$\breve{v}$ does not change sign either, so that $\rho(v,s)\neq(2k-3)$.
Applying the local structure theorem to $v$ at $s$, and using the same proof
as in Lemma 9.2, Part (C), (with the disc $D^{+}(s,r_{0})$ and circle
$C^{+}(s,r_{0})$ centered at $s$), we infer that $\rho(v,s)=(2k-2)$. Indeed,
the nodal sets $\mathcal{Z}(v_{n})$ consist either is $(k-2)$ loops at $y_{n}$
(including a special loop touching $s$), or $(k-3)$ loops at $y_{n}$ and a
loop at $s$. Since $y_{n}$ tends to $s$, for $r_{0}$ small enough, these loops
must intersect $C^{+}(s;r_{0})$ at $(2k-2)$ distinct points, and we can
conclude as in the proof of Lemma 9.2.

To prove the second statement, we can now choose a sequence
$\left\\{y_{n}\right\\}$ such that $\rho(v_{n},s)=1$ for $n$ large enough, so
that $\mathcal{S}_{b}(v_{n})=\left\\{y_{n},s,s^{\prime}_{n}\right\\}$, with
$s^{\prime}_{n}\neq s$. An argument similar to the previous one, shows that no
subsequence of $\left\\{s^{\prime}_{n}\right\\}$ can tend to $s$. Since
$s\in\Gamma_{(2k-3)}$, the only remaining possibility is that
$\rho(v,s)=(2k-3)$, and hence that $v\in W_{s}$. Assertion (iv) is proved.

The proof of Lemma 9.11 is complete. ∎

Figure 9.16. Proof of Lemma 9.11(iv)

We can enhanced the previous lemma by the following properties.

######  Properties 9.12.

Assume that $\Omega$ is simply connected, and that $\dim U=(2k-2)$. Given
$(y,s)\in\Gamma_{(2k-3)}\times\Gamma$ with $y\neq s$, there exists
$\varepsilon_{0}$ such that

  1. (i)

for all
$s\in\mathcal{A}_{c}(z(y);\varepsilon_{0})\cup\mathcal{A}_{c}(y;\varepsilon_{0})\\!\setminus\\!\left\\{y,z(y)\right\\}$,
$\rho(v_{y,s},s)=1$;

  2. (ii)

for all $\varepsilon<\varepsilon_{0}$, there exists $\eta>0$ such that for all
$s\in\mathcal{A}_{c}(z(y);\eta)\\!\setminus\\!\left\\{z(y)\right\\}$,
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}(y;\varepsilon)\\!\setminus\\!\left\\{y\right\\}$;

  3. (iii)

for all $\varepsilon<\varepsilon_{0}$, there exists $\eta>0$ such that for all
$s\in\mathcal{A}_{c}(y;\eta)\\!\setminus\\!\left\\{y\right\\}$,
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}_{c}(z(y);\varepsilon)\\!\setminus\\!\left\\{z(y)\right\\}$.

Let $s_{1}\not\in\left\\{y,z(y)\right\\}$. Assume that the function
$v_{y,s_{1}}$ satisfies $\rho(v_{y,s_{1}},s_{1})=1$, ie
$\mathcal{S}_{b}(v_{y,s_{1}})=\left\\{y,s_{1},s^{\prime}_{1}\right\\}$, with
$s^{\prime}_{1}\neq s_{1}$. Then,

  1. (a)

there exists $\varepsilon_{1}>0$, such that for all
$s\in\mathcal{A}_{c}(s_{1};\varepsilon)$, $\rho(v_{y,s},s)=1$;

  2. (b)

for all $\varepsilon>0$, there exists $\eta<\varepsilon_{1}$ such that for all
$s\in\mathcal{A}_{c}(s_{1};\eta)$,
$s^{\prime}\in\mathcal{A}_{c}(s^{\prime}_{1};\varepsilon)$, i.e. the map
$s\mapsto s^{\prime}$ is continuous.

######  Lemma 9.13.

Assume that $\Omega$ is simply connected, and that $\dim U=(2k-2)$. Let
$y\in\Gamma_{(2k-3)}$. Then, the following properties hold.

  1. (i)

For any $s\in\mathcal{A}(y,z(y))$, the function $v_{y,s}$ satisfies
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}(y,z(y))$, possibly with $s=s^{\prime}$.

  2. (ii)

There exists a unique $s_{1}\in\mathcal{A}(y,z(y))$ such that $v_{y,s_{1}}$
satisfies $\rho(v_{y,s_{1}},s_{1})=2$.

  3. (iii)

For all $s\in\mathcal{A}(s_{1},z(y))$,
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}(y,s_{1})$. Furthermore when $s$ moves counter-
clockwise in $\mathcal{A}(s_{1},z(y))$, $s^{\prime}$ moves clockwise in
$\mathcal{A}(y,s_{1})$.

  4. (iv)

For all $s\in\mathcal{A}(y,s_{1})$,
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}(s_{1},z(y))$. Furthermore when $s$ moves counter-
clockwise in $\mathcal{A}(y,s_{1})$, $s^{\prime}$ moves clockwise in
$\mathcal{A}(s_{1},z(y))$.

Similar statements hold for the arc $\mathcal{A}(z(y),y)$.

The statements in Lemma 9.13 are illustrated in Figure 9.17 (for the arc
$\mathcal{A}(y,z(y))$). The corresponding nodal patterns appear in Figure
9.22.

Figure 9.17.  Lemma 9.13: Assertions (ii) and (iii)

_Proof._ Choose a generator $w_{y}$ of $W_{y}$ such that $\breve{w}_{y}$ is
positive in $\mathcal{A}(y,z(y))$, and negative in $\mathcal{A}(z(y),y)$.

_Assertion (i)._ Assume that the assertion is false: there exists some
$s_{0}\in\mathcal{A}(y,z(y))$ such that $v_{0}:=v_{y,s_{0}}$ satisfies
$\mathcal{S}_{b}(v_{0})=\left\\{y,s_{0},s^{\prime}_{0}\right\\}$, with
$s^{\prime}_{0}\in\mathcal{A}(z(y),y)$. Since $s_{0}\neq s^{\prime}_{0}$,
Lemma 9.11(i) implies that $\breve{v}_{0}$ vanishes on $\partial\Omega$ only
at the points $y$, $s_{0}$ and $s^{\prime}_{0}$, does not change sign at $y$,
and changes sign at $s_{0}$ and $s^{\prime}_{0}$. Choose $v_{0}$ such that
$\breve{v}_{0}<0$ in $\mathcal{A}(s_{0},s^{\prime}_{0})$. For $s\neq y,z(y)$,
introduce the function

$\xi_{s}=a_{0}(s)w_{y}-b_{0}(s)v_{0}\,,$

where

$\left\\{\begin{array}[]{l}a_{0}(s)=\breve{v}_{0}(s)\left(\breve{v}_{0}^{2}(s)+\breve{w}_{y}^{2}(s)\right)^{-\frac{1}{2}}\,,\\\\[5.0pt]
b_{0}(s)=\breve{w}_{y}(s)\left(\breve{v}_{0}^{2}(s)+\breve{w}_{y}^{2}(s)\right)^{-\frac{1}{2}}\,.\end{array}\right.$

Since $b_{0}(s)\neq 0$, $\rho(\xi_{s},y)=2k-4$ and $\rho(\xi_{s},s)\geq 1$, so
that $\xi_{s}\in V_{y,s}$.

Choose $s\in\mathcal{A}(s_{0},z(y))$. Since $\breve{v}_{0}$ only changes sign
at $s_{0}$ and $s^{\prime}_{0}$, $\breve{\xi}_{s}(s^{\prime}_{0})>0$. By Lemma
3.14, at $y$ along $\partial\Omega$, $\breve{w}_{y}$ vanishes at order
$(2k-3)$, while $\breve{v}_{0}$ vanishes at order $(2k-4)$. In a pointed
neighborhood $J\\!\setminus\\!\left\\{y\right\\}$ of $y$ in $\partial\Omega$,
we have that $\breve{\xi}_{s}\sim-b_{0}(s)\breve{v}_{0}<0$, and hence
$\breve{\xi}_{s}$ vanishes at some
$s^{\prime}\in\mathcal{A}(s^{\prime}_{0},y)$. It follows that $\xi_{s}\in
V_{y,s}$ with $\mathcal{S}_{b}(\xi_{s})=\left\\{y,s,s^{\prime}\right\\}$,
$\rho(\xi_{s},s)=1$, $\rho(\xi_{s},s^{\prime})=1$.

Similarly, choosing $t\in\mathcal{A}(y,s_{0})$, we have
$\breve{\xi}_{t}(s^{\prime}_{0})<0$ and $\breve{\xi}_{t}(z(y))>0$, and
$\breve{\xi}_{t}$ vanishes at some
$t^{\prime}\in\mathcal{A}(z(y),s^{\prime}_{0})$.
Finally, we conclude as above that $\xi_{t}\in V_{y,t}$ with

$\mathcal{S}_{b}(\xi_{t})=\left\\{y,t,t^{\prime}\right\\}\,,\,\rho(\xi_{t},t)=1\,,\,\rho(\xi_{t},t^{\prime})=1\,.$

These arguments can be visualized on Figure 9.18.

Figure 9.18. Proof of Lemma 9.13(i)

From the assumed existence of $s_{0}$, we conclude that, for all
$s\in\mathcal{A}(y,z(y))$, $\xi_{s}\in V_{y,s}$ and
$\mathcal{S}_{b}(\xi_{s})=\left\\{y,s,s^{\prime}\right\\}$, with
$s^{\prime}\in\mathcal{A}(z(y),y)$, $\rho(\xi_{s},s)=1$,
$\rho(\xi_{s},s^{\prime})=1$. Because $s\in\mathcal{A}(s_{0},z(y))$ implies
that $s^{\prime}\in\mathcal{A}(s^{\prime}_{0},y)$, with a parallel statement
for $t$, the previous argument also shows that when the point $s$ moves
counter-clockwise in $\mathcal{A}(y,z(y))$, the point $s^{\prime}$ moves
counter-clockwise in $\mathcal{A}(z(y),y)$. According to Lemma 9.11,
Assertions (ii) and (iii), $\xi_{s}$ tends to $w_{y}$ when $s$ tends to $y$ or
to $z(y)$ in $\mathcal{A}(y,z(y))$. Looking at the behaviour of the nodal
sets, we reach a contradiction as Figure 9.19 shows. Assertion (i) is proved.
$\checkmark$

Figure 9.19. Proof of Lemma 9.13(i)

Remark. The previous arguments implicitly use the fact that the combinatorial
type of $v_{y,s}$ does not change when $y$ is fixed and $s$ varies in
$\Gamma\\!\setminus\\!\left\\{y\right\\}$ (the proof is similar to the proof
of Lemma 8.4(iv)), see Paragraph 9.2.2. Note also that the reasoning in Figure
9.19 is actually quite general, and only depends on the position of the arc
from $y$ to $z(y)$ with respect to the loops.

_Proof of Assertion (ii)._

Figure 9.20. Proof of Lemma 9.13(ii)

Let $\gamma_{1}:[0,\ell_{1}]\to\Gamma$ be an arc-length parametrization of
$\Gamma$, such that $\gamma_{1}(t)$ moves counter-clockwise, and
$\gamma_{1}(0)=\gamma_{1}(\ell_{1})=y$, $\gamma_{1}(\ell)=z(y)$.

According to Lemma 9.11(i), and to the above Assertion (i), taking
$s\in\mathcal{A}(y,z(y))$ close to $y$, we have
$\mathcal{S}_{b}(v_{y,s})=\left\\{y,s,s^{\prime}\right\\}$ with
$s^{\prime}\in\mathcal{A}(y,z(y))$ close to $z(y)$. If $s=\gamma_{1}(t)$ and
$s^{\prime}=\gamma_{1}(t^{\prime})$, for $t$ positive small enough, we have
$t<t^{\prime}$. Choose any such point $s_{0}=\gamma_{1}(t_{0})$ such that
$v_{0}=v_{y,s_{0}}$ satisfies
$\mathcal{S}_{b}(v_{0})=\left\\{y,s_{0},s^{\prime}_{0}\right\\}$, with
$s^{\prime}_{0}=\gamma_{1}(t^{\prime}_{0})$ and $t_{0}<t^{\prime}_{0}$. By
Lemma 9.11 (i), the function $\breve{v}_{0}$ vanishes precisely at the points
$s_{0}$ and $s^{\prime}_{0}$ and changes sign at these points. We choose it so
that it is positive on $\mathcal{A}(s_{0},s^{\prime}_{0})$. Define $\xi_{s}$
as in the proof of Assertion (i) (now with
$s_{0}^{\prime}\in\mathcal{A}(y,z(y))$). Take $s\in\mathcal{A}(y,s_{0})$,
$s=\gamma_{1}(t)$ with $0<t<t_{0}$. With our previous choice of signs for
$\breve{w}_{y}$, we find that $\breve{\xi}_{s}(z(y))>0$ and
$\breve{\xi}_{s}(s^{\prime}_{0})<0$, so that $\breve{\xi}_{s}$ vanishes at
some $s^{\prime}$ in $\mathcal{A}(s^{\prime}_{0},z(y))$. Since $\xi_{s}\in
V_{y,s}$, we have $\mathcal{S}_{b}(\xi_{s})=\left\\{y,s,s^{\prime}\right\\}$,
with $s\in\mathcal{A}(y,s_{0})$,
$s^{\prime}\in\mathcal{A}(s^{\prime}_{0},z(y))$,
$\rho(\xi_{s},s)=\rho(\xi_{s},s^{\prime})=1$. Then,
$s^{\prime}=\gamma_{1}(t^{\prime})$ with
$0<t<t_{0}<t^{\prime}_{0}<t^{\prime}<\ell$, so that the map $(0,t_{0})\ni
t\mapsto t^{\prime}$ is decreasing.

Introduce the set

$J:=\left\\{t\in(0,\ell)\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathcal{S}_{b}(v_{y,\gamma_{1}(t)})=\left\\{y,\gamma_{1}(t),\gamma_{1}(t^{\prime})\right\\}\text{\leavevmode\nobreak\
with\leavevmode\nobreak\ }t<t^{\prime}\right\\}\,.$

If $t\in J$,
$\rho(v_{y,\gamma_{1}(t)},\gamma_{1}(t))=\rho(v_{y,\gamma_{1}(t)},\gamma_{1}(t^{\prime}))=1$.
We have $(0,t_{0})\subset J$ so that $J\neq\emptyset$, and hence $s_{1}:=\sup
J$ exists. Since $\rho(v_{y,s},s)=1$ is an open condition, $s_{1}\not\in J$.
Take a subsequence $\left\\{t_{n}\right\\}\subset J$ tending to $s_{1}$, and
choose corresponding functions $v_{y,t_{n}}\in\mathbb{S}(U)\cap V_{y,t_{n}}$.
A subsequence converges to some function $v_{1}$ in $\mathbb{S}(U)$ which
satisfies $\rho(v_{1},y)\geq(2k-4)$ and $\rho(v_{1},s_{1})\geq 1$. By Lemma
9.8, this implies that $v_{1}\in V_{y,s_{1}}$ (use Remark 3.6). Since
$s_{1}\not\in J$, we must have $\rho(v_{1},s_{1})=2$, so that $v_{1}$ is a
generator $v_{y,s_{1}}$ of $V_{y,s_{1}}$.

