Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Abstract

Let $f=B_1/B_2$ be a ratio of finite Blaschke products having no critical points on $\partial \mathbb {D}$. Then $f$ has finitely many critical level curves (level curves containing critical points of $f$) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $f$, one needs only understand the configuration of the finitely many critical level curves of $f$. In this paper, we show that in fact such a function $f$ is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $f_1$ and $f_2$ have the same configuration of critical level curves, then there is a conformal map $\phi$ such that $f_1\equiv f_2\circ \phi$. We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves.

Several Lemmata

Lemma 8.1

Let $\lambda$ be a finite connected graph embedded in the plane. Suppose that $\lambda$ has the following properties.

• Each vertex of $\lambda$ is incident to more than one bounded face of $\lambda$.

• Each edge of $\lambda$ is incident to both a bounded face of $\lambda$ and the unbounded face of $\lambda$.

Then, some bounded face of $\lambda$ has a single edge of $\lambda$ as its boundary.

Proof

Construct an auxiliary graph $\mathcal {T}$ from $\lambda$ as follows. Start with $\lambda$ and place a vertex in each bounded face of $\lambda$. For each bounded face $F$ of $\lambda$, draw an edge from the $F$-vertex to each vertex of $\lambda$ in $\partial F$. Deleting all the original edges of $\lambda$, we let $\mathcal {T}$ denote the remaining graph, which is connected by the assumption that each edge of $\lambda$ is incident to both the unbounded face of $\lambda$ and a bounded face of $\lambda$. Any cycle in $\mathcal {T}$ would contradict the same assumption on the edges of $\lambda$, so $\mathcal {T}$ is a tree. Finally, the assumption that each vertex is incident to more than one bounded face of $\lambda$ implies that each leaf of $\mathcal {T}$ arises from a bounded face of $\lambda$. Finally, if $v$ is a leaf of $\mathcal {T}$, the corresponding face of $\lambda$ can have only one edge of $\lambda$ in its boundary. $\square$

Lemma 8.2

Let $v\in \mathbb {C}^{n-1}$and $\rho >0$ be given. Then, there exists a $\nu >0$ such that if $\widehat{v}\in \mathbb {C}^{n-1}$ and $|v-\widehat{v}|<\nu$, and $\widehat{u}\in \Theta ^{-1}(\widehat{v})$, then there is a $u\in \Theta ^{-1}(v)$ such that $|u-\widehat{u}|<\rho$.

Proof

Suppose by way of contradiction that the desired result fails. Thus, there exists a sequence $\{v_k\}_{k=1}^{\infty }\subset \mathbb {C}^{n-1}$ such that $v_k\rightarrow v$, and for each $k$ we may choose a $u_k\in \Theta ^{-1}(v_k)$ such that $|u_k-u|>\rho$ for each $u\in \Theta ^{-1}(v)$.

Define $K:=\{v_k\}_{k=1}^\infty \cup \{v\}$. $K$ is compact, and $\Theta$ is proper, so $\Theta ^{-1}(K)$ is compact. $\{u_k\}_{k=1}^\infty \subset \Theta ^{-1}(K)$, so there is a subsequence $\{u_{k_l}\}_{l=1}^\infty$ which converges to some point $u$. Since $\Theta$ is continuous,

$$\begin{aligned} \Theta (u)=\Theta \left( \lim _{l\rightarrow \infty }u_{k_l}\right) =\lim _{l\rightarrow \infty }\Theta (u_{k_l})=v. \end{aligned}$$

Thus, $\{u_{k_l}\}_{l=1}^\infty$ converges to a point in $\Theta ^{-1}(v)$, which is a contradiction of the choice of $\{u_k\}_{k=1}^\infty$. $\square$

Definition

If $\gamma :[\alpha ,\beta ]\rightarrow \mathbb {C}$ is a path, and $f$ is a function which is analytic and non-zero on the image of $\gamma$, then we say that $\gamma$ is parameterized according to $\arg (f)$ if for each $r\in [\alpha ,\beta ]$, $\arg (f(\gamma (r)))=r$.

