# On compact pseudoconcave sets

Zbigniew Slodkowski

(Date: October 2022)

###### Abstract.

Replying to three questions posed by N. Shcherbina, we show that a compact
psudoconcave set can have the core smaller than itself, that the core of a
compact set must be pseudoconcave, and that it can be decomposed into compact
pseudoconcave sets on which all smooth plurisubharmonic functions are
constant.

0\. Introduction

This paper is inspired by the recent work of N. Shcherbina [Sh55], where he
asks under what condition on a compact subset $K$ of a complex manifold $M$
there is a smooth strictly plurisubharmonic function defined in some
neighborhood of $K$. In the spirit of [ST04] and [HST17], he defines a subset
$c(K)$ of $K$, which (if nonempty) constitutes an obstacle to existence of
such a function. This subset, called the core of K, consists of all points of
$p$ of $K$, such that no smooth plurisubharmonic function on $K$ can be
strictly plurisubharmonic at $p$. The main result of Shcherbina in [Sh21] is
that a strictly plurisubharmonic function on $K$ exists (ie. $c(K)$ is
nonempty), if and only if $K$ does not contain any compact pseudoconcave set
(cf. Def. 1.1). However, he does not prove this by studying the core,but by
examining another subset of $K$, the nucleus, $n(K)$, which turns out to be
the maximal pseudoconcave subset of $K$.

Shcherbina poses three questions concerning the core. Question 5.1 [Sh21]: ”Is
$n(K)\subset c(K)$?” We give an example showing it is not true in general (Ex.
2.3). This circumstance makes pertinent his Question 5.2.:”Is the core of a
compact set always pseudoconcave?” We show this is true, and we also show,
answering his Question 5.3, that the core decomposes into pairwise disjoint
pseudoconcave sets on which every smooth plurisubharmonic function is constant
(in analogy with similar results in [PS19], and [Sl19]). These facts
constitute our main result, Th. 2.1. It might be of some interest that with
these properties of the core established, an alternative proof of Shcherbina’s
main theorem follows easily, cf. Cor. 2.3. in Section 1 we collect some facts
about compact pseudoconcave sets that we need.

0.1. Some conventions in terminology and notation. Following Shcherbina, we
say that a funcion is smooth plurisubharmonic on set $K$, if it has a smooth
plurisubharmonic extension to some neighborhood of $K$. By smooth we mean any
smoothness class $C^{k}$, with $k=0,1,...,\infty$. All arguments in Sections 1
and 2 are are correct for all these classes. (In case of $k=0,1$, strictly
plurisubharmonic should be understood as strongly plurisubharmonic, in the
sens of Richberg [Ri68]). Although nucleus is clearly independent on the
smoothness class under consideration, this is unknown for the core (and seems
to me unlikely). So the notation $c^{k}(K)$ might be more accurate, but, like
Shcherbina, we suppress the $k$.

A neighborhood will always be an open set of $M$.

$M$ will denote a complex manifold.

For a subset $A$ of $M$, $bA$ denotes $\overline{\setminus}A$.

Acknowledgement. I am grateful to Nikolai Shcherbina for attracting my
attention to these problems, and helpful correspondence concerning Question
5.1.

1\. Background on pseudoconcave sets and Liouville sets

1.1. Definition. (a) Let Let $X$ be a locally closed subset of M (ie.
$X\cap(\overline{X}\setminus X)=\emptyset)$. We say that $X$ is a local
maximum set, if there do not exist: a point $p\in X$, a neighborhood $V$ of
$p$ in $M$, and a $C^{\infty}$-smooth strictly plurisubharmonic function $v$
on $V$, such that $v|V\cap X$ has strict maximum at $p$.

(b) If a local maximum set is closed in $M$, we call it pseudoconcave.

Below we give just several facts about pseudoconcave sets, that we use in this
article. For more background we refer first of all to [Sh21](since we continue
that paper), and then to [Ro55], [Sl86],[ST04],[HST17, 20, 21], [PS19],and
[Sl19].

Although pseudoconcavity is the principal concept, the more relaxed notion of
a local maximum set maximum set ads occasionally some flexibility to arguments
that would be cumbersome otherwise. See Example 2.3, where the following two
properties are used. (They follow immediately from the definition.)