_Proof of Assertions (iii) and (iv)._ Take $v_{1}=v_{y,s_{1}}$ given by
Assertion (ii). By Lemma 9.11 (i), we may choose this function such that
$\breve{v}_{1}\geq 0$ and vanishes only at $y$ and $s_{1}$. For $s\neq
y,z(y)$, introduce the functions,

$\xi_{s}=a_{1}(s)w_{y}-b_{1}(s)v_{1}\,,$

where

$\left\\{\begin{array}[]{l}a_{1}(s)=\breve{v}_{1}(s)\left(\breve{v}_{1}^{2}(s)+\breve{w}_{y}^{2}(s)\right)^{-\frac{1}{2}}\,,\\\\[5.0pt]
b_{1}(s)=\breve{w}_{y}(s)\left(\breve{v}_{1}^{2}(s)+\breve{w}_{y}^{2}(s)\right)^{-\frac{1}{2}}\,.\end{array}\right.$

Figure 9.21. Proof of Lemma 9.13, Assertion (iii)

When $s=s_{1}$, we have $\breve{\xi}_{s_{1}}=-b_{1}(s_{1})\breve{v}_{1}\leq
0$, and $\breve{\xi}_{s_{1}}$ only vanishes at $y$ and $s_{1}$;
$\xi_{s_{1}}\in V_{y,s_{1}}$. When
$s\in\mathcal{A}(y,z(y))\\!\setminus\\!\left\\{s_{1}\right\\}$, taking into
account the properties of the functions involved, we find that

$\left\\{\begin{array}[]{l}\breve{\xi}_{s}(z(y))<0\,,\\\\[5.0pt]
\breve{\xi}_{s}(s_{1})>0\,,\\\\[5.0pt]
\breve{\xi}_{s}<0\text{\leavevmode\nobreak\ in\leavevmode\nobreak\
}J_{y}\\!\setminus\\!\left\\{y\right\\}\,,\\\\[5.0pt] \end{array}\right.$

where $J_{y}$ is a small arc on $\Gamma_{1}$ centered at $y$.

It follows that, for
$s\in\mathcal{A}(y,z(y))\\!\setminus\\!\left\\{s_{1}\right\\}$, $\xi_{s}\in
V_{y,s}$, $\mathcal{S}_{b}(\xi_{s})=\left\\{y,s,s^{\prime}\right\\}$,
$\rho(\xi_{s},s)=\rho(\xi_{s},s^{\prime})=1$, with the point $s$ on one side
of $s_{1}$ in $\mathcal{A}(y,z(y))$, and the point $s^{\prime}$ on the other
side. For these points $s,s^{\prime}$, we have $V_{y,s}=V_{y,s^{\prime}}$, so
that $\xi_{s}$ and $\xi_{s^{\prime}}$, must be proportional (Lemma 9.8 (i)),
and since the functions $\breve{\xi}_{s}$ and $\breve{\xi}_{s^{\prime}}$ both
take a positive value at $s_{1}$, $\xi_{s^{\prime}}=a\,\xi_{s}$, with $a>0$.

We also have $\breve{\xi}_{s}(t)\breve{\xi}_{t}(s)\leq 0$, since

$\breve{\xi}_{s}(t)\breve{\xi}_{t}(s)=-\left(\breve{v}_{1}^{2}(s)+\breve{w}_{y}^{2}(s)\right)^{-\frac{1}{2}}\left(\breve{v}_{1}^{2}(t)+\breve{w}_{y}^{2}(t)\right)^{-\frac{1}{2}}\left(\breve{v}_{1}(s)\breve{w}_{y}(t)-\breve{w}_{y}(s)\breve{v}_{1}(t)\right)^{2}\,.$

Choose $s\in\mathcal{A}(s_{1},z(y))$, and $t\in\mathcal{A}(s,z(y))$. Then
$s^{\prime},t^{\prime}\in\mathcal{A}(y,s_{1})$. The functions
$\breve{\xi}_{s}$ and $\breve{\xi}_{t}$ are positive at $s_{1}$, and hence
positive respectively on the arcs $\mathcal{A}(s^{\prime},s)$ and
$\mathcal{A}(t^{\prime},t)$. We have $\breve{\xi}_{s}(t)<0$, and hence, using
the above properties, $\breve{\xi}_{s^{\prime}}(t)>0$, and
$\breve{\xi}_{t}(s^{\prime})<0$. This implies that
$t^{\prime}\in\mathcal{A}(y,s^{\prime})$. We have proved that when $s$ moves
counter-clockwise in $\mathcal{A}(s_{1},z(y))$, $s^{\prime}$ move clockwise in
$\mathcal{A}(y,s_{1})$. This is coherent with Lemma 9.11(ii). The proof of
Assertion (iii) is similar. ∎

######  Remarks 9.14.

(i) Lemma 9.13 corresponds to the first part of the proof of Lemma 3.5 in
[HoMN1999] (from p. 1181, line (-7), “We consider the function” to p. 1182,
line (+5), “the following nodal domain”).

Note however that, in Lemma 9.13, we do not assume that the point $z(y)$ is in
the same connected component of $\Gamma_{(2k-3)}$ as the point $y$.

(ii) Figure 9.22 displays the possible nodal patterns of $w_{y}$, and the
corresponding nodal patterns for the function $v_{y,s_{1}}$ with
$s_{1}\in\mathcal{A}(y,z(y))$ (see Lemma 9.13(ii)), and for the function
$v_{y,s}$ with $s\in\mathcal{A}(s_{1},z(y))$ (see Lemma 9.13(iii)).

Figure 9.22. Lemma 9.13 (ii) and (iii): nodal patterns for $w_{y}$ and
$v_{y,s_{1}}$

##  10\.  Possible steps towards a proof that
$\operatorname{mult}(\lambda_{k})\leq(2k-3)$ for $k\geq 3$

Let $U:=U(\lambda_{k})$. throughout this section, we assume that

(10.1) $\left\\{\begin{array}[]{l}\Omega\text{\leavevmode\nobreak\ is simply
connected,}\\\\[5.0pt] k\geq 3\text{\leavevmode\nobreak\ \leavevmode\nobreak\
and\leavevmode\nobreak\ \leavevmode\nobreak\ }\dim
U=(2k-2)\,.\end{array}\right.$

In [HoMN1999, Theorem B] the authors claim that under the above assumption,
the bound $\operatorname{mult}(\lambda_{k})\leq(2k-3)$ holds for $k\geq 3$
(see [Berd2018, Section 4] for comments on the non simply connected case).

The purpose of this section is to revisit their scheme of proof, and point out
some difficulties. We use the notation of the preceding sections.

###  10.1. Preamble

We fix a direct orthonormal frame $\left\\{\vec{e}_{1},\vec{e}_{2}\right\\}$
in $\mathbb{R}^{2}$, and the associated coordinates $\xi=(\xi_{1},\xi_{2})$.
Without loss of generality, we may assume that the length of
$\Gamma:=\partial\Omega$ is equal to $2\pi$. We parametrize $\Gamma$ by arc-
length, counter-clockwise, by some $\gamma_{1}:[0,2\pi]\to\Gamma_{1}$, with
unit tangent $\tau_{1}(t)$, and unit normal $\nu_{1}(t)$, so that the frame
$\left\\{\tau_{1}(t),\nu_{1}(t)\right\\}$ is direct and $\nu_{1}$ points
inside $\Omega$. For $\varepsilon$ small enough, we define the curve
$c(t,\varepsilon)=\gamma_{1}(t)+\varepsilon\,\nu_{1}(t)$.

We also fix $\left\\{\phi_{1},\ldots,\phi_{(2k-2)}\right\\}$, an orthonormal
basis of the eigenspace $U$. According to Lemma 8.1, we have a $C^{\infty}$
map $x\mapsto[u_{x}]$ from (the interior of) $\Omega$ to $\mathbb{P}(U)$,
where the function $u_{x}$ is a generator of the space $U_{x}$ defined by
(8.3). The function $u_{x}$ is given by

(10.2) $u_{x}=\sum_{j=1}^{(2k-2)}\alpha_{j}(x)\phi_{j}$

where the nonzero vector
$\alpha(x):=(\alpha_{1}(x),\ldots,\alpha_{(2k-2)}(x))$ is well defined up to
scaling, given locally by a linear system, and $C^{\infty}$. Since $\Omega$ is
simply connected, we can lift the map $x\mapsto\alpha(x)$ to a $C^{\infty}$
map from $\Omega$ to the unit sphere $\mathbb{S}^{(2k-3)}$ or, equivalently,
lift the map $x\mapsto[u_{x}]$ to a map $x\mapsto u_{x}$ from $\Omega$ to
$\mathbb{S}(U)$. The main property of this map $x\mapsto u_{x}$ is that
$\nu(u_{x},x)=(2k-2)$ for all $x\in\Omega$. The least order homogeneous term
in the Taylor expansion of the function $u_{x}$ near $x$ has degree $(k-1)$.
More precisely, according to Bers’s Theorem,

(10.3) $h_{x}(\xi):=\frac{d^{(k-1)}}{dt^{(k-1)}}u_{x}(x+t\xi)\big{|}_{t=0}$

is a nonzero, harmonic, homogeneous polynomial of degree $(k-1)$ in $\xi$. The
expression (10.2) shows that the map $x\mapsto h_{x}$ is $C^{\infty}$. Since
$h_{x}\neq 0$ for all $x$, we also obtain a map $\bar{h}$

(10.4) $\bar{h}:\left\\{\begin{array}[]{l}x\mapsto\bar{h}_{x}\\\\[5.0pt]
\Omega\to S_{\mathcal{H}}\end{array}\right.$

where $S_{\mathcal{H}}$ is the unit circle in the $2$-dimensional vector space
$\mathcal{H}$ of harmonic, homogenous polynomials of degree $(k-1)$ in
$\mathbb{R}^{2}$.

Let $\gamma$ be any Jordan curve in $\Omega$. Restricting the map $\bar{h}$ to
$\gamma$, we obtain a map $s\mapsto\bar{h}_{\gamma(s)}$ from $\gamma$ to
$S_{\mathcal{H}}$. This map has a degree $\deg(\bar{h},\gamma)$. Since
$\gamma$ is homotopic to a point, and since $\bar{h}$ is smooth, we conclude
that $\deg(\bar{h},\gamma)=0$.

In order to show that the assumption (10.1) is absurd, hence that
$\operatorname{mult}(\lambda_{k})\leq(2k-3)$ for any $k\geq 3$, it suffices to
show that $\deg(\bar{h},c(\cdot,\varepsilon))\neq 0$ for $\varepsilon$ small
enough. As we already pointed out in Remark 9.3, there is a difficulty here:
the map $x\mapsto u_{x}$ extends to the boundary (see Lemma 9.2), and so does
the map $x\mapsto h_{x}$. However, for $y\in\Gamma$, $\lim_{x\to y}h_{x}=0$,
and we have to look more closely at what happens near $\Gamma$, using
Assertions (ii) and (iii) of Lemma 9.2.

The steps to achieve the previous goal are given in the next subsection,
unfortunately with yet incomplete proofs.

###  10.2. Steps for the proof that
$\operatorname{mult}(\lambda_{k})\leq(2k-3)$ when $k\geq 3$

The following two lemmas do not appear in [HoMN1999] and seem to be implicitly
assumed.

######  Lemma 10.1.

Assume that $\Omega$ is simply connected. Assume that $k\geq 3$ and that $\dim
U=(2k-2)$. Then, $\Gamma_{(2k-2)}$ is not empty.

_Proof._ We are only able to give a partial proof.

_Claim 1. Assume that $\Gamma_{(2k-3)}=\Gamma$. Then,
$\inf_{y\in\Gamma}d(y,z(y))>0$._

Indeed, the map $y\mapsto z(y)$ (see Lemma 8.4) is continuous on $\Gamma$ and
$z(y)\neq y$. The result follows by compactness. Another argument is as
follows. Assume that $\inf_{y\in\Gamma}d(y,z(y))=0$. Then, there exists a
sequence $y_{n}$ such that $0<d(y_{n},z(y_{n}))<1/n$. To the sequence
$\left\\{y_{n}\right\\}$ we associate a sequence of functions
$\left\\{w_{n}:=w_{y_{n}}\right\\}\in\mathbb{S}(U)$ (Lemma 8.3). Up to taking
subsequences, we may assume that $y_{n}$ converges to some $\bar{y}$, and that
$w_{n}$ converges to some $w\in\mathbb{S}(U)$. Since $y\mapsto z(y)$ is
continuous, $z(y_{n})$ converges to $z(\bar{y})$, and the assumption on the
sequence implies that $z(y)=y$. By Lemma 9.2, $w\in W_{\bar{y}}$, and the
arguments in the proof of this lemma imply that $\rho(w,\bar{y})=(2k-2)$ which
contradicts our assumption. $\checkmark$

_Claim 2. Assume that $\Gamma_{(2k-3)}=\Gamma$. Then, there exists
$\varepsilon_{0}>0$ such that for all $\varepsilon<\varepsilon_{0}$, and all
$t\in[0,2\pi]$, $\mathcal{S}_{b}(u_{c(t,\varepsilon)})\neq\emptyset$._

Assume this is not the case. Then, there exists a sequence
$\left\\{\varepsilon_{n}\right\\}$ tending to zero, and a sequence
$\left\\{t_{n}\right\\}\subset[0,2\pi]$ such that
$u_{n}:=u_{c(t_{n},\varepsilon_{n})}$ satisfies
$\mathcal{S}_{b}(u_{c(t_{n},\varepsilon_{n})})=\emptyset$. Without loss of
generality, we may assume that $\left\\{t_{n}\right\\}$ tends to some
$\bar{t}$. Then, $x_{n}:=c(t_{n},\varepsilon_{n})$ tends to
$y=c(\bar{t},0)\in\Gamma$. It follows that $[u_{n}]$ tends to $[w_{y}]$ and
the proof of Lemma 9.2 implies that $ord(w_{y},y)=(2k-2)$, contradicting our
assumption. $\checkmark$

_Claim 3. Assume that $\Gamma_{(2k-3)}=\Gamma$ Then, there exists
$\varepsilon_{0}>0$ and $\delta_{0}>0$ such that for all
$\varepsilon<\varepsilon_{0}$, and all $t\in[0,2\pi]$,
$\mathcal{S}_{b}(u_{c(t,\varepsilon)})$ has two elements whose distance on
$\Gamma$ is bounded from below by $\delta_{0}$._

Assume this is not the case. From Claim 1, we already know that
$\mathcal{S}_{b}(u_{c(t,\varepsilon)})\neq\emptyset$. Then there exists
sequences $\left\\{\varepsilon_{n}\right\\}$ and $\left\\{\delta_{n}\right\\}$
tending to zero, and $\left\\{t_{n}\right\\}$ tending to some $\bar{t}$ such
that $u_{n}:=u_{c(t_{n},\varepsilon_{n})}$ has two boundary singular points
$y_{n,1}$ and $y_{n,2}$ (possibly equal) whose distance is less than
$\delta_{n}$. Taking subsequences, we may assume that these sequences tend to
some $y\in\Gamma$, and the proof of Lemma 9.2 shows that $u_{n}$ tends to
$w_{y}$ with $\rho(w_{y},y)=(2k-2)$, a contradiction. $\checkmark$

Figure 10.1 displays the nodal sets $\mathcal{Z}(u_{n})$ (blue) and
$\mathcal{Z}(w_{y})$ (red).