Lemma 8.3

Let $v\in {V_{n-1}}$, and $\tau >0$ and be given. Then, there exists a $\rho >0$ such that if $u\in \Theta ^{-1}(v)$, and $\widehat{u}\in \Theta ^{-1}(V_{n-1})$ such that $|u-\widehat{u}|<\rho$, then the following holds. $G_{p_{\widehat{u}},1}\subset {G_{p_u,2}}$, and $|p_u(z)-p_{\widehat{u}}(z')|<\tau$ for all $z,z'\in {G_{p_u,2}}$ satisfying $|z-z'|<\rho$.

Proof

This follows from the fact that the coefficients of $p_u$ are polynomials in the components of $u$. $\square$

Lemma 8.4

Let $v\in {V_{n-1}}$, and $\delta ^{(1)}>0$ be given. There exists some $\delta ^{(2)}\in (0,\delta ^{(1)})$ such that if $u\in \Theta ^{-1}(v)$, and $\lambda$ is a critical level curve of $(p_u,G_{p_u})$ (with $|f|\equiv \epsilon >0$ on $\lambda$), and $x\in \lambda$ is a critical point of $p_u$, then if $y\in {B_{\delta ^{(2)}}(x)}$ satisfies $|f(y)|=\epsilon$, then there is a path $\sigma$ from $y$ to $x$ which is contained in $\lambda \cap {B_{\delta ^{(1)}}(x)}$. Moreover, we may choose $\sigma$ so that $\arg (p_u)$ is strictly increasing or strictly decreasing along $\sigma$, and parameterized according to $\arg (p_u)$.

Proof

Since $\Theta ^{-1}(v)$ is finite ([1]), we need only show the result for some fixed $u\in \Theta ^{-1}(v)$. Let $u\in \Theta ^{-1}(v)$ and $\lambda$ be one of the critical level curves of $(p_u,G_{p_u})$, (with $|f|\equiv \epsilon >0$ on $\lambda$). Let $x\in \lambda$ be a given critical point of $p_u$. Let $k\in \mathbb {N}$ denote the multiplicity of $x$ as a zero of ${p_u}'$. Then, there is some neighborhood $D\subset {B_{\delta ^{(1)}}(x)}$ of $x$ and $S>0$ and conformal map $\phi :D\rightarrow {B_S(p_u(0))}$ such that $p_u(z)=\phi (z)^{k+1}+p_u(x)$ for all $z\in {D}$. Define $f(w):={w}^{k+1}+p_u(x)$. The level curves of $f$ are well understood. Let $L$ denote the level curve of $f$ which contains $0$. Then if $w\in {L}$, there is a path in $L$ from $w$ to $0$ which is contained in $B_{|w|}(0)$, along which $\arg (f)$ is either strictly increasing or strictly decreasing. Choose some $r>0$ such that $B_{r}(x)\subset {D}$. Let $y\in {B_{r}(x)}$ be any point such that $|p_u(y)|=\epsilon$. Then $\phi (y)\in L$. Let $\sigma ^{(1)}$ denote the path in $B_{|\phi (y)|}(0)$ from $\phi (y)$ to $0$ along which $\arg (f)$ is strictly increasing or strictly decreasing. Then, if we define $\sigma :=\phi ^{-1}\circ \sigma ^{(1)}$, $\sigma \subset \phi ^{-1}(B_{|\phi (y)|}(0))\subset {D}\subset {B_{\delta ^{(1)}}(x)}$, and for each $t\in [0,1]$, $p_u(\sigma (t))=f(\sigma ^{(1)}(t))$, so $\arg (p_u)$ is either strictly increasing or strictly decreasing along $\sigma$.