1.2. Proposition. (a) The union of a family of local maximum subsets of $M$ is
pseudoconcave, provided it is closed.

(b) If $Z\subset X\subset M$, $Z$ relatively closed in $X$, and $X$ a local
maximum subset of M, then $X\setminus Z$ is a local maximum subset of M as
well.

1.3. Definition [HST20]. We say that a closed subset $X$ of $M$ is a Liouville
set, if every smooth plurisubharmonic function $u$ on $X$, such that $u|X$ is
bounded from the above, must be constant on $X$.

1.4. Proposition. (a) Every Liouville set is pseudoconcave.

(b) If $K\subset M$ is compact, and $L$ is a compact Liouvile subset of $M$,
then $L\subset c(K)$. In particular, $c(L)=L$.

Comments on proof. For (a) see [HST20, Lemma 4.1].

(b) Suppose, to the contrary, that there is a smooth plurisubharmonic function
$u$ on a neighborhood $U$ of $X$, which is strictly plurisubharmonic at some
neighborhood $V$ of $p\in X$. We can assume $\overline{V}\subset U$ and
$X\setminus V\neq\emptyset$. Choose a nonnegative $C^{\infty}$ function $\rho$
on $U$, such that $\rho(p)>0$ and $\overline{supp(\rho)}\subset V$. By the
standard argument, for some positive $\epsilon$, the function
$v:=u+\epsilon\rho$ is plurisubharmonic on $U$. Since $X$ is a compact
Liouville set, both $u|X$, and $v|X$ must be constant, which is not possible
(as they differ by $\epsilon\rho)$.

We will say that $X$ is a minimal pseudoconcave set, if it does not contain
any proper pseudoconcave subset $Z$. Bellow we will consider only compact
minimal pseudoconcave sets. (A pseudoconcave subset of a compact set is
automatically compact.)

1.5. Proposition. Every compact pseudoconcave subset of manifold $M$ contains
a minimal compact pseudoconcave subset.

This is very similar to [Sl19, Cor.1.11], and has practically the same proof,
but we need this formally stronger statement. (It will be only a consequence
of Th.2.1 that the two facts are actually equivalent.)

Proof. Let $K\subset M$ be a given compact pseudoconcave set. We will define
sets $L_{t}$ by transfinite induction. Let $L_{0}=K$. If sets $L_{s}$ up to
ordinal $t$ have already been defined, we let $L_{t+1}$ to be any proper
compact pseudoconcave subset of $L_{t}$, if such exists; if not the induction
stops and $L_{t}$ is the answer. If $t$ is a limit ordinal, we let

$L_{t}:=\bigcap\\{L_{s}:s<t\\}.$

Clearly $L_{t}$ is compact and nonempty; we have to check it is a local
maximum set. Suppose not, and let $p,V,v:V\leftarrow R$ be as in Def.1.1(i).
Choose $B:=B(p,r),r>0$, a ball in a coordinate neighborhood of $p$, and such
that $\overline{B}\subset V$.Let $F:=\\{x\in\overline{B}:v(x)\geq v(p)\\}$.
Since $F$ is compact, and

$\\{p\\}=F\cap L_{t}=F\cap\bigcap_{s<t}L_{s},$

there is $s_{0}<t$, such that $F\cap L_{s_{0}}\subset B(p,r/2)$.Thus

$\max v|L_{s_{0}}\cap\overline{B(p,r/2)}\geq v(p)>\max v|L_{s_{0}}\cap
bB(p,r).$

But this violates another, equivalent, form of local maximum property, namely
(ii) in [Sl86, Prop.2.3], so $L_{s_{0}}$ would not be pseudoconcave. This
contradiction shows that $L_{t}$ is pseudoconcave. For cardinality reasons,
the induction has to stop at some ordinal $t$, yielding a compact minimal
pseudoconcave subset of $K$.

1.6. Proposition. Let $Z$ be a pseudoconcave subset of $M$ and let $u$ be a
plurisubharmonic function on $Z$. Suppose $u$ is bounded from the above on $Z$
and attains maximum value at some point of $Z$. Then the (nonempty) set
$Y:=\\{x\in Z:u(x)=\sup u|Z\\}$ is pseudoconcave.

