C_i, \quad F_{n-1}:= F|_{\Sigma _{n-1}}. \end{align*}

Since every edge and ray in $C_n$ is contracted by $F$ , the subgraph $C_n$ is mapped to a single vertex $V_n\in \Delta$ . By the inductive hypothesis, $F_{n-1}$ admits an $H$ -modification

\begin{align*} \widehat F_{n-1}\colon \widehat \Sigma _{n-1}\to \widehat \Delta _{n-1} \end{align*}

with the required properties.

Define

\begin{align*} \Gamma _n=\widehat \Sigma _{n-1}\cup C_n \end{align*}

and a map $G_n\colon \Gamma _n\to \widehat \Delta _{n-1}$ by

\begin{align*} G_n(x)= \begin{cases} \widehat F_{n-1}(x), & x\in \widehat \Sigma _{n-1},\\ V_n, & x\in C_n. \end{cases} \end{align*}

This is well-defined: for $i\lt n$ we have $C_i\cap C_n\subset \Sigma$ , hence $ \widehat \Sigma _{n-1}\cap C_n=\Sigma \cap C_n,$ and on this intersection both definitions agree with $F$ . The map $G_n$ is clearly harmonic.

By assumption, $h_n$ admits an $H$ -modification $\widehat h_n$ . Let $T_n=\operatorname {Im}(\widehat h_n)$ , a tree rooted at $V_n$ arising from the modification, and let $\widehat C_n$ denote $C_n$ with all attached modifications. We now construct the desired modification.

• • Target. Obtain $\widehat \Delta$ from $\widehat \Delta _{n-1}$ by attaching a copy of $T_n$ , identifying its root with $V_n$ .

• • Domain. Start with $\widehat \Sigma _{n-1}\cup \widehat C_n$ , glued along $\Sigma \cap C_n$ . For each vertex $W\in \widehat \Sigma _{n-1}\smallsetminus (\Sigma \cap C_n)$ with $G_n(W)=V_n$ , attach $d_W(\widehat F_{n-1})$ copies of $T_n$ at $W$ , identifying their roots with $W$ . Let $\widehat \Sigma$ be the resulting graph.

• • Map. Define

  \begin{align*} \widehat F\colon \widehat \Sigma \to \widehat \Delta \end{align*}
  to equal $\widehat F_{n-1}$ on $\widehat \Sigma _{n-1}$ , to equal $\widehat h_n$ on $\widehat C_n$ , and to map each newly attached copy of $T_n$ identically onto the target tree with slope $1$ .

The map $\widehat F$ is well-defined. For every vertex $W$ with $G_n(W)=V_n$ , each edge or ray attached to $W$ outside $\widehat \Sigma _{n-1}$ has slope $1$ ; hence, harmonicity and local realizability hold at every vertex. By construction, the natural diagram relating $F$ and $\widehat F$ commutes: newly added rays in the domain map to the corresponding components in the target.

Thus, $\widehat F$ is an $H$ -modification of $F$ with the desired properties.

Proof. We induct on the number of edges in the contracted subtree $T_1$ . The base case is when $T_1$ consists of a single vertex, possibly with finitely many rays attached; the claim holds clearly. For the inductive step, we consider several different cases. In the course of the proof, we will call the vertices $A_1,\ldots ,A_n$ the critical vertices.

1. (i) Suppose there exists a critical vertex of valency at least $2$ in the tree $T_1$ , and relabel so that this vertex is $A_1$ . Since $T_1$ is a tree, we can disconnect $T_1$ at $A_1$ , resulting in two strictly smaller subtrees, which we will call $T_1'$ and $T_1''$ , such that $T_1' \cap T_1'' = \{A_1\}$ . Let $S'$ be the union of $T_1'$ with all the stars in $T$ of the critical points that lie in $T_1'$ , excluding the interior of $T_1''$ . Let $S''$ be obtained analogously from $T_1''$ . Since $F$ contracts any edges in $T_1'$ and $T_1''$ , $F$ restricts to harmonic maps on $S'$ and $S''$ , with targets being the respective images. Furthermore, these restrictions both admit $H$ -modifications with the required properties, by the inductive hypothesis. Hence, the hypothesis of Lemma3.5 is satisfied, so we get the required $H$ -modification.