Since $p_u$ has only finitely many critical points in $\lambda$, we may choose $\delta ^{(2)}>0$ to be smaller than the $r$ chosen above for each critical point $x$ of $p_u$ in $\lambda$, and this $\delta ^{(2)}$ has the desired property. $\square$

Lemma 8.5

Let $v\in {V_{n-1}}$, and $\delta ^{(1)}>0$ be given. There exists some $\delta ^{(2)}\in (0,\delta ^{(1)})$ such that the following holds. Let $u\in \Theta ^{-1}(v)$ be given. Let $\lambda$ be a critical level curve of $(p_u,G_{p_u})$ (with $|p_u|\equiv \epsilon >0$ on $\lambda$) and $x\in \lambda$ be a critical point of $p_u$. Then if $y\in {B_{\delta ^{(2)}}(x)}$ satisfies $\arg (p_u(y))=\arg (p_u(x))=0$, then there is a path $\sigma$ from $y$ to $x$ which is contained in $B_{\delta ^{(1)}}(x)$ and such that $\arg (p_u(\sigma (r)))=\arg (p_u(x))$ for all $r$. Moreover, we may choose $\sigma$ so that $|p_u|$ is strictly increasing or strictly decreasing along $\sigma$, and parameterized according to $|p_u|$.

Proof

Essentially the same argument for Lemma 8.4 works here. $\square$

Lemma 8.6

Given any GFBP $(f,G)$ and $\eta >0$, and any compact set $G'\subset {G}$ which does not contain any critical points of $f$, there exists $\tau >0$ such that if $g$ is analytic on $G$, and $|f(z)-g(z)|<\tau$ for all $z\in {G}$, then the following holds:

• 1. If $z^{(0)}\in {G'}$, and $w^{(1)}\in {B_{\tau }(f(z^{(0)}))}$, then there is a point $z^{(1)}\in {B_{\eta }(z^{(0)})}$ such that $g(z^{(1)})=w^{(1)}$. $($In particular, we may put $w^{(1)}=f(z^{(0)}).)$

• 2. If $z^{(0)}\in {G'}$ and $w^{(1)}\in {B_{\tau }(g(z^{(0)}))}$, then there is a point $z^{(1)}\in {B_{\eta }(z^{(0)})}$ such that $f(z^{(1)})=w^{(1)}$. $($In particular, we may put $w^{(1)}=g(z^{(0)}).)$

Proof

This follows from elementary properties of an analytic function of one complex variable, including primarily the maximum modulus principle.

Definition

For $u\in \mathbb {C}^{n-1}$, if $\gamma :[0,1]\rightarrow \mathbb {C}$ is a path, and $0<a<b<1$, then for $0<\rho <\delta$, we say that $\gamma$ takes an $(\rho ,\delta )$ trip on $[a,b]$ with respect to $p_u$ if the following holds.

• There is some $\iota >0$ such that for all $r\in (a-\iota ,a)\cup (b,b+\iota )$, $\gamma (r)$ is less than $\rho$ away from some critical point of $p_u$ (possibly different critical points of $p_u$ for different values of $r$).

• For each $r\in (a,b)$, $\gamma (r)$ is greater than or equal to $\rho$ away from every critical point of $p_u$.

• There is some $r\in (a,b)$ such that $\gamma (r)$ is greater than $\delta$ away from every critical point of $p_u$.

Lemma 8.7

Fix some $v=(v^{(1)},\ldots ,v^{(n-1)})\in {V_{n-1}}$ not the zero vector, and $\delta ^{(1)}>0$. Then, there exists a constant $\rho >0$ such that for each $u\in \Theta ^{-1}(v)$, the following holds. Fix some $\widehat{u}\in {B_{\rho }}(u)$ such that, if we define $\widehat{v}=(\widehat{v^{(1)}},\ldots ,\widehat{v^{(n-1)}}):=\Theta (\widehat{u})$, then $\arg (\widehat{v^{(k)}})=\arg (v^{(k)})$ for each $k\in \{1,\ldots ,n-1\}$. For some $k\in \{1,\ldots ,n-1\}$ with $|v^{(k)}|\ne 0$, let $\widehat{\lambda }$ denote the level curve of $p_{\widehat{u}}$ which contains $\widehat{u^{(k)}}$. Let $\widehat{E}$ denote some edge of $\widehat{\lambda }$ which is incident to $\widehat{u^{(k)}}$, and $\widehat{\gamma }$ denote a parameterization of $\widehat{E}$ according to $\arg (p_{\widehat{u}})$ beginning with $\widehat{u^{(k)}}$, with domain $[\alpha ,\beta ]$ (where $\alpha <\beta$ if $\arg (p_{\widehat{u}})$ is increasing on $\widehat{E}$, and $\alpha >\beta$ otherwise). Then if we let $\lambda$ denote the critical level curve of $p_u$ containing $u^{(k)}$, there is a path $\gamma :[\alpha ,\beta ]\rightarrow \lambda$ such that $\gamma (\alpha )=u^{(k)},$ and for each $t\in [\alpha ,\beta ]$, $\arg (p_u(\gamma (t)))=t$ and $|\gamma (t)-\widehat{\gamma }(t)|<\delta ^{(1)}$.