. Note $u$ is does not have to be smooth, just uppersemicontinuous. The other
assumptions about it hold automatically when $Z$ is compact.

Proof. Since $u$ is uppersemicontinuous, $Y$ is closed. Assume WLOG that $\max
u|Z=0$, and suppose that Y is not a local maximum set. By Def.1.1 there are: a
point $p\in Y$, a neighborhood $V$ of $p$, a smooth plurisubfarmonic function
$v$ on $V$, and a coordinate ball $B(p,r),r>0$, such that : $\overline{V}\cap
Z$ is compact, $\overline{B(p,r)}\subset V$, $v(p)=0$ , and $v|Y\cap
bB(p,r)\leq-1$. So there is a neighborhood $W$ of $Y\cap bB(p,r)$, such that
$v|W\leq-1/2$. Let $F:=Z\cap bB(p,r)\setminus W$ . Since $F$ is disjoint from
$Y$ , $u$ is negative on $F$. Let $c_{1}:=\max u|F$, then $c_{1}<0$ (as $F$ is
compact and $u$ usc). Similarly, $v$ is bounded on $F$ by some finite $c$ .

Choose now $n\in N$ large enough so that $c+nc_{1}<-1/2$, and define a
plurisubharmonic function on $V$ by $\phi:=v+nu$. Now $\phi\leq v\leq-1/2$ on
$W\cap bB(p,r)$, but $\phi\leq c+nc_{1}<-1/2$ on $F$ . Thus $\phi|Z\cap
bB(p,r)<-1/2$ , while $\phi(p)=0$. This contradicts (one of the forms) of
local maximum property of $Z$, cf.[Sl86, Prop.2.3(ii)].

1.7. Corollary. Every minimal compact pseudoconcave set $Z\subset M$ has the
Liouville property (with respect to all plurisubharmonic functions on $Z$.

Proof. Let $u$ be a plurisubharmonic function defined in a neighborhood of
$Z$. As $Z$ is compact, $u|Z$ has maximum value $c$. If $u$ were not constant
on Z, then the set $Y:=\\{x\in Z:u(x)=c\\}$ would be different from $Z$.
Since, by last proposition, $Y$ is pseudoconcave, we would obtain
contradiction with the minimality of $Z$.

We Recall the properties of the regularized maximum function.

1.8. Proposition.[HM88,4.13-14]. For every $\delta>0$ there is a smooth convex
function $M_{\delta}$ on $R^{2}$, with both partial nonnegative partial
derivatives, and such that

$M_{\delta}(t,s)=\max(t,s)\hbox{ , when }|t-s|\geq\delta.$

2\. Pseudoconcave decomposition of the core

2.1. Theorem. Let $K\subset M$ be compact, and $p\in K$. Let $Y$ be the set of
all points $x\in K$ such that $u(x)=u(p)$ for all smooth plurisubharmonic
functions $u$ on $K$.

_(a)_ If $Y=\\{p\\}$ , then then $p$ does not belong to the core $c(K)$.

_(b)_ If $Y$ is not a singleton, then $Y$ is a (compact) Liouville set. In
particular, $Y$ is pseudoconcave, and $c(Y)=Y$.

Proof. Suppose that we are given a neighborhood $V$ of $Y$, and a smooth
plurisubharmonic function $v:V\rightarrow[1/2,1]$. During most of the proof we
will construct, for the pair $(V,v)$, a smooth plurisubharmonic function
$\chi^{*}$ on $K$, with special properties (cf. Conclusion below). Then a pair
$(V,v)$ will be chosen differently in cases (a) and (b), and using
corresponding $\chi^{*}$, we will obtain (a) and (b) immediately.

By the definition of $Y$, there is a family of smooth plurisubharmonic
functions $\phi_{t}:U_{t}\rightarrow R$, with $U_{t}$ neighborhoods of $K$,
for $t\in T$, such that

$Y=\bigcap_{t\in T}\\{x\in K:\phi_{t}(x)=0\\}.$

Since the sets in this intersection are compact, we can choose finitely many
of them so that (simplifying the notation slightly) we get

$Y\subset Z:=\bigcap_{i=1}^{n}\\{x\in K:\phi_{i}(x)=0\\}\subset V,$

with $\phi_{i}:U_{i}\rightarrow R$, $K\subset U_{i}$, $i=1,...,n$. Choose a
relatively compact open set $U$, and such that

$K\subset U\subset\overline{U}\subset U_{0}:=\bigcap_{i=1}^{n}U_{i}.$

Shrinking $U_{i}$’s if needed, we we can assume WLOG that all $\phi_{i}$’s are
uniformly bounded on $U_{0}$.