2. (ii) We now consider the case where every critical vertex $A_i$ is a leaf of $T_1$ , and let $e_i$ denote the corresponding leaf edges. If any $e_i$ is incident to two critical vertices, then by connectivity of $T_1$ , there must only be two critical vertices $A_1$ and $A_2$ , and the edge $e_1$ between them is all of $T_1$ . In this case, we modify as follows. Let $M$ be the midpoint of $e_1$ and let $a = |A_1 M|$ , the length of $A_1 M$ . For the domain modification, add two rays $\ell _1, \ell _2$ to the domain graph based at $M$ . For the target, add a new ray $\ell$ to the target graph at the image of $M$ , i.e. $0 \in \mathbb{R}$ . Place a vertex $N$ on this new ray $\ell$ , such that its distance to $0$ is $a$ . Map the segments from $A_i$ to $M$ , for both $i$ , to the segment from $0$ to $N$ . Map $\ell _1, \ell _2$ to the ray emanating from $N$ along $\ell$ with slope $1$ . Finally, at each critical vertex $A_i$ , add $d_{A_i} - 1$ rays to $A_i$ , and map each to $\ell$ with slope $1$ .

3. (iii) Assume that we are in neither of the previous situations, and that the critical vertices $A_i$ are all leaves with associated leaf edges $e_i$ . In particular, none of the $e_i$ connect distinct $A_j$ . Without loss of generality, let $e_1$ have minimal length among these leaf edges, and denote this length by $a$ .

4. Construction of the $H$ -modification: For each $i$ , let $B_i$ be the point on $e_i$ at distance $a$ from $A_i$ , and let $\mathcal{B}$ be the set of these points. For each distinct $P \in \mathcal{B}$ , attach $d_P$ rays $\ell _{P,1},\ldots ,\ell _{P,d_P}$ to the domain at $P$ . This produces a modification $T'$ of the domain $T$ .

   Add a ray $\ell$ to the target graph at $0 \in \mathbb{R}$ , and place a vertex $N$ on $\ell$ at distance $a$ from $0$ . This yields a modification $\Sigma'$ of the target. Map each newly added ray in the domain to $\ell$ with slope $1$ , starting at $N$ .

5. Applying the inductive hypothesis: Remove the half-open segments $[A_i,B_i)$ from $T$ , giving a disjoint union of trees $S_1,\ldots ,S_m$ , each containing strictly fewer edges than $T_1$ . For each $j$ , form $S'_j$ by attaching the stars of all vertices in $\mathcal{B} \cap S_j$ to $S_j$ .

   Viewing the interior of $\ell$ as a copy of $\mathbb{R}$ , each $S'_j$ induces a map $h_j \colon S_j \to \mathbb{R}$ that contracts $S_j$ to $N$ . By the inductive hypothesis, each $h_j$ admits an $H$ -modification satisfying the required properties. Applying the gluing construction of Lemma3.5, we obtain an $H$ -modification of the original map $T \to \mathbb{R}$ with the desired properties.

The three situations exhaust the inductive step, so the result follows by induction on the number of vertices of the contracted subtree.

Proof. Any $F$ -contracted rays in $\Gamma$ can be regarded as arising from a modification, so by Lemma3.4 we may assume that $\Gamma$ has no such rays. Let $\Sigma$ be the largest metric subgraph of $\Gamma$ without contracted edges, and let $C$ be the closure of its complement. By hypothesis, $C$ is a union of disjoint compact metric trees $T_1, \ldots , T_n$ , and if $C$ is non-empty, then $\Sigma \cap C$ is finite. By Lemma3.3, the restriction $F|_{\Sigma }$ admits an $H$ -harmonic modification $F|_{\Sigma }'$ that agrees with $F$ on $\Sigma$ .