Proof

We will show that the result of the lemma holds for any fixed $u\in \Theta ^{-1}(v)$, which will suffice because $\Theta ^{-1}(v)$ is finite by [1]. Broadly speaking, the idea of the proof is that close to any critical point of $p_u$, $\gamma$ can be defined using Lemma 8.4, and far from the critical points of $p_u$, $\gamma$ can be defined using Lemma 8.6. The notion of a $(\rho ,\delta )$ trip defined above is how we will quantify “close” and “far” from the critical points of $p_u$.

Reduce $\delta ^{(1)}>0$ if necessary so that $\delta ^{(1)}<\dfrac{\text {mindiff}({u_1}^{(1)},\ldots ,{u_1}^{(n-1)})}{2}$ and for each $l\in \{1,\ldots ,n-1\}$ with $|v^{(l)}|\ne 0$, if $|z-u^{(l)}|<\delta ^{(1)}$, then $|p_u(z)-v^{(l)}|<\frac{\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)}{4}$. Of course $\frac{\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)}{4}\le \frac{|v^{(l)}|}{4}$, so by geometry, if $|z-u^{(l)}|<\delta ^{(1)}$ then $|\arg (p_u(z))-\arg (v^{(l)})|<\frac{\pi }{4}$. Note that this also implies that if $u^{(l)}$ is a critical point of $p_u$ at which $p_u\ne 0$, and $\sigma$ is a path contained in $B_{\delta ^{(1)}}(u^{(l)})$, then $\Delta _{\arg }(p_u,\sigma )<\frac{\pi }{2}$.

By Lemma 8.4, we may choose $\delta ^{(2)}\in (0,\frac{\delta ^{(1)}}{4})$ such that the following holds. If $y\in {B_{\delta ^{(2)}}(u^{(l)})}$ for some $l\in \{1,\ldots ,n-1\}$ such that $|p_u(y)|=|v^{(l)}|\ne 0$, then there is a path $\sigma$ from $y$ to $u^{(l)}$ contained in $B_{\frac{\delta ^{(1)}}{2}}(u^{(l)})\cap {E_{p_u,|v^{(l)}|}}$ such that $\arg (p_u)$ is strictly monotonic along $\sigma$.

Since $p_u$ is an open mapping, we may choose some $M>0$ small enough so that for each $k\in \{1,\ldots ,n-1\}$, $B_{2M}(v^{(k)})\subset {p_u}(B_{\delta ^{(2)}}(u^{(k)}))$. By Lemma 8.3, we may choose a $\rho ^{(1)}>0$ so that $\rho ^{(1)}<\frac{\delta ^{(2)}}{2}$, and if $\widehat{u}\in {B_{\rho ^{(1)}}(u)}$, then $|p_u(z)-p_{\widehat{u}}(\widehat{z})|<M$ for all $z,\widehat{z}\in {G_{p_u}}$ such that $|z-\widehat{z}|<\rho ^{(1)}$.

Let $K$ denote the set of all points $x$ in $G_{p_u}$ such that the following holds.

• $x\in {E_{p_u,|v^{(k)}|}}$ for some $k\in \{1,\ldots ,n-1\}$.