In this part of the proof we will follow the beginning of the proof of Lemma
3.5 in[Sl19] (preserving the notation). Define first two smooth
plurisubharmonic functions $\phi$ and $\mu$ on $U_{0}$. Let

$\phi(x):=\phi_{1}(x)+...+\phi_{n}(x)\hbox{ , for }x\in U_{0}.$

To define $\mu$, let first
$v_{\epsilon}(t_{1},...,t_{n}):=t_{1}+...+t_{n}+\epsilon(t_{1}^{2}+...+t_{n}^{2})$
which defines a smooth convex function on $R^{n}$, for $\epsilon>0$. Since
$m_{0}:=\inf\\{\phi_{i}(x):x\in U_{0},i=1,...,n\\}$ is finite (by the choice
of $U_{0}$), we can choose an $\epsilon>0$, such that $1+\epsilon m_{0}>0$ .
Then $\frac{\partial v_{\epsilon}}{\partial t_{i}}(t)>0$ when
$\min(t_{1},...,t_{n})>m_{0}$, in particular on the joint range of the vector
valued function $(\phi_{1},...,\phi_{n})$. So, by Lemma 1.13(iii) in [Sl19],
the function $\mu(x):=v_{\epsilon}(\phi_{1}(x),...,\phi_{n}(x))$ is smooth
plurisubharmonic on $U_{0}$. Similarly as in [Sl19], we have
$\phi(x)\leq\mu(x)$, for $x\in U_{0}$, and

$\\{x\in U_{0}:\phi(x)=\mu(x)\\}=\bigcap_{i=1}^{n}\\{x\in
U_{0}:\phi_{i}(x)=0\\}.$

Observe that

$Z=K\cap\bigcap_{c>0}\\{x\in\overline{U}:|\phi_{i}(x)|\leq c,i=1,...,n\\}.$

Since the sets $\\{x\in\overline{U}:|\phi_{i}(x)|\leq c,i=1,...,n\\}$ are
compact and decreasing with $c$, there is $c>0$, such that
$K\cap\overline{H}\subset V$, where
$H:=\bigcap_{i=1}^{n}\\{x\in\overline{U}:|\phi_{i}(x)|<c\\}$. Finally, there
is an open set $W$ such that

$K\subset W\subset\overline{W}\subset U\hbox{ , and
}\overline{W}\cap\overline{H}\subset V.$

Observe that

$U\setminus H\subset\bigcup_{i+1}^{n}\\{x\in
U:|\phi_{i}(x)|=1\\}\subset\\{x\in U:\mu(x)>\phi(x)\\}.$

Now, as $W\setminus H$ is relatively compact in $U$, we conclude that
$\Delta:=\inf\\{(\mu-\phi)(x):x\in W\setminus H\\}>0$. So, choosing a
$\delta\in(0,\Delta/5)$, we obtain

$\mu\geq\phi+\Delta\geq(\phi+(\Delta/2)v)+\delta\hbox{ , on }W\cap V\setminus
H.$

By this inequality and Prop.1.8, the function $\chi$ defined as
$M_{\delta}(\phi+(\Delta/2)v,\mu)$ on $W\cap V$, equals $\mu$ on $W\cap
V\setminus\overline{H}$. Function $\chi$ is smmoth plurisubharmonic on $W\cap
V$. Define now function $\chi^{*}$ as $\chi$ on $V\cap W$ and as $\mu$ on
$W\setminus\overline{H}$. Since the latter two functions are equal on the
intersection of their domains $(V\cap W)\cap(W\setminus\overline{H})=W\cap
V\setminus\overline{H}$, function $\chi^{*}$ is well defined smooth
plurisubharmonic function on the union of these domains:

$(V\cap W)\cup(W\setminus\overline{H})=W\hbox{ , since
}W\cap\overline{H}\subset V.$

Consider now $\chi^{*}$ near $Y$. Since $\phi|Y=\mu|Y=0$, it follows that
$\phi+(\Delta/2)v>\mu+\delta$, on $Y$, and so on some neighborhood of $Y$,
thus $\chi^{*}=\phi+(\Delta/2)v$ near $Y$.