Figure 10.1. $\mathcal{Z}(u_{n})$ (blue) and $\mathcal{Z}(w_{y})$ (red)

The assumption of the lemma implies that the combinatorial type of $w_{y}$ is
constant along $\Gamma$, see Lemma 8.4, Assertion (iv). This means that the
nodal arc from $y$ to $z(y)$ makes a constant angle $\theta$ with the tangent
to $\Gamma$ at $y$.

Here is a _heuristic_ argument to proceed. _For $\varepsilon$ small enough,
the nodal set $\mathcal{Z}(u_{c(t,\varepsilon)})$ contains a nodal arc which
goes from $c(t,\varepsilon)$ to some $y_{c(t,\varepsilon),2}$ close to the
point $z(\gamma_{1}(t))$. Furthermore, this nodal arc is close to the arc from
$\gamma_{1}(t)$ to $z(\gamma_{1}(t))$ (in the Hausdorff distance). As
illustrated in Figure 10.1, its tangent at $c(t,\varepsilon)$ should make an
angle $\theta_{t}$ with the tangent to $\Gamma$ at $\gamma_{1}(t)$, and
$\theta_{t}$ should be close to $\theta$. For $\varepsilon$ small enough, the
degree of the map $\bar{h}$ restricted to the curve $c(\cdot,\varepsilon)$
should therefore be one. A contradiction._ We however do not know how to turn
this argument into a rigorous proof of the lemma.

The (unproven) arguments in [HoMN1999] are as follows.

p. 1184, line (-2),
“We want to analyze how the star of tangents at $c(t,\varepsilon)$ to the
$(k-1)$ nodal lines of $u_{c(t,\varepsilon)}$ crossing at $c(t,\varepsilon)$
turns if we follow $t$ from $0$ to $2\pi$."

p. 1185, line (+1),
“To make this precise, we consider the continuous function
$f:\Omega\to\mathbb{S}^{1}$ given by $f(x)=(2k-2)\alpha(x)$ modulo $2\pi$,
where

$t\mapsto t\exp\left(\frac{2\pi
ik}{(2k-2)}+i\alpha(x)\right),\leavevmode\nobreak\ \leavevmode\nobreak\
k=0,\ldots,(2k-3),\leavevmode\nobreak\ t\geq 0$

are the tangents rays of the nodal lines through $x\in\mathcal{Z}(u_{x})$. We
want to analyze $f(c(t,\varepsilon))$."

p. 1185, line (+ 9),
“If $x\in\Omega$ is near enough to some point $y$ in the open set
$\Gamma_{(2k-3)}$ then the nodal pattern $\mathcal{Z}(u_{x})$ can be read off
the nodal pattern $\mathcal{Z}(w_{y})$, since the nodal domains move
continuously. The nodal line leaving $\partial\Omega$ vertically stays
connected to $\partial\Omega$ near $y$."

p. 1185, line (+ 22),
“This already implies that $f(c(t))$ follows the angle of $\partial D$ along
each arc $(\partial D)_{2\ell-3}$." ∎

######  Lemma 10.2.

Assume that $\Omega$ is simply connected. Assume that $k\geq 3$ and that $\dim
U=(2k-2)$. Then, $\Gamma_{(2k-2)}$ is not reduced to one point.

_Proof._ The proof would be clear if we knew that when $y\in\Gamma_{(2k-3)}$
tends to $\eta\in\Gamma_{(2k-2)}$, both $y$ and $z(y)$ are close to $\eta$,
one on each side, compare with Remark 10.4. For the time being, we do not have
a proof of this lemma. ∎

The next lemma appears as Lemma 3.5 in [HoMN1999]. It implicitly assumes that
$\Gamma_{(2k-2)}$ contains at least two points.

######  Lemma 10.3.

Assume that $\Omega$ is simply connected. Assume that $k\geq 3$ and that $\dim
U=(2k-2)$. Let $C$ be a connected component of $\Gamma_{(2k-3)}$, and $y\in
C$. Then, $z(y)\not\in C$.

_Proof._ According to Lemmas 10.1 and 10.2, $\Gamma_{(2k-2)}$ has at least two
elements.

Let $C:=\mathcal{A}(\eta_{1},\eta_{2})$ be a connected component of
$\Gamma_{(2k-3)}$, and $y\in\mathcal{A}(\eta_{1},\eta_{2})$. According to
Lemma 9.13 (ii), there exists (a unique) $s_{1}\in\mathcal{A}(y,z(y))$ such
that the function $v_{y,s_{1}}$ satisfies $\rho(v_{y,s_{1}},s_{1})=2$.
Similarly, there is a unique $s_{2}$ in the arc $\mathcal{A}(z(y),y)$ such
that $\rho(v_{y,s_{1}},s_{1})=2$. If $z(y)\in\mathcal{A}(\eta_{1},\eta_{2})$,
one of the points $s_{1}$ or $s_{2}$ also belongs to
$\mathcal{A}(\eta_{1},\eta_{2})$.

Without loss of generality, we may start out with a point $y_{1}\in C$, such
that $z(y_{1})\in C$, and there is a unique
$s_{1}\in\mathcal{A}(y_{1},z(y_{1}))$ such that
$\rho(v_{y_{1},s_{1}},s_{1})=2$. According to Assertion (iv) of Lemma 9.11,
for $y\in\mathcal{A}(y_{1},s_{1})$ close enough to $s_{1}$, we have
$\rho(v_{y,s_{1}},s_{1})=1$.

We now come back to the analysis of Lemma 3.5 in [HoMN1999] and consider the
validity of the arguments given in [HoMN1999], starting from p. 1182, line
(+6).

The first one claims (in our notation):
“Then we move $y\in\mathcal{A}(y_{1},s_{1})$ towards $s_{1}$ and consider
$\mathcal{Z}(v_{y,s_{1}})$. One of the nodal arcs hitting at $s_{1}$ must move
away before $y$ hits $s_{1}$ since there is no point of $\Gamma_{(2k-2)}$ in
between,…”.

Assuming $y$ is close enough to $s_{1}$ so that $\rho(v_{y,s_{1}},s_{1})=1$,
applying Lemma 9.11, Assertion (iv), we have
$\mathcal{S}_{b}(v_{y,s_{1}})=\left\\{y,s_{1},s^{\prime}(y)\right\\}$, and
there are two possibilities, either $s^{\prime}(y)\in\mathcal{A}(y,s_{1})$ or
$s^{\prime}(y)\not\in\mathcal{A}(y,s_{1})$. (We write $s^{\prime}(y)$ to
mention the dependence with respect to $y$ ($s_{1}$ being fixed).) _This claim
is correct_.

Then, [HoMN1999] claims that only the first case can occur for $y$ close
enough to $s_{1}$, see p. 1182, line (+8),
“and it must move eventually towards $y$ […] since the nodal type of $w_{y}$
[in our notation] is constant in $C$ [the connected component]”.

This claim is questionable but let us continue the proof (see below the
discussion of the second possibility) .

First possibility. Assuming that the first possibility actually holds, namely
that for $y$ close enough to $s_{1}$, $s^{\prime}(y)\in\mathcal{A}(y,s_{1})$.
Letting $y$ tend to $s_{1}$, we find some
$y_{2}\in\mathcal{A}(y_{1},z(y_{1}))$ such that $s^{\prime}(y_{2})=y_{2}$
which implies that $\rho(v_{y_{2},s_{1}},y_{2})=2k-3$, hence $z(y_{2})=s_{1}$,
see Figure 10.2. Summing up, we have
$y_{2},z(y_{2})\in\mathcal{A}(y_{1},z(y_{1}))$ and, measuring distances on the
arc $\mathcal{A}(y_{1},z(y_{1}))$, $d(y_{2},z(y_{2}))<d(y_{1},z(y_{1}))$.

To end up the argument in [HoMN1999], p. 1182, line (+10),
“We have then the same situation …a contradiction.”,
we can consider the set

$K:=\left\\{y\in\mathcal{A}[y_{1},z(y_{1})]\leavevmode\nobreak\
|\leavevmode\nobreak\ z(y)\in\mathcal{A}[y_{1},z(y_{1})]\right\\}$

and the number $d:=\inf\left\\{d(y,z(y))\leavevmode\nobreak\
|\leavevmode\nobreak\ y\in K\right\\}$. Since the set $K$ is non-empty and
closed, the infimum is achieved. The previous argument shows that $d=0$, and
hence that there exists some $y_{0}$ in $K$ such that $z(y_{0})=y_{0}$ which
contradicts the fact that $y_{0}\in\Gamma_{(2k-3)}$. So finally, if
$y\in\Gamma_{(2k-3)}$, $y$ and $z(y)$ cannot belong to the same connected
component of $\Gamma_{(2k-3)}$. This would end the proof of Lemma 10.3, i.e.,
the proof of [HoMN1999], Lemma 3.5, assuming that at each step only the first
possibility occurs.

Figure 10.2. 1st possibility

Second possibility. [HoMN1999] seems to claim that this possibility leads to
an immediate contradiction and refers to the nodal type of $w_{y}$. Figure
10.3 seems to invalidate this claim. As far as we understand, Lemma 10.3
remains unproven. ∎

Figure 10.3. 2nd possibility: apparent counterexample to the claim on the
nodal type

######  Remark 10.4.

It might be sufficient to prove the following weaker statement: if
$C=\mathcal{A}(\eta_{1},\eta_{2})$, when $y$ is close enough to $\eta_{i}$,
the point $z(y)$ is not in $C$ (so that $y$ and $z(y)$ are not on the same
side of $\eta_{i}$).

Lemma 3.6 in [HoMN1999] contains the following assertions.

  1. (i)

If $y$ moves clockwise to $\eta_{1}$ then $z(y)$ moves anticlockwise to
$\eta_{1}$.

  2. (ii)

If $y$ moves anti-clockwise to $\eta_{2}$ then $z(y)$ moves clockwise to
$\eta_{2}$.

  3. (iii)

In particular the nodal pattern of $\mathcal{Z}(w_{\eta_{1}})$ and
$\mathcal{Z}(w_{\eta_{2}})$ are different.

  4. (iv)

Moreover $\Gamma_{(2k-3)}$ consists of only finitely many open arcs and
$\Gamma_{(2k-2)}$ is a finite set of $\Gamma$.

Figure 10.4. $Z(w_{\eta_{1}})$ and $Z(w_{\eta_{2}})$ have different
combinatorial types

Some comments.

  1. $\diamond$

If _clockwise_ and _anticlockwise_ are meant to indicate some monotonic
behavior, Assertions (i) and (ii) are not proved in [HoMN1999]. It is not
clear whether knowing this precise behavior of the map $y\mapsto z(y)$ is
necessary.

  2. $\diamond$

Assuming Lemma 10.3 (their Lemma 3.5) is true, and taking Lemma 8.4(iii) into
account, we see that when $y\in\Gamma_{(2k-3)}$ tends to some
$\eta\in\Gamma_{(2k-2)}$ the points $y$ and $z(y)$ do not lie on the same side
of $\eta$. According to Remark 10.4, this property seems sufficient to prove
that the eigenfunctions $w_{\eta_{1}}$ and $w_{\eta_{2}}$ have different
combinatorial types (same proof as in Subsection 7.6), giving a precise
meaning to Assertion (iii). This is illustrated in Figure 10.4. This fact does
not seem to be used in the sequel.

  3. $\diamond$

More interestingly, assuming that when $y\in\Gamma_{(2k-3)}$ tends to
$\eta\in\Gamma_{(2k-3)}$ the points $y$ and $z(y)$ do not lie on the same side
of $\eta$, we see that the combinatorial type of $w_{y}$ changes when $y$
passes through some $\eta\in\Gamma_{(2k-2)}$. This is illustrated in Figure
10.5

  4. $\diamond$

Assertion (iv) is proved in our Lemma 8.4(ii).

Figure 10.5. The combinatorial types of $Z(w_{y})$ are different on the right
and on the left of $\eta\in\Gamma_{(2k-2)}$

The last argument in the proof should be as follows.

######  Lemma 10.5.

For $\varepsilon$ small enough, the smooth mapping
$t\mapsto\bar{h}_{c(t,\varepsilon})$ from the circle $c(\cdot,\varepsilon)$ to
$\mathbb{S}_{\mathcal{H}}$ has nonzero degree.

_Proof._ We do not know how to prove this lemma.

The argument given in [HoMN1999, p. 1186] is:

p. 1186, line (-5),
“So finally, the smooth mapping $t\mapsto
f(c(t,\varepsilon)),\mathbb{S}^{1}\to\mathbb{S}^{1}$ has mapping degree
$2\pi\\#(\partial D)_{2\ell-2}+2\pi(2\ell-2)>0$ and cannot be null homotopic.
But by construction it is continuously extended into the interior of the
circle and this is null homotopic, a contradiction."

Note that we have the same difficulty as in the proof of Lemma 10.1. ∎

Concluding remarks. Unless we modify the strategy of proof, we are led to two
main questions.

_Question 1._ Given $\eta\in\Gamma_{(2k-2)}$ and
$y\in\Gamma\\!\setminus\\!\left\\{\eta\right\\}$ close to $\eta$, $z(y)$ is
close to $\eta$ as well, so that $y$ and $z(y)$ are both in a small arc
$\mathcal{A}$ centered at $\eta$. Is it true that $y$ and $z(y)$ are not on
the same connected component of
$\mathcal{A}\\!\setminus\\!\left\\{\eta\right\\}$ (i.e., that they are on
either sides of $\eta$)?

If so, we could prove that the combinatorial type of $w_{y}$ (which is
constant in each connected component of $\Gamma_{(2k-3)}$) changes each time
$y$ crosses some $\eta\in\Gamma_{(2k-2)}$. This would in particular prove
Lemma 10.2.

One way to prove Lemma 10.1 and Lemma 10.5 would be to better understand how
the _star_ at $x$, i.e. the zero set of the harmonic polynomial $h_{x}$,
changes (rotates) when $x\in\Omega$ tends to $y\in\Gamma_{(2k-3)}$. If $x_{n}$
tends to $y$, the stars at $x_{n}$ have limit points because they can be
identified to sets of equidistant points on the unit circle. This leads us to
ask the following question.