For each $T_i$ , let $T_i'$ be the union of $T_i$ with the stars of those points in $T_i$ that meet $\Sigma$ . By Proposition3.6, the restriction of $F$ to $T_i'$ admits an $H$ -modification with slope $1$ on all edges or rays that were contracted by $F$ or newly added. Applying Lemma3.5 to glue these modifications to $F|_{\Sigma }'$ produces an $H$ -modification of $F$ , as required.

Proof. Let $w$ be a path from $A$ to $B$ consisting of edges $e_1,\ldots ,e_k$ . Then $F(w)$ is a path in $\Sigma$ . By injectivity, the edges $F(e_1),\ldots ,F(e_k)$ are distinct, with vertices $F(A),P_2,\ldots ,P_k,F(B)$ .

Since $F(B') = F(B)$ , there exists an edge $e'_k \in \operatorname {Star}_{\Gamma }(B)$ with

\begin{align*} F(e'_k) = F(e_k), \end{align*}

which follows from the local surjectivity of harmonic maps. If $B' = B$ and $B$ is a turning point, we can ensure that $e'_k \neq e_k$ . This guarantees the second statement of the lemma.

Let $F(e'_k)$ have vertices $F(B)$ and $P'_k$ . Since $F(e_{k-1})$ is incident to $P'_k$ , harmonicity of $F$ ensures the existence of an edge $e'_{k-1}$ with vertices $P'_k$ and $P'_{k-1}$ satisfying

\begin{align*} F(e'_{k-1}) = F(e_{k-1}). \end{align*}

Inductively, this produces edges $e'_1,\ldots ,e'_k$ with $B$ a vertex of $e'_k$ and $F(e_i) = F(e'_i)$ . Then the path $v$ in $\Gamma$ consisting of this sequence of edges is a lifting of $w$ that terminates at $B'$ .

Proof. Let $w$ be the path described in the statement, with vertices $A, P_1, \ldots , P_{k-1}, B$ and edges $e_1, \ldots , e_k$ . We first examine the local structure at $A$ . Since $A$ is a critical point, the balancing condition implies there are at least two edges or rays, $\ell _1$ and $\ell _2$ , incident to $A$ that are not contracted by $F$ . Because $A$ is a simple critical vertex, there are exactly two such edges, and the slopes of $F$ along $\ell _1$ and $\ell _2$ coincide; denote this slope by $s$ . The local degree of $\hat F$ at $A$ is the same as that of $F$ , which we also denote by $s$ .

Next, we consider the slope of $\hat F$ along $e_1$ and show that it must be $1$ . Analyzing the corresponding local map of algebraic curves at $A$ , we have a degree- $s$ map with two fully ramified points corresponding to $\ell _1$ and $\ell _2$ . By the Riemann–Hurwitz formula, no further ramification is possible, so the slope of $e_1$ under $\hat F$ is necessarily $1$ .

We now turn to the interior vertices and edges of $w$ and claim that their local degrees under $\hat F$ are also $1$ . Suppose, for contradiction, that $P_i$ is the first vertex where the local degree is not $1$ . The edge $e_i$ has slope $1$ by assumption, so there must be some other edge attached to $P_i$ with the same image under $\hat F$ . This would make $P_i$ a turning vertex, contradicting our assumptions. Hence, $\hat F$ has slope $1$ along the entire path $w$ , proving the first claim.

For the second claim, Lemma 4.6 guarantees a path $v$ in $\hat \Gamma$ starting at a vertex $A'$ and ending at $B'$ that lifts $\hat F(w)$ . Write $v = \overline {A', Q_1, \ldots , Q_{k-1}, B'}$ . We claim that $A'$ lies in the original $\Gamma$ . Indeed, since $A$ is a critical point, $F(A)$ lies in the original target $\mathbb{R} \subset \hat \Sigma$ , so $\hat F(A') \in \mathbb{R}$ and $A'$ must lie in $\Gamma$ . If any of the vertices $Q_i$ left $\Gamma$ , the path could not return to $B'$ , so they must also lie in $\Gamma$ .

Since $w$ has no interior critical vertices, for all interior vertices $P_i$ and $Q_i$ , we have $\hat F(P_i) = \hat F(Q_i)$ lying outside the original copy of $\mathbb{R}$ in $\hat \Sigma$ . Thus, the edge $A'Q_1$ is contracted by $F$ , and since $\hat F(A') \in \mathbb{R}$ , $A'$ is an $F$ -critical vertex.