• $|x-u^{(k)}|\ge \frac{\delta ^{(2)}}{2}$ for each $k\in \{1,\ldots ,n-1\}$.

By the compactness of $K$, we may choose an $\eta >0$ such that the following holds.

• $\eta <\min (d(\{z\},\partial {G_{p_u}}):z\in {K})$.

• $p_u$ is injective on $B_{\eta }(x)$ for each $x\in {K}$. (Since $K$ does not contain any critical point of $p_u$.)

• $\eta <\rho ^{(1)}$. (Thus $|x-u^{(l)}|>\eta$ for each $l\in \{1,\ldots ,n-1\}$, since $\rho ^{(1)}<\frac{\delta ^{(2)}}{2}$.)

Define $G':=\{x\in {G_{p_u}}:d(x,\partial {G_{p_u}})\ge \eta ,d(x,u^{(k)})\ge \eta \text { for each }k\}$. By Lemma 8.6, we may choose $\tau >0$ so that $\tau <\min (M,\frac{\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)}{2})$, and if $f$ is analytic on $G'$ with $|f-p_u|<\tau$ on $G'$, then for all $x$ in $G'$, the following holds.

• For any $w\in {B_{\tau }(p_u(x))}$, there is a $y\in {B_{\eta }(x)}$ with $f(y)=w$.

• For any $w\in {B_{\tau }(f(x))}$, there is a $y\in {B_{\eta }(x)}$ with $p_u(y)=w$.

By Lemma 8.3 and the continuity of $\Theta$, we may choose $\rho \in (0,\rho ^{(1)})$ so that if $\widehat{u}\in {B_{\rho }(u)}$, then $|p_u(z)-p_{\widehat{u}}(\widehat{z})|<\tau$ for all $z,\widehat{z}\in {G_{p_u}}$ such that $|z-\widehat{z}|<\rho$, and for $\widehat{v}=(\widehat{v^{(1)}},\ldots ,\widehat{v^{(n-1)}}):=\Theta (\widehat{u})$, $|\widehat{v}-v|<\tau$. We now show that the statement of the lemma holds for the chosen $\rho$.

Let $\widehat{u}\in {B_{\rho }}(u)$ be chosen. Fix some $k\in \{1,\ldots ,n-1\}$ such that $|v^{(k)}|\ne 0$. Note that since $\tau <\frac{\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)}{2}$, and $|v-\widehat{v}|<\tau$, we have $|\widehat{v^{(k)}}|>\frac{\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)}{2}$. Let $\widehat{\lambda }$ denote the level curve of $p_{\widehat{u}}$ which contains $\widehat{u^{(k)}}$. Let $\widehat{E}$ denote some edge of $\widehat{\lambda }$ which is incident to $\widehat{u^{(k)}}$. Let $\alpha$ denote some choice of the argument of $p_{\widehat{u}}(\widehat{u^{(k)}})$, and $\widehat{\gamma }:[\alpha ,\beta ]\rightarrow \widehat{\lambda }$ be a path which parameterizes $\widehat{E}$ according to the argument of $p_{\widehat{u}}$ beginning with $\widehat{u^{(k)}}$. (Here, we assume that $\arg (p_{\widehat{u}})$ is increasing as $\widehat{E}$ is traversed beginning with $\widehat{u^{(k)}}$, and thus $\beta >\alpha$. Otherwise make the appropriate minor changes.)

We will now define a path $\gamma$ with domain $[\alpha ,\beta ]$ which has the desired properties. We first identify the sub-intervals of $[\alpha ,\beta ]$ on which $\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}{/}2)$ trip (with respect to $p_u$). On these sub-intervals we will define $\gamma$ in one way, and on the intervening sub-intervals we will define $\gamma$ in another way. Note that by the definition of a $(\rho ^{(1)},\delta ^{(1)}{/}2)$ trip over an interval, if $\widehat{\gamma }$ takes $(\rho ^{(1)},\delta ^{(1)}/2)$ trips over two sub-intervals $I^{(1)},I^{(2)}\subset [\alpha ,\beta ]$, then either $I^{(1)}=I^{(2)}$, or $I^{(1)}$ and $I^{(2)}$ are disjoint. Therefore since $\widehat{\gamma }$ is a rectifiable path, and any sub-interval on which $\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}/2)$ trip must have length at least $\dfrac{\delta ^{(1)}}{2}-\rho ^{(1)}$, $\widehat{\gamma }$ takes at most finitely many distinct $(\rho ^{(1)},\delta ^{(1)}/2)$ trips.