_Question 2._ Does the star at $x$ have a (unique) limit when $x$ tends to
$y\in\Gamma_{(2k-3)}$, and can we identify one branch of the limit star as the
ray at $y$ corresponding to the nodal arc from $y$ to $z(y)$ ?

One difficulty here is that Hausdorff convergence of nodal sets is not
sufficient.

##  Appendix A A remark on upper bounds for multiplicities vs Courant-sharp
eigenvalues

For simplicity, let us only consider Dirichlet eigenvalues in a smooth bounded
domain $\Omega\subset\mathbb{R}^{2}$. The upper bounds on the eigenvalue
multiplicities strongly rely on Courant’s nodal domain theorem. As observed in
[HoMN1999], they are actually a consequence of the following 3-step result,
$n\in\left\\{1,2,3\right\\}$, provided that the third step ($n=3$) is correct,
see the final paragraph in Section 7.

######  Proposition A.1.

Let $U(\lambda)$ is a Dirichlet eigenspace. Assume that

$\sup\left\\{\kappa(u)\leavevmode\nobreak\ |\leavevmode\nobreak\ u\in
U(\lambda)\right\\}=\ell$

for some $\ell\geq 3$. Then $\dim(U(\lambda))\leq(2\ell-n)$.

Indeed, since $u\in U(\lambda_{k})$ implies that $\kappa(u)\leq k$:

  1. (1)

The first step, $n=1$, yields the upper bound
$\operatorname{mult}(\lambda_{k}\leq(2k-1)$ (Theorem 1 in [Nadi1987]).

  2. (2)

The second step, $n=2$, yields the upper bound
$\operatorname{mult}(\lambda_{k})\leq(2k-2)$ (Lemma 2.13 in [HoMN1999] or
Proposition 7.17).

  3. (3)

The third step, $n=3$, yields the upper bound
$\operatorname{mult}(\lambda_{k})\leq(2k-3)$ (Theorem B in [HoMN1999]).

According to Pleijel [Plej1956], the equality in Courant’s theorem can only
occur for finitely many eigenvalues, the so-called _Courant-sharp_
eigenvalues. If $\lambda_{k}$ is not a Courant-sharp eigenvalue, in particular
if $k$ is large enough (depending on the geometry of the domain), $u\in
U(\lambda_{k})$ implies that $\kappa(u)\leq(k-1)$, and the above proposition
implies that $\operatorname{mult}(\lambda_{k})\leq(2k-2-n)$. When
$\lambda_{k}$ is not Courant-sharp, the upper bound for the multiplicity is
improved by $2$.

The above proposition can be restated as

(A.1) $\operatorname{mult}(\lambda_{k})\leq 2\sup_{u\in
U(\lambda_{k})}\kappa(u)\,-1\,.$

Note that this implies that

$\limsup_{k\rightarrow+\infty}\frac{\operatorname{mult}(\lambda_{k})}{k}\leq
2\,.$

We can continue the discussion a little further recalling Pleijel’s proof. Let
$u$ be an eigenfunction in $U(\lambda_{k})$ with precisely
$\kappa(\lambda_{k}):=\sup_{u\in U(\lambda_{k})}\kappa(u)$ nodal domains.
Applying the Faber-Krahn inequality to each nodal domain, we obtain

$\lambda_{k}|\Omega|\geq\kappa(\lambda_{k})\pi j_{0,1}^{2}\,,$

where $|\Omega|$ denotes the area of $\Omega$. Then,

(A.2) $\operatorname{mult}(\lambda_{k})\leq\frac{2}{\pi
j_{0,1}^{2}}\lambda_{k}|\Omega|\,-1\,.$

For a regular bounded domain $\Omega$ in $\mathbb{R}^{2}$, Weyl’s asymptotic
formula reads

(A.3) $N(\lambda):=\\#\left\\{j\leavevmode\nobreak\ |\leavevmode\nobreak\
\lambda_{j}<\lambda\right\\}=\frac{|\Omega|}{4\pi}\,\lambda+O(\sqrt{\lambda})\,.$

######  Remark A.2.

For this formula, we refer to Ivrii’s papers [Ivri1980r, Ivri1980], and to
Hörmander’s book [Horm2009d, Theorem 29.3.3]. These authors actually give a
much more precise formula which yields a two-term asymptotic formula under an
assumption on the set of periodic billiard trajectories (see Corollary 29.3.4
in Hörmander’s book).

It follows from (A.3) that

(A.4)
$\limsup_{k\rightarrow+\infty}\frac{\kappa(\lambda_{k})}{k}\leq\frac{|\Omega|}{\pi
j_{0,1}^{2}}\,\lim_{k\rightarrow+\infty}\frac{\lambda_{k}}{k}=\frac{4}{j_{0,1}^{2}}=:\gamma<1.$

As a consequence, we obtain the better estimate (see [Plej1956])

(A.5)
$\limsup_{k\rightarrow+\infty}\frac{\operatorname{mult}(\lambda_{k})}{k}\leq
2\,\gamma<2\,.$

On the other hand, for any $\delta$ small enough,
$\operatorname{mult}(\lambda_{k})=N(\lambda_{k}+\delta)-N(\lambda_{k}-\delta)$.
By Weyl’s asymptotic formula, we should have

$\operatorname{mult}(\lambda_{k})=O(\sqrt{\lambda_{k}})\,,$

and hence

$\limsup_{k\rightarrow+\infty}\frac{\operatorname{mult}(\lambda_{k})}{k}=0\,,$

which shows that the inequality (A.5) is not pertinent.

For a bounded domain $\Omega\subset\mathbb{R}^{n}$ with $C^{2}$ boundary, and
Dirichlet boundary condition, one can show that there exists a constant
$C(\Omega)$, depending on $\Omega$ and invariant under dilations, such that
for $k>C(\Omega)$, the $k$th Dirichlet eigenvalue $\lambda_{k}$ is not
Courant-sharp. In dimension $2$, the constant $C(\Omega)$ can be estimated in
terms of the area $|\Omega|$, the length $|\partial\Omega|$ of the boundary,
the curvature and the cut-distance of $\partial\Omega$. We refer to [BeHe2016,
Theorem 1.3] for more details, and to [BeGi2016, Theorem 1] for an extension
to less regular domains. The proof of this result makes use of a lower bound
on the remainder term $R(\lambda):=N(\lambda)-\frac{|\Omega|}{4\pi}\,\lambda$
in Weyl’s asymptotic estimate, as given for example in [BeLi2001].

######  Remark A.3.

For the case of domains with Neumann or Robin boundary condition, we refer to
[GiLe2020, Proposition 1.1]. This paper also considers the special case of
convex domains.

######  Remark A.4.

As in Bérard-Helffer [BeHe2016] (see also [BeGi2016]) we can think of a
geometrical control of the constants (see Safarov [Safa2001] or Van den Berg–
Lianantonakis [BeLi2001]) who give estimates in the form

$|N(\lambda)-C_{n}|\Omega|\lambda^{n/2}|\leq
C_{geom}(\Omega)\lambda^{(n-1)/2}\ln\lambda\,.$

Although we only use lower bounds of the remainder in the proof of Pleijel’s
theorem, we will also need here upper bounds.

##  Appendix B The local structure of the nodal set in the neighborhood of a
critical zero

In this Appendix, we provide a proof of the local structure theorem for the
nodal set of an eigenfunction $u$ in the neighborhood of an interior singular
point $x$. Cheng [Chen1976] proposes a more precise result (existence of a
local diffeomorphism which sends the nodal set of $u$ in a neighborhood of $x$
onto the nodal set of the higher order term in the Taylor expansion of $u$ at
$x$ (a harmonic polynomial), which consists of rays. Our proof has the
advantage of being more quantitative in particular when applied to
eigenfunctions depending on a parameter. This proof is probably known although
we did not find any reference except the analysis in § 128 of the book
_Georges Valiron, Cours d’analyse mathématique. Théorie des fonctions. Masson,
1966._ cited in [Bess1980].

Let $(M,g)$ be a compact Riemannian surface, and let $V:M\to\mathbb{R}$ be a
real valued smooth potential. Let $u\not=0$ be a real valued function,
satisfying

(B.1) $(-\Delta_{g}+V)u=\lambda u$

for some real number $\lambda$. Here $\Delta_{g}$ denotes the Laplace-Beltrami
operator of $(M,g)$. Let $x$ be a given (interior) point of $M$ (in case $M$
has a boundary).

Choose some $r_{0}>0$ such that the exponential map $\exp_{x}:T_{x}M\to M$ is
a diffeomorphism from the disk $D(2r_{0})$, with center $0$ and radius
$2r_{0}$ in $T_{x}M$, onto the geodesic disk $D(x,2r_{0})$, with center $x$
and radius $2r_{0}$ in $M$. Choose an orthonormal frame in $T_{x}M$, call
$(\xi_{1},\xi_{2})$ the corresponding coordinates in $T_{x}M$, and
$(r,\omega)$ the associated polar coordinates. In the normal coordinates
$(\xi_{1},\xi_{2})$, the Riemannian metric is given by the $2\times 2$ matrix
$G=(g_{ij})$, where
$g_{ij}=g(\frac{\partial}{\partial\xi_{i}},\frac{\partial}{\partial\xi_{j}})$;
the Riemannian measure is given by $v_{g}\,d\xi_{1}d\xi_{2}$, where
$v_{g}=\sqrt{\det G}$. Write the matrix $G^{-1}$ as $G^{-1}=(g^{ij})$. Then
(see for example [GaHuLa2004, § 2.89bis, p. 87]),

(B.2)
$\left\\{\begin{array}[]{l}G(0,0)=\operatorname{Id}\,,\text{\leavevmode\nobreak\
\leavevmode\nobreak\ i.e.\leavevmode\nobreak\ \leavevmode\nobreak\
}g_{ij}(0,0)=\delta_{ij}\,,\quad 1\leq i,j\leq 2\,,\\\\[5.0pt] \frac{\partial
G}{\partial\xi_{k}}(0,0)=0\,,\text{\leavevmode\nobreak\ \leavevmode\nobreak\
i.e.\leavevmode\nobreak\ \leavevmode\nobreak\ }\frac{\partial
g_{ij}}{\partial\xi_{k}}(0,0)=0\,,\quad 1\leq i,j,k\leq 2.\end{array}\right.$

It follows that

(B.3) $\left\\{\begin{array}[]{l}v_{g}(0,0)=1\,,\\\\[5.0pt]
G^{-1}(0,0)=\operatorname{Id}\,,\text{\leavevmode\nobreak\
\leavevmode\nobreak\ i.e.\leavevmode\nobreak\ \leavevmode\nobreak\
}g^{ij}(0,0)=\delta_{ij}\,,\quad 1\leq i,j\leq 2\,,\\\\[5.0pt] \frac{\partial
v_{g}}{\partial\xi_{k}}(0,0)=0\,,\quad 1\leq k\leq 2.\\\\[5.0pt]
\frac{\partial G^{-1}}{\partial\xi_{k}}(0,0)=0\,,\text{\leavevmode\nobreak\
\leavevmode\nobreak\ i.e.\leavevmode\nobreak\ \leavevmode\nobreak\
}\frac{\partial g^{ij}}{\partial\xi_{k}}(0,0)=0\,,\quad 1\leq i,j,k\leq
2.\end{array}\right.$

Given a function $u$ on $M$, let $f=u\circ\exp_{x}$. In the local coordinates
$(\xi_{1},\xi_{2})$, the Laplace-Beltrami operator $\Delta_{g}$ is given (see
[BeGM1971, §G.III, p. 126]) by

(B.4) $\left\\{\begin{array}[]{ll}\Delta_{g}f&=v_{g}^{-1}\,\sum_{1\leq i,j\leq
2\,}\frac{\partial}{\partial\xi_{i}}\left(v_{g}\,g^{ij}\frac{\partial
f}{\partial\xi_{j}}\right)\,,\\\\[5.0pt] &=\sum_{1\leq i,j\leq
2\,}g^{ij}\,\frac{\partial^{2}f}{\partial\xi_{i}\partial\xi_{j}}+\sum_{1\leq
j\leq 2\,}b_{j}\,\frac{\partial f}{\partial\xi_{j}}\,,\\\\[5.0pt]
&\text{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
where\leavevmode\nobreak\ }b_{j}=\sum_{1\leq i\leq
2\,}v_{g}^{-1}\,\frac{\partial}{\partial\xi_{i}}(v_{g}g^{ij})\,,\quad 1\leq
j\leq 2\,.\end{array}\right.$

Letting $\Delta_{0}=\sum_{1\leq j\leq
2\,}\frac{\partial^{2}}{\partial\xi_{j}^{2}}$ denote the Laplacian in the
Euclidean space $(T_{x}M,g_{x})$, and taking relations (B.2) and (B.3) into
account, we obtain the following expression for the Laplace-Beltrami operator,

(B.5) $\left\\{\begin{array}[]{l}\Delta_{g}=\Delta_{0}+\sum_{1\leq i,j\leq
2\,}a_{ij}\,\frac{\partial^{2}}{\partial\xi_{i}\partial\xi_{j}}+\sum_{1\leq
j\leq 2\,}b_{j}\,\frac{\partial}{\partial\xi_{j}}\,,\\\\[5.0pt]
\text{where\leavevmode\nobreak\ \leavevmode\nobreak\
}\operatorname{ord}\left(a_{ij},(0,0)\right)=2\text{\leavevmode\nobreak\
and\leavevmode\nobreak\
}\operatorname{ord}\left(b_{j},(0,0)\right))=1\,.\end{array}\right.$

If $u$ satisfies (B.1) and $u(x)=0$, the unique continuation theorem
[Aron1957, DoFe1990a] implies that $f$ does not vanish at infinite order at
$0$. If $ord(u,x)=ord(f,0)=p$, Taylor’s formula at $0$, gives

(B.6)
$\left\\{\begin{array}[]{l}f(\xi_{1},\xi_{2})=\sum_{|\alpha|=p}\frac{1}{\alpha!}\,D^{\alpha}\\!f(0,0)\,(\xi_{1},\xi_{2})^{\alpha}+R_{p+1}(\xi_{1},\xi_{2})\,,\text{\leavevmode\nobreak\
where}\\\\[8.0pt]
R_{p+1}(\xi_{1},\xi_{2})=\sum_{|\alpha|=p+1}\frac{p+1}{\alpha!}\,\,(\xi_{1},\xi_{2})^{\alpha}\,{\displaystyle\int_{0}^{1}(1-t)^{p}\,D^{\alpha}\\!f(t\,\xi_{1},t\,\xi_{2})\,dt}\,.\end{array}\right.$

Here, as usual,

(B.7)
$\left\\{\begin{array}[]{l}\alpha=(\alpha_{1},\alpha_{2})\,,\quad|\alpha|=\alpha_{1}+\alpha_{2}\,,\quad(\xi_{1},\xi_{2})^{\alpha}=\xi_{1}^{\alpha_{1}}\,\xi_{2}^{\alpha_{2}}\,,\text{and}\\\\[5.0pt]
D^{\alpha}\\!f=\dfrac{\partial^{|\alpha|}\\!f}{\partial\xi_{1}^{\alpha_{1}}\partial\xi_{2}^{\alpha_{2}}}\,.\end{array}\right.$

Using relations (B.1) and (B.5), and identifying the terms with lowest order,
we find that the polynomial
$P_{p}(\xi_{1},\xi_{2}):=\sum_{|\alpha|=p}\frac{1}{\alpha!}\,D^{\alpha}\\!f(0,0)\,(\xi_{1},\xi_{2})^{\alpha}$
is homogenous of degree $p$, and harmonic with respect to $\Delta_{0}$.