Conclusion. For a smooth plurisubharmonic function $1/2<v<1$ on a neighborhood
of $Y$, there is a smooth plurisubharmonic function $\chi^{*}$ on $K$ that is
equal to $\phi+(\Delta/2)v$ near $Y$, and to $(\Delta/2)v$ on $Y$.

Proof of (a). If $Y=\\{p\\}$, we choose as $V$ a small coordinate ball
centered at $P$, and as $v$ a smooth strictly plurisubharmonic function on
$V$, bounded by $1/2$ and $1$. By the Conclusion, we obtain a smooth
plurisubharmonic function $\chi^{*}$ on $K$, that is strictly plurisubharmonic
near $p$. Thus $p$ is not in $c(K)$.

Proof of (b).Suppose $Y$ is not a Liouville set. Then there is a smooth
plurisubharmonic function $v$ on some open $V$ containing $Y$, and $p,q\in Y$,
such that $v(p)\neq v(q)$ (WLOG $1/2<v<1$). By the Conclusion, there is a
smooth strictly plurisubharmonic function $\chi^{*}$ on $K$, that is equal to
$(\Delta/2)v$ on $Y$. But then $\chi^{*}(p)\neq\chi^{*}(q)$, which contradicts
the definition of $Y$.

Observe that the last theorem gives positive answer to Shcherbina’s
Question5.3. Also, since $c(K)$ is the union of sets $Y$ having local maximum
property by (b), $c(K)$ is pseudoconcave by Proposition 1.2(a). Finally,
Theorem 2.1 allows for an alternative proof of Shcherbina’s Main Theorem.

2.2. Corollary. Let $K$ be a compact subset of a complex manifold. Then there
is a smooth strictly plurisubharmonic function on $K$ , if and only if $K$
does not contain any compact pseudoconcave subset.

Proof. If $K$ contains a compact pseudoconcave set $X$, then, by by
Proposition 1.5, $X$ contains a minimal compact pseudoconcave set $Z$, which
by Proposition 1.7 is a Liouville set, and so, by Proposition 1.4(b), must be
containd in the core $c(K)$. Core being nonempty, a smooth strictly
plurisubharmonic fuction on $K$ cannot exist.

If $K$ does not contain any compact pseudoconcave set, then $c(K)$ must be
empty (it would be otherwise pseudoconcave), which means that a smooth
strictly plurisubharmonic fuction on $K$ does exist.

The next example illustrates Theorem 2.1, and gives answer to Shcherbina’s
Question 5.1.

2.3. Example. Let $M:=\overline{C}\times C$ and $K:=\overline{C}\times
S^{1}\cup\\{0\\}\times\overline{D(0,2)}$. Consider pluriharmonic functions
$u(z,w)=\Re w$, and $v(z,w)=\Im w$, and let $Y_{a,b}:=\\{(z,w)\in
K:u(z,w)=a,v(z,w)=b\\}$. If $a^{2}+b^{2}=1$, then
$Y_{a,b}=\overline{C}\times\\{a+ib\\}$ and is a compact Liouville set.
Otherwise $Y_{a,b}=\\{(0,a+ib)\\}$ is a single point (when $a^{2}+b^{2}<1$, or
$1<a^{2}+b^{2}\leq 4$). By Theorem 2.1 the core is the union of Liouville sets
contained in $K$. Hence $c(K)=\overline{C}\times S^{1}$.

Let now $K_{1}:=c(K)\cup\\{0\\}\times D(0,1)$. Since this set is closed, and
is the union of local maximum sets (the second not closed), it is
pseudoconcave by Proposition 1.2(a), and so is contained in $n(K)$. Thus in
this case the nucleus and the core are not equal (which answers [Sh21,Question
5.1]). Observe yet that $n(K)=K_{1}$. Suppose $n(K)$ contains $K_{1}$
properly. By Proposition 1.2(b), $n(K)\setminus K_{1}$ would be a relatively
closed local maximum subset of the annulus
$\\{0\\}\times\\{a+ib:1<a^{2}+b^{2}\leq 4\\}$. But this is not possible.