Case 8.0.1

$\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}/2)$ trip on some sub-interval of $[\alpha ,\beta ]$.

Let $[r^{(1)},s^{(1)}],\ldots ,[r^{(N)},s^{(N)}]\subset [\alpha ,\beta ]$ be the disjoint sub-intervals of $[\alpha ,\beta ]$ over which $\gamma$ takes $(\rho ^{(1)},\delta ^{(1)}/2)$ trips, ordered so that $s^{(k)}<r^{(k+1)}$ for each $k\in \{1,\ldots ,N-1\}$. Fix for the moment some $j\in \{1,\ldots ,N\}$ and some $r\in [r^{(j)},s^{(j)}]$.

For all $t\in [r^{(j)},s^{(j)}]$, define $w(t)\!:=\!|v^{(k)}|e^{it}$. Then since $\tau <\text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|)$, $|w(t)-p_{\widehat{u}}(\widehat{\gamma }(t))|<\tau$, so there is some $y\in {B_{\eta }(\widehat{\gamma }(r))}$ such that $p_u(y)=w(r)$. Moreover, since $p_u$ is injective in $B_{\eta }(\widehat{\gamma }(r))$, this choice of $y$ is unique. Define $\gamma (r)=y$.

Since $p_u$ is injective on $B_{\eta }(\widehat{\gamma }(r))$ for each $r\in [r^{(j)},s^{(j)}]$, and $p_u$ is an open mapping, it is easy to show that $\gamma$ is a continuous function, and thus a path from $\gamma (r^{(j)})$ to $\gamma (s^{(j)})$. Further, if $r\in [r^{(j)},s^{(j)}]$, $|p_u(\gamma (r))|=|w(r)|=|v^{(k)}|$. Therefore, we conclude that $\gamma |_{[r^{(j)},s^{(j)}]}$ is a path in $E_{p_u,|v^{(k)}|}$, and by construction, for each $r\in [r^{(j)},s^{(j)}]$, $|\widehat{\gamma }(r)-\gamma (r)|<\eta$ and $\arg (p_u(\gamma (r)))=r$. Having done this for each $j\in \{1,\ldots ,N\}$, we now wish to define $\gamma$ on $(s^{(j)},r^{(j+1)})$ for each $j\in \{1,\ldots ,N-1\}$.

Again fix for the moment some new $j\in \{1,\ldots ,N\}$. Since there is no sub-interval of $(s^{(j)},r^{(j+1)})$ on which $\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}/2)$ trip, $\widehat{\gamma }(r)$ is within $\delta ^{(1)}/2$ of some critical point of $p_u$ for each $r\in (s^{(j)},r^{(j+1)})$. However $\delta ^{(1)}<\frac{\text {mindiff}(u)}{2}$, thus there is some unique $l\in \{1,\ldots ,n-1\}$ such that for each $r\in (s^{(j)},r^{(j+1)})$, $|\widehat{\gamma }(r)-u^{(l)}|\le \delta ^{(1)}/2$. Since $\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}/2)$ trip over $[r^{(j)},s^{(j)}]$, $|\widehat{\gamma }(s^{(j)})-u^{(l)}|=\rho ^{(1)}$. Therefore

$$\begin{aligned} |\gamma (s^{(j)})-u^{(l)}|\le |\gamma (s^{(j)})-\widehat{\gamma }(s^{(j)})|+|\widehat{\gamma }(s^{(j)})-u^{(l)}|<\eta +\rho ^{(1)}<\delta ^{(2)}. \end{aligned}$$