######  Remark B.1.

As a matter of fact, we may write a $2$-term Taylor formula for the function
$f$,

$f(\xi_{1},\xi_{2})=P_{p}(\xi_{1},\xi_{2})+P_{p+1}(\xi_{1},\xi_{2})+R_{p+1}(\xi_{1},\xi_{2})\,,$

where $P_{p}$ and $P_{p+1}$ are homogeneous polynomials of degrees $p$ and
$(p+1)$ respectively, and where the remainder term $R_{p+2}$ vanishes at order
at least $(p+2)$. Then, we actually have that $\Delta_{0}P_{p}=0$ and
$\Delta_{0}P_{p+1}=0$.

Writing the harmonicity condition $\Delta_{0}P_{p}=0$ in polar coordinates
$(r,\omega)$ in $T_{x}M$, we find that the polynomial $P_{p}$ has the form

(B.8)
$P_{p}(r\,\cos\omega,r\,\sin\omega)=a\,r^{p}\,\sin(p\,\omega-\omega_{0})\,.$

for some $0\not=a\in\mathbb{R}$ and some $\omega_{0}\in[0,2\pi]$.

Multiplying the function $f$ by some constant, and rotating the coordinates
$(\xi_{1},\xi_{2})$ in $\mathbb{R}^{2}$ if necessary, we can assume that $a=1$
and $\omega_{0}=0$. It follows that $f$ can be written as

(B.9)
$f(r\,\cos\omega,r\,\sin\omega)=r^{p}\,\sin(p\,\omega)+R_{p+1}(r\,\cos\omega,r\,\sin\omega)\,,$

and we can write the second term as

(B.10)
$\left\\{\begin{array}[]{l}R_{p+1}(r\,\cos\omega,r\,\sin\omega)=r^{p+1}T_{p+1}(r\,,\omega)\,,\text{\leavevmode\nobreak\
where\leavevmode\nobreak\ }T_{p+1}(r,\omega)\text{\leavevmode\nobreak\ is
given by}\\\\[5.0pt]
\sum_{|\alpha|=p+1}\frac{p+1}{\alpha!}\,\,(\cos\omega,\sin\omega)^{\alpha}\,{\displaystyle\int_{0}^{1}(1-t)^{p}\,D^{\alpha}\\!f(tr\,\cos\omega,tr\,\sin\omega)\,dt}\,.\end{array}\right.$

Define

(B.11) $W(r,\omega):=\sin(p\,\omega)+r\,T_{p+1}(r,\omega)\,.$

The function $W(0,\omega)$ vanishes precisely for the values

(B.12) $\omega_{j}:=j\frac{\pi}{p}\,,\quad
j\in\left\\{0,\ldots,2p-1\right\\}.$

Choose $\alpha_{1}\in\left(0,\frac{\pi}{8}\right)$ and define
$\alpha_{p}:=\frac{\alpha_{1}}{p}$. We have the following relations.

(B.13)
$\left\\{\begin{array}[]{l}\sin\left(p\,(\omega_{j}\pm\alpha_{p})\right)=\pm(-1)^{j}\,\sin(\alpha_{1})\,,\\\\[5.0pt]
|\sin(p\,\omega)|\geq\sin\alpha_{1}\,,\text{\leavevmode\nobreak\
for\leavevmode\nobreak\
}\omega\not\in\bigcup_{j=0}^{2p-1}\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)\end{array}\right.$

Define

(B.14)
$r_{1}:=\min\left\\{r_{0}\,,\,\frac{1}{2}\,\sin(\alpha_{1})\,\|T_{p+1}\|^{-1}_{\infty,D(r_{0})}\,,\frac{1}{2}\,\cos(\alpha_{1})\,\|\partial_{\omega}T_{p+1}\|^{-1}_{\infty,D(r_{0})}\right\\}\,,$

where $\|\cdot\|_{\infty,D(\frac{1}{2})}$ denotes the $L^{\infty}$ norm of
functions on the disk $D(r_{0})$ of radius $r_{0}$ in $TxM$.

######  Proposition B.2.

For any $r\leq r_{1}$,

  1. (i)

the function $\omega\mapsto W(r,\omega)$ does not vanish in

$[0,2\pi]\\!\setminus\\!\bigcup_{j=0}^{2p-1}\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)=\bigcup_{j=0}^{2p-1}\left[\omega_{j}+\alpha_{p},\omega_{j+1}-\alpha_{p}\right]\,;$

  2. (ii)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the function $\omega\mapsto
W(r,\omega)$ has exactly one zero
$\widetilde{\omega}_{j}(r)\in\left(\omega_{j}-\alpha_{p}\,,\,\omega+\alpha_{p}\right)$;

  3. (iii)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the function
$r\mapsto\widetilde{\omega}_{j}(r)$ is $C^{\infty}$ in $(0,r_{1})$ and tends
to $\omega_{j}$ as $\rho$ tends to zero;

  4. (iv)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the curve

$\left(0,r_{1}\right)\ni r\mapsto
a_{j}(r)=\left(r\,\cos(\widetilde{\omega}_{j}(r)\,,\,r\,\sin(\widetilde{\omega}_{j}(r)\right)$

is smooth and has semi-tangent $\omega_{j}$ at the origin.

_Proof._ To prove (i), we observe that in each interval
$\left\\{r\right\\}\times\left[\omega_{j}+\alpha_{p},\omega_{j+1}-\alpha_{p}\right]$,
$|W(r,\omega)|\geq\frac{1}{2}\sin(\alpha_{1})$. To prove (ii), we observe that
the function $W(r,\omega)$ changes sign in
$\left\\{r\right\\}\times\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)$
and that its partial derivative with respect to $\omega$ does not vanish.
Assertion (iii) follows from the implicit function theorem. Assertion (iv)
follows from the previous ones. ∎

######  Remark B.3.

Later on, we will encounter the following situation. There exist two
eigenfunctions $v_{1}$ and $v_{2}$ such that the functions
$h_{i}=v_{i}\circ\exp_{x}$ satisfy the relations

(B.15)
$\left\\{\begin{array}[]{l}h_{1}(r\,\cos\omega,r\,\sin\omega)=r^{p}\,\sin(p\,\omega)+R_{1,p+1}(r\,\cos\omega,r\,\sin\omega)\,,\\\\[5.0pt]
h_{2}(r\,\cos\omega,r\,\sin\omega)=r^{p}\,\cos(p\,\omega)+R_{2,p+1}(r\,\cos\omega,r\,\sin\omega)\,.\end{array}\right.$

Defining the family of functions
$w_{\theta}=\cos\theta\,v_{1}-\sin\theta\,v_{2}$, the associated family of
functions $h_{\theta}=w_{\theta}\circ\exp_{x}$, satisfies

(B.16)
$\left\\{\begin{array}[]{l}h_{\theta}(r\,\cos\omega,r\,\sin\omega)=r^{p}\,\sin(p\,(\omega-\theta))+R_{\theta,p+1}(r\,\cos\omega,r\,\sin\omega)\,,\text{\leavevmode\nobreak\
with}\\\\[5.0pt]
R_{\theta,p+1}=\cos\theta\,R_{1,p+1}-\sin\theta\,R_{2,p+1}.\end{array}\right.$

Then, Proposition B.2 remains valid for the family $h_{\theta}$, uniformly
with respect to the variable $\theta\in[0,2\pi]$, with $\omega_{j}$ replaced
by $(\omega_{j}-\theta)$, and the corresponding functions
$r\mapsto\widetilde{\omega}_{j}(r,\theta)$ and $r\mapsto a_{j}(r,\theta)$ are
smooth in $(r,\theta)$.

######  Remark B.4.

Proposition B.2 tells us that, in a neighborhood of the critical zero $x$ of
$u$, the nodal set $\mathcal{Z}(u)$ consists of $p$ smooth semi-arcs emanating
from $x$ tangentially to the rays $\omega_{j}$.

##  Appendix C Local structure of the nodal set near a critical zero, another
approach

In this Appendix, we use a local conformal chart to determine the local
structure of the nodal set $\mathcal{Z}(u)$ near an interior critical zero of
$u$, instead of normal coordinates as in Appendix B.

######  Remark C.1.

Using “boundary isothermal coordinates” as given in [YaZh2021, Section 2], one
can also determine the local structure of the nodal set $\mathcal{Z}(u)$ near
a boundary singular point of $u$ (assuming Dirichlet, Neumann or Robin
boundary condition).

Let $(M,g)$ be a compact Riemannian surface, and let $V:M\to\mathbb{R}$ be a
real valued smooth (ie, $C^{\infty}$) potential. Let $u\not=0$, a real valued
function, satisfying $(-\Delta+V)u=\lambda u$, for some real number $\lambda$.
Let $x$ be a given (interior) point of $M$ (in case $M$ has a boundary).
Assume that $u(x)=0$.

Choose some $r_{0}>0$ such that the geodesic disk $D(x,r_{0})$ is contained in
a conformal coordinate chart centered at $x$: there exists some open
neighborhood $\mathcal{U}_{x}$ of $x$, such that
$D(x,r_{0})\subset\mathcal{U}_{x}\subset M$, and a conformal diffeomorphism
$\phi:\mathcal{U}_{x}\to\mathbb{D}$, such that $\phi(x)=0$ and
$\phi^{*}(d\xi_{1}^{2}+d\xi_{2}^{2})=\sigma(\psi(\xi_{1},\xi_{2}))\,g$. Here
$\mathbb{D}$ is the unit disk in $\mathbb{R}^{2}$, $(\xi_{1},\xi_{2})$ denote
the coordinates in $\mathbb{R}^{2}$, $\psi=\phi^{-1}$, and $\sigma$ is a
smooth positive function on $\mathcal{U}_{x}$.

Then, we have

(C.1) $-\Delta_{0}f=\sigma\,\left(\lambda-V\circ\psi\right)\,f\,,$

where, $f=u\circ\psi$, and $\Delta_{0}$ is the Laplacian in $\mathbb{R}^{2}$
with the metric $d\xi_{1}^{2}+d\xi_{2}^{2}$. To determine the local structure
of the nodal set $\mathcal{Z}(u)$ in a neighborhood of
$x\in\operatorname{int}(M)$, is sufficient to determine the structure of the
zero set of $f$ in a neighborhood of $0$ in $\mathbb{D}$.

The unique continuation theorem [Aron1957, DoFe1990a] implies that $f$
(equivalently $u$) does not vanish at infinite order at $0$. If
$ord(u,x)=ord(f,0)=p$, Taylor’s formula at $0$, gives

(C.2)
$\left\\{\begin{array}[]{l}f(\xi_{1},\xi_{2})=\sum_{|\alpha|=p}\frac{1}{\alpha!}\,D^{\alpha}\\!f(0,0)\,(\xi_{1},\xi_{2})^{\alpha}+R_{p+1}(\xi_{1},\xi_{2})\,,\text{\leavevmode\nobreak\
where}\\\\[8.0pt]
R_{p}{p+1}(\xi_{1},\xi_{2})=\sum_{|\alpha|=p+1}\frac{p+1}{\alpha!}\,\,(\xi_{1},\xi_{2})^{\alpha}\,{\displaystyle\int_{0}^{1}(1-t)^{p}\,D^{\alpha}\\!f(t\,\xi_{1},t\,\xi_{2})\,dt}\,.\end{array}\right.$

Here, as usual,

(C.3)
$\left\\{\begin{array}[]{l}\alpha=(\alpha_{1},\alpha_{2})\,,\quad|\alpha|=\alpha_{1}+\alpha_{2}\,,\quad(\xi_{1},\xi_{2})^{\alpha}=\xi_{1}^{\alpha_{1}}\,\xi_{2}^{\alpha_{2}}\,,\text{and}\\\\[5.0pt]
D^{\alpha}\\!f=\dfrac{\partial^{|\alpha|}\\!f}{\partial\xi_{1}^{\alpha_{1}}\partial\xi_{2}^{\alpha_{2}}}\,.\end{array}\right.$

Using (C.1), and identifying the terms with lowest order, we find that the
polynomial
$P_{p}(\xi_{1},\xi_{2}):=\sum_{|\alpha|=p}\frac{1}{\alpha!}\,D^{\alpha}\\!f(0,0)\,(\xi_{1},\xi_{2})^{\alpha}$
is homogenous of degree $p$, and harmonic (with respect to $\Delta_{0}$).
Writing the harmonicity condition in polar coordinates $(\rho,\omega)$ in
$\mathbb{R}^{2}$, we find that the polynomial $P_{p}$ has the form

(C.4)
$P_{p}(\rho\,\cos\omega,\rho\,\sin\omega)=a\,\rho^{p}\,\sin(p\,\omega-\omega_{0})\,.$

for some $0\not=a\in\mathbb{R}$ and some $\omega_{0}\in[0,2\pi]$.

Multiplying the function $f$ by some constant, and rotating the coordinates
$(\xi_{1},\xi_{2})$ in $\mathbb{R}^{2}$ if necessary, we can assume that $a=1$
and $\omega_{0}=0$. It follows that $f$ can be written as

(C.5)
$f(\rho\,\cos\omega,\rho\,\sin\omega)=\rho^{p}\,\sin(p\,\omega)+R_{p+1}(\rho\,\cos\omega,\rho\,\sin\omega)\,,$

and we can write the second term as

(C.6)
$\left\\{\begin{array}[]{l}R_{p+1}(\rho\,\cos\omega,\rho\,\sin\omega)=\rho^{p+1}T_{p+1}(\rho\,,\omega)\,,\text{\leavevmode\nobreak\
where\leavevmode\nobreak\ }T_{p+1}(\rho,\omega)\text{\leavevmode\nobreak\ is
given by}\\\\[5.0pt]
\sum_{|\alpha|=p+1}\frac{p+1}{\alpha!}\,\,(\cos\omega,\sin\omega)^{\alpha}\,{\displaystyle\int_{0}^{1}(1-t)^{p}\,D^{\alpha}\\!f(t\rho\,\cos\omega,t\rho\,\sin\omega)\,dt}\,.\end{array}\right.$

Define

(C.7) $W(\rho,\omega):=\sin(p\,\omega)+\rho\,T_{p+1}(\rho,\omega)\,.$

The function $W(0,\omega)$ vanishes precisely for the values

(C.8) $\omega_{j}:=j\frac{\pi}{p}\,,\quad j\in\left\\{0,\ldots,2p-1\right\\}.$

Choose $\alpha_{1}\in\left(0,\frac{\pi}{8}\right)$ and define
$\alpha_{p}:=\frac{\alpha_{1}}{p}$. We have the following relations.