References

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[HST20]Harz, T., Shcherbina, N., Tomassini, G.: On defining functions and
cores for unbounded domains II. J. Geom. Anal. 30 (2020), 2293-2325.

[HST21]Harz, T., Shcherbina, N., Tomassini, G.: On defining functions and
cores for unbounded domains III. Mat. Sb. 212, 6 (2021), 126-156.

[HL88]Henkin,G.-M ., Leiterer, J., G.: Andreootti-Grauert theory by integral
formulas. Progress in Mathematics 74 (1988).

[MST18]Mongodi, S., Slodkowski, Z., Tomassini, G.: Weakly complete complex
surfaces. Indiana Univ. Math. J. 67 (2018), 899-935.

[PS19]Poletsky, E. A., Shcherbina, N.: Plurisubharmonically separable complex
manifolds.  Proc. Amer. Math. Soc. 147 (2019), 2413 - 2424.

[Ri68]Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175
(1968), 257–286.

[Ro55]Rothstein, W.: Zur Theorie der analytischen Manningfaltigkeiten in Raume
von n komplexen Veranderlichen. Math. Ann. 129 (1955), 96-138.

[Sh21]Shcherbina, N.V.: On compacts possessing strictly plurisubharmonic
functions.  Izvestiya : Mathematics 85 (2021), 605-618.

[ST04]Slodkowski, Z., Tomassini, G.: Minimal kernels of weakly complete
spaces.  J. Funct. Anal. 210 (2004), 125–147.

[Sl19]Slodkowski, Z.:Pseudoconcave decompositions in complex manifolds.
Contemp. Math. 735 (2019), 239-259.

Zbigniew Slodkowski Department of Mathematics, University of Illinois at
Chicago
851 South Morgan Street, Chicago, IL 60607, USA
_e-mail_<EMAIL_ADDRESS>


††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>hamburg.de.††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

#  Comment on “Electron spin resonance and collective excitations in magic-
angle twisted bilayer graphene”.

Lars Tiemann  Center for Hybrid Nanostructures (CHyN), Universität Hamburg,
Luruper Chaussee 149, Hamburg 22761 Germany.  M. Prada  I. Institute for
Theoretical Physics, Universität Hamburg D-22761 Hamburg, Germany  Vincent
Strenzke  Center for Hybrid Nanostructures (CHyN), Universität Hamburg,
Luruper Chaussee 149, Hamburg 22761 Germany.  Robert Blick  Center for Hybrid
Nanostructures (CHyN), Universität Hamburg, Luruper Chaussee 149, Hamburg
22761 Germany.

###### Abstract

This comment pertains the recent manuscript by Morissette et al.
https://arxiv.org/abs/2206.08354v1. The authors claim to have found signatures
of collective excitations in electron spin resonance experiments that would be
linked to the correlated structure of magic angle bilayer graphene. However,
identical resonance features have already been reported in previous works on
mono- and few-layer graphene, voiding their theoretical framework. A straight
forward theoretical picture within the single-particle topologically non-
trivial band structure of graphene delivers satisfactory explanations for the
observation of the resonant features and applies as well to the data presented
by Morissette et al.. However, this intuitive picture has been disregarded by
the authors.

Topological Insulators, Magic angle bilayer graphene, graphene, electron spin
resonance, resistively detected electron spin resonance, bilayer graphene

Introduction:  The recent work by Morissette et al. [1] reports on resistively
detected electron spin resonance (RD-ESR), where it is claimed that collective
excitations linked to the correlated states in magic angle bilayer graphene
(MABG) cause particular resonance features. However, previous work in graphene
mono- and few-layers [2, 3, 4, 5, 6] observe similar signatures in RD-ESR. The
origin of these lines has been reported and is accepted within a single-
particle description of spin bands. The existing theoretical framework can
explain all the experiments consistently, whereas the correlated MABG is
inconsistent with the experimental results mentioned above [2, 3, 4, 5, 6].
The aim of this comment is to present evidence that the experimental data in
reference [1] simply confirm and reproduce the existing literature, which
incidentally was not cited in their work.