In addition to this, $|p_u(\gamma (s^{(j)}))|=|v^{(k)}|$, so by choice of $\delta ^{(1)}$,

$$\begin{aligned} ||v^{(k)}|-|v^{(l)}||= & {} ||p_u(\gamma (s^{(j)}))|-|v^{(l)}||<|p_u(\gamma (s^{(j)}))-v^{(l)}|\\< & {} \text {mindiff}(0,|v^{(1)}|,\ldots ,|v^{(n-1)}|). \end{aligned}$$

Therefore, we conclude that $|p_u(\gamma (s^{(j)}))|=|v^{(l)}|=|v^{(k)}|$. Then by choice of $\delta ^{(2)}$, there is some path $\sigma ^{(1)}$ from $\gamma (s^{(j)})$ to $u^{(l)}$ contained in $B_{\frac{\delta ^{(1)}}{2}}(u^{(l)})\cap {E_{p_u,|v^{(l)}|}}$ such that $\arg (p_u)$ is strictly monotonic on $\sigma ^{(1)}$, and $\sigma ^{(1)}$ is parameterized according to $\arg (p_u)$. Since $\arg (p_{\widehat{u}})$ is increasing along $\widehat{\gamma }$, $\arg (p_u)$ is increasing along the portions of $\gamma$ which have already been defined. Let $D$ denote an open region containing $\gamma (s^{(j)})$ on which $p_u$ is injective. Choose some $t^{(0)}\in (r^{(j)},s^{(j)})$ such that $\gamma (t^{(0)},s^{(j)})\subset {D}$. If $\arg (p_u)$ is decreasing on $\sigma ^{(1)}$, then since $p_u$ is injective on $D$, for each $r\in (s^{(j)},t^{(0)})$, $\sigma ^{(1)}(r)=\gamma (r)$. Furthermore, since $p_u$ is injective in a neighborhood of each point of $\gamma ([r^{(j)},s^{(j)}])$, $\sigma ^{(1)}$ must continue to trace back along the entire length of $\gamma ([r^{(j)},s^{(j)}])$. This is because both $\sigma ^{(1)}$ and $\gamma$ are parameterized according to $\arg (p_u)$, so any branching off of $\sigma ^{(1)}$ from $\gamma$ would have to be a critical point of $p_u$. However, $\sigma ^{(1)}$ may not trace back along $\gamma ([r^{(j)},s^{(j)}])$ because the image of $\sigma ^{(1)}$ is contained in $B_{\frac{\delta ^{(1)}}{2}}(u^{(l)})$. Therefore, we conclude that $\arg (p_u)$ is increasing on $\sigma ^{(1)}$.

By very similar reasoning we may obtain a path $\sigma ^{(2)}$ from $u^{(l)}$ to $\gamma (r^{(j+1)})$ contained in $B_{\frac{\delta ^{(1)}}{2}}(u^{(l)})\cap {E_{p_u,|v^{(l)}|}}$ parameterized according to $\arg (p_u)$, and along which $\arg (p_u)$ is increasing. Moreover, the choice of $\delta ^{(1)}$ may now be used to show that the concatenation of these two paths may be assumed to have domain $(s^{(j)},r^{(j+1)})$. Therefore, we define $\gamma (r):=\sigma (r)$ for each $r\in (s^{(j)},r^{(j+1)})$. With this definition, we have that for each $r\in (s^{(j)},r^{(j+1)})$, $\arg (p_u(\gamma (r)))=r$, and

$$\begin{aligned} |\gamma (r)-\widehat{\gamma (r)}|\le |\gamma (r)-u^{(l)}|+|u^{(l)}-\widehat{\gamma (r)}|<\dfrac{\delta ^{(1)}}{2}+\dfrac{\delta ^{(1)}}{2}=\delta ^{(1)}. \end{aligned}$$

We extend $\gamma$ in this manner to $(s^{(j)},r^{(j+1)})$ for each $j\in \{1,\ldots ,N-1\}$. Moreover, we may extend $\gamma$ using the exactly similar construction to $[\alpha ,r^{(1)})$ and $(s^{(N)},\beta ]$, and this extended $\gamma$ has all of the desired properties.