(C.9)
$\left\\{\begin{array}[]{l}\sin\left(p\,(\omega_{j}\pm\alpha_{p})\right)=\pm(-1)^{j}\,\sin(\alpha_{1})\,,\\\\[5.0pt]
|\sin(p\,\omega)|\geq\sin\alpha_{1}\,,\text{\leavevmode\nobreak\
for\leavevmode\nobreak\
}\omega\not\in\bigcup_{j=0}^{2p-1}\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)\end{array}\right.$

Define

(C.10)
$\left\\{\begin{array}[]{l}\rho_{0}:=\min\left\\{\frac{1}{2}\,,\,\frac{1}{2}\,\sin(\alpha_{1})\,\|T_{p+1}\|^{-1}_{\infty,D(\frac{1}{2})}\right\\}\,,\\\\[5.0pt]
\rho_{1}:=\min\left\\{\frac{1}{2}\,,\,\frac{1}{2}\,\cos(\alpha_{1})\,\|\partial_{\omega}T_{p+1}\|^{-1}_{\infty,D(\frac{1}{2})}\right\\}\,,\\\\[5.0pt]
\end{array}\right.$

where $\|\cdot\|_{\infty,D(\frac{1}{2})}$ denotes the $L^{\infty}$ norm of
functions on the disk $D(\frac{1}{2})$ of radius $\frac{1}{2}$.

######  Proposition C.2.

For any $\rho\leq\min\left\\{\rho_{0},\rho_{1}\right\\}$,

  1. (i)

the function $\omega\mapsto W(\rho,\omega)$ does not vanish in

$[0,2\pi]\\!\setminus\\!\bigcup_{j=0}^{2p-1}\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)=\bigcup_{j=0}^{2p-1}\left[\omega_{j}+\alpha_{p},\omega_{j+1}-\alpha_{p}\right]\,;$

  2. (ii)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the function $\omega\mapsto
W(\rho,\omega)$ has exactly one zero
$\widetilde{\omega}_{j}(\rho)\in\left(\omega_{j}-\alpha_{p}\,,\,\omega+\alpha_{p}\right)$;

  3. (iii)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the function
$\rho\mapsto\widetilde{\omega}_{j}(\rho)$ is $C^{\infty}$ in the interval
$\left(0\,,\,\min\left\\{\rho_{0},\rho_{1}\right\\}\right)$ and tends to
$\omega_{j}$ as $\rho$ tends to zero;

  4. (iv)

for each $j\in\left\\{0,\ldots,2p-1\right\\}$, the curve

$\left(0\,,\,\min\left\\{\rho_{0},\rho_{1}\right\\}\right)\ni\rho\mapsto
a_{j}(\rho)=\left(\rho\,\cos(\widetilde{\omega}_{j}(\rho)\,,\,\rho\,\sin(\widetilde{\omega}_{j}(\rho)\right)$

is smooth and has semi-tangent $\omega_{j}$ at the origin.

_Proof._ To prove (i), we observe that in each interval
$\left\\{\rho\right\\}\times\left[\omega_{j}+\alpha_{p},\omega_{j+1}-\alpha_{p}\right]$,
$|W(\rho,\omega)|\geq\frac{1}{2}\sin(\alpha_{1})$. To prove (ii), we observe
that the function $W(\rho,\omega)$ changes sign in
$\left\\{\rho\right\\}\times\left(\omega_{j}-\alpha_{p},\omega_{j}+\alpha_{p}\right)$
and that its partial derivative with respect to $\omega$ does not vanish.
Assertion (iii) follows from the implicit function theorem. Assertion (iv)
follows from the previous ones. ∎

######  Remark C.3.

Later on, we will encounter the following situation. There exist two
eigenfunctions $v_{1}$ and $v_{2}$ such that the functions
$h_{i}=v_{i}\circ\psi$ satisfy the relations

(C.11)
$\left\\{\begin{array}[]{l}h_{1}(\rho\,\cos\omega,\rho\,\sin\omega)=\rho^{p}\,\sin(p\,\omega)+R_{1,p+1}(\rho\,\cos\omega,\rho\,\sin\omega)\,,\\\\[5.0pt]
h_{2}(\rho\,\cos\omega,\rho\,\sin\omega)=\rho^{p}\,\cos(p\,\omega)+R_{2,p+1}(\rho\,\cos\omega,\rho\,\sin\omega)\,.\end{array}\right.$

Defining the family of functions
$w_{\theta}=\cos\theta\,v_{1}-\sin\theta\,v_{2}$, the associated family of
functions $h_{\theta}=w_{\theta}\circ\psi$, satisfies

(C.12)
$\left\\{\begin{array}[]{l}h_{\theta}(\rho\,\cos\omega,\rho\,\sin\omega)=\rho^{p}\,\sin(p\,(\omega-\theta))+R_{\theta,p+1}(\rho\,\cos\omega,\rho\,\sin\omega)\,,\text{\leavevmode\nobreak\
with}\\\\[5.0pt]
R_{\theta,p+1}=\cos\theta\,R_{1,p+1}-\sin\theta\,R_{2,p+1}.\end{array}\right.$

Then, Proposition C.2 remains valid for the family $h_{\theta}$, uniformly
with respect to the variable $\theta\in[0,2\pi]$, with $\omega_{j}$ replaced
by $(\omega_{j}-\theta)$, and the corresponding functions
$\rho\mapsto\widetilde{\omega}_{j}(\rho,\theta)$ and $\rho\mapsto
a_{j}(\rho,\theta)$ are smooth in $(\rho,\theta)$.

######  Remark C.4.

Proposition C.2 tells us that, in a neighborhood of the critical zero $x$ of
$u$, the nodal set $\mathcal{Z}(u)$ consists of $p$ smooth semi-arcs emanating
from $x$ tangentially to the rays $\omega_{j}$. An alternative proof is to
take exponential coordinates at $x$, and polar coordinates in $T_{x}M$, as in
Appendix B. Cheng [Chen1976] gives a more precise, yet less quantitative,
result by showing that there exists a local diffeomorphism sending the nodal
set of $u$ in a neighborhood of the singular point $x$ onto the nodal set of
higher order terms of the Taylor expansion of $u$ at $x$ (a harmonic
polynomial) which consists of rays.

##  Appendix D Labeling loops and nodal domains

In this Appendix, $(M,g)$ is a surface of genus $0$ and we consider
eigenfunctions of the operator $-\Delta+V$, where $g$ is a $C^{\infty}$ metric
and $V$ a $C^{\infty}$ real valued potential.

Similar arguments can be applied to smooth bounded domains in
$\mathbb{R}^{2}$.

###  D.1. Labeling of loops and bouquets of loops

Let $u$ be an eigenfunction such that $\operatorname{ord}(u,x)=p$, with
$x\in\operatorname{int}(M)$. Furthermore, assume that the point $x$ is the
sole critical zero of $u$, and that the nodal set $\mathcal{Z}(u)$ is
connected.

In a small neighborhood of $x$, the nodal set $\mathcal{Z}(u)$ consists of
$2p$ nodal semi-arcs emanating from $x$, tangentially to $2p$ distinct777The
rays actually form an equiangular system, but this is not needed here. rays.
Fix an orientated $2$-frame in $T_{x}M$, with first vector supported by one of
the rays, denoted by $\omega_{0}$, and label the $2p$ rays counter-clockwise,
$\omega_{0},\omega_{1},\ldots,\omega_{2p-1}$. Denote by $(r,\omega)$ the polar
coordinates associated with this frame in $T_{x}M$.

Since nodal arcs can only meet at critical zeros, the nodal semi-arc emanating
from $x$ tangentially to the ray $\omega_{j}$ must eventually come back to
$x$, arriving tangentially to some ray $\omega_{\tau(j)}$, forming a loop
$\gamma_{j,\tau(j)}$ at $x$. Hence, we have a map,

(D.1) $\tau:L_{0}\mapsto L_{0}\,,\text{\leavevmode\nobreak\
where\leavevmode\nobreak\ \leavevmode\nobreak\
}L_{0}:=\left\\{0,\ldots,2p-1\right\\}\,,$

with the following properties,

(D.2) $\left\\{\begin{array}[]{l}\tau^{2}=\operatorname{Id}\,,\\\\[5.0pt]
\tau(j)\not=j\,,\quad\forall j\in L_{0}\,,\\\\[5.0pt]
\tau(j)-j\text{\leavevmode\nobreak\ is odd}\,,\quad\forall j\in
L_{0}\,.\end{array}\right.$

The first property is clear. The second and third ones follow from the local
structure theorem.

The global picture is that the nodal set $\mathcal{Z}(u)$ is a _$p$ -bouquet
of loops_ $W(L_{0})$ at $x$: $p$ simple loops at $x$ which do not intersect
away from $x$, and which meet transversally at $x$.

Take any loop, $\gamma_{j,\tau(j)}$. Exchanging, $j$ and $\tau(j)$ if
necessary, we may always assume that $j<\tau(j)$. Consider the sets

(D.3)
$\left\\{\begin{array}[]{l}L_{(j)}:=\left\\{j,(j+1),\ldots,(\tau(j)-1),\tau(j)\right\\}\,,\\\\[5.0pt]
J^{\prime}_{(j)}:=L_{(j)}\\!\setminus\\!\left\\{j,\tau(j)\right\\}\end{array}\right.$

Since $M$ is topologically a sphere,
$M\\!\setminus\\!\left\\{\gamma_{j,\tau(j)}\right\\}$ has two components. The
local structure of $\mathcal{Z}(u)$ at $x$ shows that the rays
$\omega_{k},k\in J^{\prime}_{(j)}$, point inside one of the components, call
it $A_{(j)}$. The corresponding nodal arcs must stay in $A_{(j)}$, so that
$J^{\prime}_{(j)}$ is invariant under $\tau$, and corresponds to a bouquet of
loops $W(J^{\prime}_{(j)})\subset A_{(j)}$. Similarly, the rays
$\omega_{k},k\in L_{0}\\!\setminus\\!L_{(j)}$, point inside the other
component, call it $A^{\prime}_{(j)}$. The corresponding nodal arcs stay in
$A^{\prime}_{(j)}$, so that $L_{0}\\!\setminus\\!L_{(j)}$ is invariant under
$\tau$, and corresponds to a bouquet of loops
$W(L_{0}\\!\setminus\\!L_{(j)})\subset A^{\prime}_{(j)}$.

In the polar coordinates $(r,\omega)$, and for $r$ small enough, the nodal
semi-arcs emanating from $x$ are given by equations
$r\mapsto\exp_{x}(r\,\widetilde{\omega}_{j}(r))$, $0\leq j\leq 2p-1$. These
arcs determine $2p$ intervals

(D.4) $I_{j}(r):=\left\\{\exp_{x}(r\,\omega)\leavevmode\nobreak\
|\leavevmode\nobreak\
\omega\in\left(\widetilde{\omega}_{j}(r),\widetilde{\omega}_{j+1}(r)\right)\right\\},\quad
0\leq j\leq 2p-1\,,$

on the circle $S_{x}(r)=\left\\{\exp_{x}(r\,\omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ \omega\in[0,2\pi]\right\\}$. The function $u$ does not
vanish in these open intervals, and changes sign while crossing an end point
$\exp_{x}(r\,\widetilde{\omega}_{j}(r))$ along the circle. In particular, each
open interval is contained in a unique nodal domain, and two contiguous
intervals are contained in different nodal domains. Note that the point $x$ is
in the closure of each nodal domain of $u$, so that each nodal domain meets at
least one interval.

######  Lemma D.1.

The complement of a $p$-bouquet of loops $W$ in $M$ has $(p+1)$ components.

_Proof._ Looking at the $p$-bouquet as a connected multi-graph with $1$ vertex
and $p$ edges, we find that the number of components in the complement of the
bouquet $W$ is
$\alpha_{2}(W,\mathbb{S}^{2})=\alpha_{1}(W)-\alpha_{0}(W)+c(W)+1=p+1$ (with
the notation of Section 4). Note that a $p$-bouquet of loops is not a graph in
the sense of Giblin [Gibl2010]. We can modify the bouquet as in Subsection 4.1
to turn it into a graph, so that we can apply Euler’s formula for the sphere.
∎

###  D.2. Labeling of nodal domains

We now define a labeling of nodal domains.

######  Definition D.2.

Label the nodal domains by introducing the map

$\delta:L_{0}\to\left\\{1,\ldots,p\right\\}$

such that the interval $I_{j}(r)$ is contained in the nodal domain
$\Omega_{\delta(j)}$. More precisely,

  1. (1)

let $\Omega_{1}$ be the nodal domain which contains the interval $I_{0}(r)$,
i.e., $\delta(0):=1$;

  2. (2)

let $\Omega_{2}$ be the nodal domain which contains the interval $I_{1}(r)$,
i.e., $\delta(1):=2$;

  3. (3)

for $j\geq 1$, assuming that $\delta(1),\ldots,\delta(j)$ are defined, let

    1. $\diamond$

$\delta(j+1)=\delta(\ell)$, if the interval $I_{j+1}(r)$ is contained in the
nodal domain $\Omega_{\delta(\ell)}$ for some $\ell\leq j$, i.e. in a nodal
domain already labeled;

    2. $\diamond$

let $\delta(j+1)=\delta(j)+1$, if the interval $I_{j+1}(r)$ is not contained
in an already labeled nodal domain.

######  Property D.3.

Once we have chosen an orientation in $T_{x}M$, an initial ray $\omega_{0}$,
and labeled the other rays counter-clockwise, the map $\tau:L_{0}\to L_{0}$
determines the map $\delta:L_{0}\to\left\\{1,\ldots,p\right\\}$, and
conversely.

####  D.2.1. Working out an example

Choose $p=8$, so that

$L_{0}=\left\\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\right\\}\,,$

and define the map $\tau$ in matrix form ($\tau$ maps the first line to the
second line) by

(D.5) $\tau\leavevmode\nobreak\ =\leavevmode\nobreak\
\setcounter{MaxMatrixCols}{16}\begin{pmatrix}0&1&2&3&4&5&6&7&8&9&{10}&11&12&13&14&15\\\
3&2&1&0&9&8&7&6&5&4&15&12&11&14&13&10\\\ \end{pmatrix}\,.$

Since $\tau(0)=3$, $\tau(4)=9$ and $\tau(10)=15$, we decompose $L_{0}$ into
three subsets $L_{1}=\left\\{0,1,2,3\right\\}$,
$L_{2}=\left\\{4,5,6,7,8,9\right\\}$ and
$L_{3}=\left\\{10,11,12,13,14,15\right\\}$,

(D.6) $\tau\leavevmode\nobreak\ =\leavevmode\nobreak\
\left(\begin{vmatrix}0&1&2&3\\\ 3&2&1&0\\\
\end{vmatrix}\begin{vmatrix}4&5&6&7&8&9\\\ 9&8&7&6&5&4\\\
\end{vmatrix}\begin{vmatrix}10&11&12&13&14&15\\\ 15&12&11&14&13&10\\\
\end{vmatrix}\right)\,,$

It is clear from the definition of $\tau$ that these subsets are
$\tau$-invariant. The reason behind this fact is explained in Subsection D.1.