For the third claim, if $B = B'$ , then $A \neq A'$ , since otherwise $w$ and $v$ would share both endpoints, forming a cycle in $\Gamma$ and contradicting the assumption that $A$ is not a cycle vertex.

Finally, regarding lengths, note that $w$ and $v$ have the same image under $\hat F$ . The slope of $\hat F$ on $w$ is $1$ , while on edges of $v$ it may be any integer $\geqslant 1$ . Therefore, the length of $w$ is at least that of $v$ .

Proof. We assume the existence of an $H$ -modification $\hat {F}\colon \hat {\Gamma } \to \hat {\Sigma }$ and show that there cannot be a simple critical point whose associated critical path has the unique minimal length among all critical paths. We argue by contradiction and, to set it up, suppose such a path exists and is given by $w$ with vertices $P_1,\ldots ,P_n$ , with $P_1$ the only simple critical point on this path, and $P_n$ is the only point on the cycle.

We first claim that none of $P_2,\ldots ,P_n$ is a turning point. If $P_i$ is a turning point, Proposition4.7 gives us a new path $\overline {Q_1,Q_2,\ldots ,Q_i}$ with $F(Q_k) = F(P_k)$ and such that $Q_1$ is a critical point. By the length considerations in this proposition, we contradict the length-minimality or simplicity hypotheses.

We now examine the algebraic map at the point $P_n$ . We claim that the local degree is $1$ . By applying again Proposition4.7, the slope of $\hat {f}$ for the edge $P_{n-1}P_n$ is $1$ . Suppose that the local degree at $P_n$ is greater than $1$ , a different edge in $\hat {\Gamma }$ attached to $P_n$ has the same image as $P_{n-1}P_n$ . This contradicts the turning point hypothesis.

Now we can conclude the argument. Consider the subgraph consisting of the following three edges, $P_{n-1}P_n$ , $P_n A$ , and $P_n B$ , where $A$ and $B$ are the two vertices on the cycle incident to $P_n$ . Since the local degree at $P_n$ is $1$ , we know the images of $P_nA$ and $P_nB$ differ under $\hat F$ . Now go around the cycle from $P_n$ . By continuity of the map $\hat F$ , there must be a point on the cycle distinct from $P_n$ , whose image has the same image under $\hat F$ as $P_n$ . By Proposition4.7, there is a path $\overline {R_1,\ldots ,R_n}$ such that $\hat {F}(R_i) = \hat {F}(P_i)$ for all $i$ , with $R_1$ is a critical point, and the length of this path is at most that of $w$ . This gives the contradiction.

Proof. This follows immediately from the closedness of the realizable locus in the moduli cone of maps containing $[F]$ , as stated in Theorem3.1.

Proof. Let $A$ and $B$ be the endpoints of the new critical path, with $A$ being the critical point. We break the proof into two cases based on whether $B$ is located on an existing critical path (Case I) or on the cycle (Case II).

Case I. Suppose $w$ is attached to a critical path connecting a critical point $A_2$ to a vertex $B_2$ on the cycle. By the trivalency assumption, $B$ and $B_2$ are distinct. Assume that $|AB| \gt |\widehat {F}(A_2B)|$ . Then there exists a unique point $C$ in the interior of $AB$ such that $|AC| = |\widehat {F}(A_2B)|$ . We proceed with the following modification.

Let $T \subset \Delta$ be the tree attached to $\mathbb{R}$ at $0$ as part of the modification. We construct a new map on $\widehat {\Gamma } \cup w$ as follows. Glue $T$ to $\widehat {\Gamma } \cup w$ by identifying $\widehat {F}(A_2B)$ with $AC$ . Define the map

\begin{align*} \widehat {\Gamma } \cup w \to \Delta \end{align*}

by mapping $\widehat {\Gamma }$ via $\widehat {F}$ , mapping $T$ identically with slope $1$ , and contracting the segment $BC$ to the single point $\widehat {F}(B)$ . This map is harmonic.