Case 8.0.2

There is no sub-interval of $[\alpha ,\beta ]$ along which $\widehat{\gamma }$ takes a $(\rho ^{(1)},\delta ^{(1)}/2)$ trip.

Then either $|\widehat{\gamma }(r)-u^{(k)}|\le \delta ^{(1)}/2$ for all $r\in [\alpha ,\beta ]$, or there is some $r^{(0)}\in (\alpha ,\beta )$ such that for all $r\in [\alpha ,r^{(0)}]$, $|\widehat{\gamma }(r)-u^{(k)}|\le \delta ^{(1)}/2$, and for all $r\in (r^{(0)},\beta ]$, $\widehat{\gamma }$ is greater than $\rho ^{(1)}$ from any critical point of $p_u$.

Sub-case 8.7.2.1

$|\widehat{\gamma }(r)-u^{(k)}|\le \delta ^{(1)}/2$ for all $r\in [\alpha ,\beta ]$.

In this case, we construct $\gamma$ using the same method as in the second part of Case 8.0.1.

Sub-case 8.7.2.2

There is some $r^{(0)}\in (\alpha ,\beta )$ such that for all $r\in [\alpha ,r^{(0)}]$, $|\widehat{\gamma }(r)-u^{(k)}|\le \delta ^{(1)}/2$, and for all $r\in (r^{(0)},\beta ]$, $\widehat{\gamma }$ is greater than $\rho ^{(1)}$ from any critical point of $p_u$.

In this case, we construct $\gamma$ on $[\alpha ,r^{(0)})$ using the same method as in the second part of Case 8.0.1 and construct $\gamma$ on $[r^{(0)},\beta ]$ using the same method as in the first part of Case 8.0.1. $\square$

Lemma 8.8

Fix some $v=(v^{(1)},\ldots ,v^{(n-1)})\in {V_{n-1}}$ not the zero vector, and $\delta ^{(1)}>0$. Then, there exist constants $\rho ,\delta ^{(2)}>0$ such that the following holds. Let $u\in \Theta ^{-1}(v)$ be chosen, and fix some $\widehat{u}\in {B_{\rho }}(u)$. Let $\widehat{x_1},\widehat{x_2}\in {G}_{p_{\widehat{u}}}$ be given such that $\arg (p_u(x_1))=\arg (p_u(x_2))=0$, and such that there is a path $\widehat{\sigma }:[0,1]\rightarrow {G_{p_u}}$ such that $\widehat{\sigma (0)}=\widehat{x_1}$ and $\widehat{\sigma (1)}=\widehat{x_2}$ and $\arg (p_{\widehat{u}}(\widehat{\sigma }(r)))=0$ for all $r\in [0,1]$. Then if $x_1,x_2\in {G_{p_{\widehat{u}}}}$ are such that $\arg (p_u(x_1))=\arg (p_u(x_1))=0$ and $|\widehat{x_1}-x_1|<\delta ^{(2)}$ and $|\widehat{x^{(2)}}-x^{(2)}|<\delta ^{(2)}$, then there is a path $\sigma :[0,1]\rightarrow {G_{p_u}}$ such that $\sigma (0)=x_1$, $\sigma (1)=x_2$, and for all $r\in [0,1]$, $\arg (p_u(\sigma (r)))=0$ and $|\widehat{\sigma }(r)-\sigma (r)|<\delta ^{(1)}$. Moreover, if $|p_{\widehat{u}}|$ is strictly increasing or strictly decreasing on $\widehat{\sigma }$, then we may assume that $|p_u|$ is strictly increasing or strictly decreasing on $\sigma ,$ respectively.

Proof

The exact same method of proof used for Lemma 8.7 works here except that instead of invoking Lemma 8.4, we would invoke the gradient line version Lemma 8.5. $\square$