$\diamond$ Since $\tau(0)=3$, the first subset we consider is
$L_{1}=\left\\{0,1,2,3\right\\}$. Let $\ell_{1}=0$, $k_{1}:=|L_{1}|/2=2$. The
set $L_{1}$ corresponds to a $k_{1}$-bouquet of loops,
$W(L_{1}):=\left\\{\gamma_{0,3},\gamma_{1,2}\right\\}$. The set
$J^{\prime}_{1}:=L_{1}\\!\setminus\\!\left\\{0,3\right\\}=\left\\{1,2\right\\}$
corresponds to a $(k_{1}-1)$-bouquet of loops,
$W(J^{\prime}_{1}):=\left\\{\gamma_{1,2}\right\\}$.

Since $M$ is a sphere, $M\\!\setminus\\!\gamma_{0,3}$ has two components
$A_{1}$ and $A^{\prime}_{1}$, and we choose $A_{1}$ such that
$W(J^{\prime}_{1})\subset A_{1}$ (see Subsection D.1). We begin labeling the
nodal domains.

We have $\delta(0)=1$, $\delta(1)=2$. Since $k_{1}=2$, $W(L_{1})$ contains two
loops, $I_{0}(r),I_{1}(r)$ and $I_{2}(r)$ are contained in $A_{1}$, and we
must have $\delta(2)=\delta(0)=1$. Furthermore, since $A_{1}$ is simply
connected, the set $A_{1}\\!\setminus\\!W(J^{\prime}_{1})$ has two components
which are nodal domains of $u$. It also follows that $I_{3}(r)\subset
A^{\prime}_{1}$, which implies that $\delta(3)=k_{1}+1=3$. We have actually
determined $\delta$ on $L_{1}$.

$\diamond$ Since $\tau(4)=9$, we turn to the second set,
$L_{2}=\left\\{4,5,6,7,8,9\right\\}$, with $\ell_{2}=4$, $k_{2}=|L_{2}|/2=3$,
and $J^{\prime}_{2}=\left\\{5,6,7,8\right\\}$. The set
$M\\!\setminus\\!\gamma_{4,9}$ has two components, $A_{2}$ and
$A^{\prime}_{2}$, and we choose $A_{2}$ such that $W(J^{\prime}_{2})\subset
A_{2}$. Notice that the nodal domain $\Omega_{3}$ (corresponding to
$\delta(3)$ is contained in $A^{\prime}_{1}\cap A^{\prime}_{2}$. Since $A_{2}$
is simply connected, the set $A_{2}\\!\setminus\\!W(J^{\prime}_{2})$ has
$k_{2}=3$ components which are nodal domains of $u$. These nodal domains are
different from the previous labeled ones ($\Omega_{1},\Omega_{2}$ and
$\Omega_{3}$). We label them beginning with $k_{1}+2=4$; furthermore
$\delta(\ell_{2})=\delta(\tau(\ell_{2})-1)=k_{1}+2=4$, i.e.
$\delta(4)=\delta(8)=4$. To continue, we look at
$J^{\prime}_{2}=\left\\{5,6,7,8\right\\}$ which is invariant by $\tau$. The
action of $\tau$ on this set is similar to the action of $\tau$ on $L_{1}$.
Taking into account the fact that we must have $\delta(5)=k_{1}+3=5$, we
conclude that $\delta(5)=5,\delta(6)=6$ and $\delta(7)=5$. Notice that
$\delta(9)=k_{1}+1=3$. We have labeled $1+k_{1}+k_{2}$ nodal domains.

$\diamond$ Since $\tau(10)=15$, we turn to the third set,
$L_{3}=\left\\{10,11,12,13,14,15\right\\}$, with $\ell_{3}=10$,
$k_{3}=|L_{3}|/2=3$ and $J^{\prime}_{3}=\left\\{11,12,13,14\right\\}$. Notice
that the actions of $\tau$ on $L_{2}$ and $L_{3}$ are different. The set
$M\\!\setminus\\!\gamma_{10,15}$ has two components, $A_{3}$ and
$A^{\prime}_{3}$, and we choose $A_{3}$ such that $W(J^{\prime}_{3})\subset
A_{3}$. Notice that the nodal domain $\Omega_{3}$ (corresponding to
$\delta(3)$ is contained in $A^{\prime}_{1}\cap A^{\prime}_{2}\cap
A^{\prime}_{3}$. Since $A_{3}$ is simply connected, the set
$A_{3}\\!\setminus\\!W(J^{\prime}_{3})$ has $k_{3}=3$ components which are
nodal domains of $u$. These nodal domains are different from the previous
labeled ones ($\Omega_{1},\ldots,\Omega_{6}$). We label them beginning with
$k_{2}+k_{1}+2=7$; furthermore
$\delta(\ell_{3})=\delta(\tau(\ell_{3})-1)=k_{2}+k_{1}+2=7$, i.e.
$\delta(10)=\delta(14)=7$. To continue, we look at the sets
$\left\\{11,12\right\\}$ and $\left\\{13,14\right\\}$, and we easily infer
that $\delta(11)=8,\delta(12)=7$, and $\delta(13)=9$.

We have proved, for this explicit example, that the map $\tau$ determines the
map $\delta$.

We can describe the map $\delta$ by the following matrix

(D.7) $\delta\leavevmode\nobreak\ =\leavevmode\nobreak\
\setcounter{MaxMatrixCols}{16}\begin{pmatrix}0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15\\\
1&2&1&3&4&5&6&5&4&3&7&8&7&9&7&3\\\ \end{pmatrix}\,,$

where $\delta$ sends the entry $j\in L_{0}$ in the first line, the label of
the interval $I_{j}(r)$, to the entry $\delta(j)$ in the second line, the
label of the nodal domain $\Omega_{\delta(j)}$.

To determine $\tau(0)$, we look at the occurrences of $1$, the label of the
first nodal domain $\Omega_{1}$: we find $1=\delta(0)=\delta(2)$, and notice
that $\delta(3)=\delta(15)=3$. We have determined that
$L_{1}=\left\\{0,1,2,3\right\\}$. This also tells us that the nodal domain
$\Omega_{2}$ is bounded by a single loop. It follows that $\tau$ on $L_{1}$ is
given by

$\tau|_{L_{1}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{pmatrix}0&1&2&3\\\ 3&2&1&0\\\ \end{pmatrix}\,.$

The next nodal domain which appears is $\Omega_{4}$. To determine $\tau(4)$,
we look at the occurrences of $4$, $4=\delta(4)=\delta(8)$, and we note that
$\delta(3)=\delta(9)=3$. This means that $\tau(4)=9$, so that
$L_{2}=\left\\{4,5,6,7,8,9\right\\}$. In $A_{2}$, the label $4$ occurs twice,
the label $5$ occurs twice as well, and the label $6$ once. We conclude that

$\tau|_{L_{2}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{pmatrix}4&5&6&7&8&9\\\ 9&8&7&6&5&4\\\ \end{pmatrix}\,.$

The next nodal domain to appear is $\Omega_{7}$. To determine $\tau(10)$, we
look at the occurrences of $7$: $7=\delta(10)=\delta(12)=\delta(14)$, while
$3=\delta(9)=\delta(15)$. This determines $L_{3}$, and actually
$\tau|_{L_{(}3)}$ as well,

$\tau|_{L_{3}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{pmatrix}10&11&12&13&14&15\\\ 15&12&11&14&13&10\\\ \end{pmatrix}\,.$

We have recovered the map $\tau$ from the map $\delta$ in the example at hand.

####  D.2.2. Proof of Property D.3

$\bullet$ We first show that the map $\tau$ determines the map $\delta$.

$\diamond$ We first consider the loop $\gamma_{0,\tau(0)}$. Note that
$0<\tau(0)\leq 2p-3$. Define

(D.8) $\left\\{\begin{array}[]{l}\ell_{1}=0\,,\\\\[5.0pt]
L_{1}=\left\\{0,\ldots,\tau(0)\right\\}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\
}L_{1}^{\prime}=L_{1}\\!\setminus\\!\left\\{0,\tau(0)\right\\}\,,\\\\[5.0pt]
k_{1}=\frac{1}{2}|L_{1}|=\frac{1}{2}\left(\tau(0)+1\right)\,.\end{array}\right.$

The loop $\gamma_{0,\tau(0)}$ divides $M$ into two components, both
homeomorphic to a disk (recall that $M$ is topologically a sphere). The rays
with index in $J^{\prime}_{1}$ point inside one of the components, call this
first component $A_{1}$; the rays with index in $L_{0}\\!\setminus\\!L_{1}$
point inside the second component $A^{\prime}_{1}$. In case
$J^{\prime}_{1}=\emptyset$, there is no ray pointing into the component
$A_{1}$, and $A_{1}$ is actually a nodal domain bounded by the loop
$\gamma_{0,1}$. Because the loops at $x$ do not intersect away from $x$, the
loops with semi-tangents in $J^{\prime}_{1}$ must stay inside $A_{1}$; the
loops with semi-tangents in $L_{0}\\!\setminus\\!L_{1}$ must stay inside
$A^{\prime}_{1}$.

In particular, this implies that the map $\tau$ leaves the sets $L_{1}$,
$J^{\prime}_{1}$ and $L_{0}\\!\setminus\\!L_{1}$ globally invariant. To the
set $L_{1}$ we can therefore associate a $k_{1}$-bouquet $W(L_{1})$ of loops
at $x$, whose complement in $M$ has $(k_{1}+1)$ components. Similarly, to the
set $J^{\prime}_{1}$ we can associate a $(k_{1}-1)$-bouquet
$W(J^{\prime}_{1})$ which is contained in $A_{1}$, and whose complement in
$A_{1}$ has $k_{1}$ components which are actually nodal domains of $u$. The
remaining nodal domains are all contained in $A^{\prime}_{1}$. The $k_{1}$
nodal domains contained in $A_{1}$ are labeled from $1$ to $k_{1}$; the
labeling of the nodal domains contained in $A^{\prime}_{1}$ begins with
$k_{1}+1$. The intervals $I_{j}(r),0\leq j\leq\tau(0)-1$ are contained in
$A_{1}$; the intervals $I_{j}(r),\tau(0)\leq j$ are contained in
$A^{\prime}_{1}$. From these facts, we infer that

(D.9) $\left\\{\begin{array}[]{l}\delta(0)=\delta(\tau(0)-1)=1\,,\\\\[5.0pt]
\delta(1)=2\,,\\\\[5.0pt] \delta(\tau(0))=\delta(2p-1)=k_{1}+1\,,\\\\[5.0pt]
\delta(\tau(0)+1)=k_{1}+2\,.\end{array}\right.$

$\diamond$ Since the labeling of the nodal domains contained in $A_{1}$ and
the labeling of the nodal domains contained in $A^{\prime}_{1}$ are
essentially independent matters, we can pursue labeling the nodal domains in
$A_{1}$ as if we only worked with the bouquet $W(L_{0}\\!\setminus\\!L_{1})$.

The domain $A_{1}$ is simply connected, with boundary $\gamma_{0,\tau(0)}$.
Any loop $\gamma$ in the bouquet $W(J^{\prime}_{1})$ is contained in $A_{1}$
and hence separates $A_{1}$ into two components, a component $A_{\gamma}$
which only touches $\gamma_{0,\tau(0)}$ at the point $x$, and a component
$A^{\prime}_{\gamma}$ one of whose boundaries is $\gamma_{0,\tau(0)}$, and
hence meets the nodal set $\Omega_{1}$.

We now look at the bouquets of nodal loops contained in $A_{1}$. If
$\tau(1)=2$, the corresponding loop $\gamma_{1,2}$ bounds the nodal domain
$\Omega_{2}$, we have $\delta(3)=3$, and we are left with analyzing the
smaller bouquet associated with the set
$\left\\{3,\ldots,(\tau(0)-1)\right\\}$. If $\tau(1)>2$, we apply the above
reasoning with the bouquet associated with the set
$\left\\{1,\ldots,\tau(1)\right\\}$.

$\diamond$ We now define

(D.10)
$\left\\{\begin{array}[]{l}\ell_{2}=\tau(\ell_{1})+1=\tau(0)+1\,,\\\\[5.0pt]
L_{2}=\left\\{\ell_{2},\ldots,\tau(\ell_{2})\right\\}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\
}L_{2}^{\prime}=L_{2}\\!\setminus\\!\left\\{\ell_{2},\tau(\ell_{2})\right\\}\,,\\\\[5.0pt]
k_{2}=\frac{1}{2}|L_{2}|=\frac{1}{2}\left(\tau(\ell_{2})-\ell_{2}+1\right)\,.\end{array}\right.$

We reason as above. The loop $\gamma_{\ell_{2},\tau(\ell_{2})}$ divides $M$
into two components. One of them $A_{2}$ contains the $(k_{1}-1)$-bouquet of
loops $W(J^{\prime}_{2})$; the other one, denoted by $A^{\prime}_{2}$,
contains the nodal domains $\Omega_{k_{0}+1}$. The complement of
$W(J^{\prime}_{2})$ in $A_{2}$ has $k_{2}$ components which are nodal domains,
which are labeled $\Omega_{k_{0}+2},\ldots\Omega_{k_{0}+k_{1}+1}$. In
particular, we have $\delta(k_{0}+1)=k_{0}+2$ and
$\delta(k_{0}+k_{1}+1)=k_{0}+1$.

We then continue as previously by analyzing smaller and smaller bouquets of
loops inside $A_{2}$.

After finitely many such steps, we exhaust the set $L_{0}$. This proves that
the map $\tau$ determines the map $\delta$.

$\bullet$ We now show that the map $\delta$ determines the map $\tau$.

It is not clear which maps $\delta$ correspond to maps $\tau$ (with the above
properties). Assume that we are given some $\delta$ associated with some
$p$-bouquet of loops as above. The $p$-bouquet actually defines some map
$\tau$ which induces $\delta$. In order to recover $\tau$ from $\delta$, the
idea is to recover the sets $L_{i}$ introduced above, and to reason by
induction on the length of the size of the bouquets.

Assume that we are given a map
$\delta:L_{0}:=\left\\{0,\ldots,2p-1\right\\}\to\left\\{1,\ldots,p\right\\}$,
induced by some map $\tau:L_{0}\to L_{0}$ such that
$\tau^{2}=\operatorname{Id}$ and, for all $j\in L_{0}$, $\tau(j)\not=j$, and
$\tau(j)-j$ is odd. Does $\delta$ determine $\tau$?

$\diamond$ We first look at the nodal domain $\Omega_{1}$, with label $1$, and
at the set $\delta^{-1}(1)$. If $\delta^{-1}(1)=\left\\{0\right\\}$, then we
must have $\tau(0)=1$, and the loop $\gamma_{0,1}$ bounds $\Omega_{1}$. If
$|\delta^{-1}(1)|>1$, we look at $m_{1}:=\max\delta^{-1}(1)$. Then, the only
possibility is that $\tau(0)=m_{1}+1$. Because of our framework, see
Subsection 7.6, the subsets $L_{1}:=\left\\{0,\ldots,(m_{1}+1)\right\\}$,
$J^{\prime}_{1}:=\left\\{1,\ldots,m_{1}\right\\}$, and
$L_{0}\\!\setminus\\!L_{1}$ must be $\tau$ invariant. Defining $A_{1}$ and
$A^{\prime}_{1}$ as above, we see that there are exactly $k_{1}$ nodal domains
inside $A_{1}$, where $2k_{1}=|L_{1}|$, and we have $\delta(m_{1}+1)=k_{1}+1$.
We can then use induction.

## References

  * [Ales1994] Giovanni Alessandrini.  Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains.  Comment. Math. Helv., 69(1):142–154, 1994.
  * [ArMa2007] Wolfgang Arendt and Rafe Mazzeo.  Spectral properties of the dirichlet-to-neumann operator on lipschitz domains.  Ulmer Seminare, 12:28–38, 2007.
  * [ArMa2012] Wolfgang Arendt and Rafe Mazzeo.  Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup.  Commun. Pure Appl. Anal., 11(6):2201–2212, 2012.
  * [Aron1957] Nachman Aronszajn.  A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order.  J. Math. Pures Appl. (9), 36:235–249, 1957.
  * [BeGM1971] Marcel Berger, Paul Gauduchon, and Edmond Mazet.  Le spectre d’une variété riemannienne.  Lecture Notes in Mathematics, Vol. 194. Springer-Verlag, Berlin-New York, 1971.
  * [BeGi2016] Michiel van den Berg and Katie Gittins.  On the number of Courant-sharp Dirichlet eigenvalues.  J. Spectr. Theory, 6(4):735–745, 2016.
  * [BeGo1988] Marcel Berger and Bernard Gostiaux.  Géométrie différentielle: variétés, courbes et surfaces.  Mathématiques. [Mathematics]. Presses Universitaires de France, Paris, 1987.
  * [BeHK2021k] Pierre Bérard, Bernard Helffer, and Rola Kiwan.  Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders.  Proc. Amer. Math. Soc., 150(1):439–453, 2021.
  * [BeHK2021m] Pierre Bérard, Bernard Helffer, and Rola Kiwan.  Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip.  Port. Math., 78(1):1–41, 2021.
  * [BeHe2014] Pierre Bérard and Bernard Helffer.  Remarks on the boundary set of spectral equipartitions.  Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372(2007):20120492, 15, 2014.
  * [BeHe2015r] Pierre Bérard and Bernard Helffer.  Nodal sets of eigenfunctions, Antonie Stern’s results revisited.  In Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2014–2015, pages 1–37. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, 2015.
  * [BeHe2015s] Pierre Bérard and Bernard Helffer.  Edited extracts from Antonie Stern’s thesis.  In Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2014–2015, pages 39–72. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, 2015.
  * [BeHe2016] Pierre Bérard and Bernard Helffer.  The weak Pleijel theorem with geometric control.  J. Spectr. Theory, 6(4):717–733, 2016.
  * [BeHe2020m] Pierre Bérard and Bernard Helffer.  An Euler-type formula for partitions on the Möbius strip.  arXiv:2005.12571, 2020.
  * [BeHe2021t] Pierre Bérard and Bernard Helffer.  Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains application to the extended Courant property.  Ann. Fac. Sci. Toulouse Math. (6), 30(3):429–462, 2021.
  * [BeLi2001] Michiel van den Berg and Maria Lianantonakis.  Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions.  Indiana Univ. Math. J., 50(1):299–333, 2001.
  * [BeNP2016] Aleksandr Berdnikov, Nikolai Nadirashvili, and Alexei Penskoi.  Bounds on multiplicities of Laplace-Beltrami operator eigenvalues on the real projective plane.  arXiv:1612.04805, 2016.
  * [Berd2018] Aleksandr Berdnikov.  Bounds on multiplicities of Laplace operator eigenvalues on surfaces.  J. Spectr. Theory, 8(2):541–554, 2018.
  * [Bess1980] Gérard Besson.  Sur la multiplicité de la première valeur propre des surfaces riemanniennes.  Ann. Inst. Fourier (Grenoble), 30(1):x, 109–128, 1980.
  * [BoHe2017] Virginie Bonnaillie-Noël and Bernard Helffer.  Nodal and spectral minimal partitions—the state of the art in 2016.  In Shape optimization and spectral theory, pages 353–397. De Gruyter Open, Warsaw, 2017.
  * [Bona2009] Francis Bonahon.  Low-dimensional geometry, volume 49 of Student Mathematical Library.  American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2009.  From Euclidean surfaces to hyperbolic knots, IAS/Park City Mathematical Subseries.
  * [BuCo1985] Marc Burger and Bruno Colbois.  À propos de la multiplicité de la première valeur propre du Laplacien d’une surface de Riemann.  C. R. Acad. Sci. Paris Sér. I Math., 300(8):247–249, 1985.
  * [Chen1976] Shiu Yuen Cheng.  Eigenfunctions and nodal sets.  Comment. Math. Helv., 51(1):43–55, 1976.
  * [CoCo1988] Bruno Colbois and Yves Colin de Verdière.  Sur la multiplicité de la première valeur propre d’une surface de Riemann à courbure constante.  Comment. Math. Helv., 63(2):194–208, 1988.
  * [ColV1986] Yves Colin de Verdière.  Sur la multiplicité de la première valeur propre non nulle du Laplacian.  Comment. Math. Helv., 61(2):254–270, 1986.
  * [ColV1987] Yves Colin de Verdière.  Construction de laplaciens dont une partie finie du spectre est donnée.  Ann. Sci. École Norm. Sup. (4), 20(4):599–615, 1987.
  * [Colb1985] Bruno Colbois.  Petites valeurs propres du Laplacien sur une surface de Riemann compacte et graphes.  C. R. Acad. Sci. Paris Sér. I Math., 301(20):927–930, 1985\.
  * [DaGH2021] Joel Dahne, Javier Gómez-Serrano, and Kimberly Hou.  A counterexample to Payne’s nodal line conjecture with few holes.  Commun. Nonlinear Sci. Numer. Simul., 103:Paper No. 105957, 13, 2021\.
  * [Dies2017] Reinhard Diestel.  Graph theory, volume 173 of Graduate Texts in Mathematics.  Springer, Berlin, fifth edition, 2017.
  * [DoFe1990a] Harold Donnelly and Charles Fefferman.  Nodal sets of eigenfunctions: Riemannian manifolds with boundary.  In Analysis, et cetera, pages 251–262. Academic Press, Boston, MA, 1990.
  * [FoBP2021] Maxime Fortier Bourque and Bram Petri.  The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian.  arXiv: 2111.14699.
  * [Four2001] Søren Fournais.  The nodal surface of the second eigenfunction of the Laplacian in ${\bf R}^{D}$ can be closed.  J. Differential Equations, 173(1):145–159, 2001.
  * [FrSc2016] Ailana Fraser and Richard Schoen.  Sharp eigenvalue bounds and minimal surfaces in the ball.  Invent. Math., 203(3):823–890, 2016.
  * [Frie1991] Leonid Friedlander.  Some inequalities between Dirichlet and Neumann eigenvalues.  Arch. Rational Mech. Anal., 116(2):153–160, 1991.
  * [GaHuLa2004] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine.  Riemannian geometry.  Universitext. Springer-Verlag, Berlin, third edition, 2004.
  * [GaXu2013] Jean Gallier and Dianna Xu.  A guide to the classification theorem for compact surfaces, volume 9 of Geometry and Computing.  Springer, Heidelberg, 2013.
  * [GiHe2019] Katie Gittins and Bernard Helffer.  Courant-sharp Robin eigenvalues for the square and other planar domains.  Port. Math., 76(1):57–100, 2019.
  * [GiLe2020] Katie Gittins and Corentin Léna.  Upper bounds for Courant-sharp Neumann and Robin eigenvalues.  Bull. Soc. Math. France, 148(1):99–132, 2020.
  * [Gibl2010] Peter Giblin.  Graphs, surfaces and homology.  Cambridge University Press, Cambridge, third edition, 2010.
  * [HeHJ2020] Bernard Helffer, Thomas Hoffmann-Ostenhof, François Jauberteau, and Corentin Léna.  On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –.  Asymptotic Analysis, 121(1):35–57, 2021.
  * [HeHN2002] Bernard Helffer, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and Nikolai Nadirashvili.  Spectral theory for the dihedral group.  Geom. Funct. Anal., 12(5):989–1017, 2002.
  * [HeHO1999] Bernard Helffer, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and Mark P. Owen.  Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains.  Comm. Math. Phys., 202(3):629–649, 1999.
  * [HeHT2009] Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini.  Nodal domains and spectral minimal partitions.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(1):101–138, 2009.
  * [HoHN1997] Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and Nikolai Nadirashvili.  The nodal line of the second eigenfunction of the Laplacian in ${\bf R}^{2}$ can be closed.  Duke Math. J., 90(3):631–640, 1997.
  * [HoHN1998] Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and Nikolai Nadirashvili.  On the nodal line conjecture.  In Advances in differential equations and mathematical physics (Atlanta, GA, 1997), volume 217 of Contemp. Math., pages 33–48. Amer. Math. Soc., Providence, RI, 1998.
  * [HoHN1999] Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and Nikolai Nadirashvili.  On the multiplicity of eigenvalues of the Laplacian on surfaces.  Ann. Global Anal. Geom., 17(1):43–48, 1999.
  * [HoMN1999] Thomas Hoffmann-Ostenhof, Peter W. Michor, and Nikolai Nadirashvili.  Bounds on the multiplicity of eigenvalues for fixed membranes.  Geom. Funct. Anal., 9(6):1169–1188, 1999.
  * [Horm2009d] Lars Hörmander.  The analysis of linear partial differential operators. IV.  Classics in Mathematics. Springer-Verlag, Berlin, 2009.  Fourier integral operators, Reprint of the 1994 edition.
  * [Ivri1980] Victor Ja. Ivriĭ.  Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary.  Funct Anal Its Appl., 14(2):98–106, 1980.
  * [Ivri1980r] Victor Ja. Ivriĭ.  The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary.  Funktsional. Anal. i Prilozhen., 14(2):25–34, 1980.
  * [Jam2016] Pierre Jammes.  Multiplicité du spectre de Steklov sur les surfaces et nombre chromatique.  Pacific J. Math., 282(1):145–171, 2016.
  * [KaKP2014] Mikhail Karpukhin, Gerasim Kokarev, and Iosif Polterovich.  Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces.  Ann. Inst. Fourier (Grenoble), 64(6):2481–2502, 2014.
  * [Ken2011] James B. Kennedy.  The nodal line of the second eigenfunction of the Robin Laplacian in $\mathbb{R}^{2}$ can be closed.  J. Differential Equations, 251(12):3606–3624, 2011.
  * [Ken2013] James B. Kennedy.  Closed nodal surfaces for simply connected domains in higher dimensions.  Indiana Univ. Math. J., 62(3):785–798, 2013.
  * [Kiwa2018] Rola Kiwan.  On the nodal set of a second Dirichlet eigenfunction in a doubly connected domain.  Ann. Fac. Sci. Toulouse Math. (6), 27(4):863–873, 2018.
  * [Lena2019] Corentin Léna.  Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions.  Ann. Inst. Fourier (Grenoble), 69(1):283–301, 2019.
  * [Lewy1977] Hans Lewy.  On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere.  Comm. Partial Differential Equations, 2(12):1233–1244, 1977.
  * [LiNi1988] Chang Shou Lin and Wei-Ming Ni.  A counterexample to the nodal domain conjecture and a related semilinear equation.  Proc. Amer. Math. Soc., 102(2):271–277, 1988.
  * [Lin1987] Chang Shou Lin.  On the second eigenfunctions of the Laplacian in ${\bf R}^{2}$.  Comm. Math. Phys., 111(2):161–166, 1987.
  * [Liq1995] Zhang Liqun.  On the multiplicity of the second eigenvalue of Laplacian in ${\bf R}^{2}$.  Comm. Anal. Geom., 3(1-2):273–296, 1995.
  * [Maz1991] Rafe Mazzeo.  Remarks on a paper of L. Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues.  Internat. Math. Res. Notices, 1991(4):41–48, 1991.  eigenvalues” [Arch. Rational Mech. Anal. 116 (1991), no. 2, 153–160; MR1143438 (93h:35146)].
  * [Mela1992] Antonios D. Melas.  On the nodal line of the second eigenfunction of the Laplacian in ${\bf R}^{2}$.  J. Differential Geom., 35(1):255–263, 1992.
  * [Miln1997] John W. Milnor.  Topology from the differentiable viewpoint.  Princeton University Press, Princeton, NJ, 1997.  Revised reprint of the 1965 original.
  * [Nadi1987] Nikolai S. Nadirashvili.  Multiple eigenvalues of the Laplace operator (Russian).  Mat. Sb. (N.S.), 133(175)(2):223–237, 272, 1987.  Translation, Math USSR-Sb. 61 225-238 1988.
  * [Payn1967] Lawrence E. Payne.  Isoperimetric inequalities and their applications.  SIAM Rev., 9:453–488, 1967.
  * [Payn1973] Lawrence E. Payne.  On two conjectures in the fixed membrane eigenvalue problem.  Z. Angew. Math. Phys., 24:721–729, 1973.
  * [Plej1956] Åke Pleijel.  Remarks on Courant’s nodal line theorem.  Comm. Pure Appl. Math., 9:543–550, 1956.
  * [Polt2009] Iosif Polterovich.  Pleijel’s nodal domain theorem for free membranes.  Proc. Amer. Math. Soc., 137(3):1021–1024, 2009.
  * [Putt1990] Rolf Pütter.  On the nodal lines of second eigenfunctions of the fixed membrane problem.  Comment. Math. Helv., 65(1):96–103, 1990.
  * [Putt1991] Rolf Pütter.  On the nodal lines of second eigenfunctions of the free membrane problem.  Appl. Anal., 42(3-4):199–207, 1991.
  * [Safa2001] Yuri Safarov.  Fourier Tauberian theorems and applications.  J. Funct. Anal., 185(1):111–128, 2001.
  * [Seve2002] Bruno Sévennec.  Multiplicity of the second Schrödinger eigenvalue on closed surfaces.  Math. Ann., 324(1):195–211, 2002.
  * [Shen1988] Chao Liang Shen.  On the nodal sets of the eigenfunctions of the string equation.  SIAM J. Math. Anal., 19(6):1419–1424, 1988.
  * [Ster1925] Antonie Stern.  Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen.  Math-Naturwiss. Diss., Universität Göttingen, 1925.
  * [YaZh2021] Yunyan Yang and Jie Zhou.  Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface.  J. Math. Anal. Appl., 504(2):Paper No. 125440, 39, 2021.