Figure 1:  a. Data points extracted from Fig. 3(b) of Ref. [1] with
corresponding linear fits (solid lines). A zero-field splitting consistent
with the previously reported value of 42$\mu$eV can be obtained with the two
upper lines (green and light blue), yielding, respectively, $\Delta_{\rm
Blue}$ and $\Delta_{\rm Orange}$. The $g$-factor is extracted from the Zeeman
line that crosses the origin (dark blue). An additional line with $g$ = 5.2
can be extracted (green line), which compares with the data obtained by
Strenzke [10] (green dashed lines of b.) with a $g$-factor of 4.66. b. The
dashed lines correspond to the linear fits to the data previously reported by
Strenzke et al. in 13C graphene [10, 4], whereas the solid lines correspond to
Singh et al. [3].

Summary of experimental observations:  RD-ESR is a spin-sensitive probing
technique that couples carriers of opposite spin by microwave excitation and
detects the response resistively [7, 2, 6]. In a nutshell, a signal in the
longitudinal resistance of a Hall bar is expected whenever a resonance
condition is met, typically in the frequency-magnetic field plane. Four
resonance lines are observed by Morissette et al. which have already been
reported by Singh et al. [3] and by Strenzke et al. [4], where a common slope
can be identified with the Zeeman term $g\mu_{B}B$. The respective intercepts
at zero field have been subject of extensive study: The most revealing one,
first measured by Mani et al. and subsequently reproduced [2, 3, 4] has been
identified with intrinsic spin-orbit coupling (SOC)[2]. The theoretically
predicted [8] zero-field splitting of 42 $\mu$eV has been systematically
reproduced, not only in RD-ESR experiments but also employing quantum point
contacts [9].

Discussion of data in Ref. [1]:  The data of Fig. 1a. (dots) were extracted
using a customary program from Fig. 3(b) of Ref. [1], where the solid lines
are the corresponding linear fits. For comparison, Fig. 1b. shows the linear
fits to the data previously reported by Strenzke [10, 4] (dashed) and by Singh
et al. [3] (solid). The intercept of the blue line with $h\nu$ axis
$\Delta_{Blue}$ in Fig. 1a. has been identified by Morissette and co-workers
with the $k_{1}$ mode of a magnon [1], $\Delta_{\rm Blue}=(2\pi
a/L_{x})^{2}J$, where $J$ would be the spin stiffness, $a$ the Moiré lattice
constant and $L_{x}$ the sample dimensions. In order to proof the validity of
their model, it would be necessary to reproduce the data with a sample of
different dimensions and deduce a scaling law, but only one sample was
measured in their work. We could extract a value of $\Delta_{\rm Blue}=$
(41.85 $\pm$ 1.04)$\mu$eV, which coincides with the previously reported value
by Mani et al. [6] and Sichau et al. [2] of $\Delta_{\rm ISOC}=$ (42.2 $\pm$
0.8)$\mu$eV, also resolved with RD-ESR and identified with intrinsic SOC gap.
This value was confirmed in bilayer graphene by Banszerus et al.[9]. We thus
argue here that the blue line simply reproduces previous data on Zeeman
resonance shifted by intrinsic SOC, and thus, $\Delta_{\rm Blue}=\Delta_{\rm
ISOC}$.

Under special conditions, two additional lines with finite intercepts (or
zero-field splittings) were observed by Singh et al., owing to sublattice
splitting [3]. Two lines with intercepts at $\pm\Delta_{\gamma}$ can be
observed as long as $\Delta_{\gamma}$, which accounts for sublattice
splitting, does not exceed the intrinsic SOC splitting [3]. These findings
were later confirmed by Strenzke et al. [4] on ${}^{13}C$ graphene. They
observed an additional shift of these lines owing to the nuclei-induced field,
in perfect agreement with the theory. We argue here that these results by
Singh et al. and Strenzke et al. are simply reproduced in Ref. [1]: Sublattice
splitting results on the orange and red lines of Fig. 1a., as twisted bilayer
graphene can give rise to a small asymmetry between the two sublattices,
comparable to the CVD-grown samples in Singh et al. and Strenzke et al.. The
asymmetry in the splittings with respect to the Zeeman line that cross the
origin in the $\nu-B$ plane could be related to MABG, but this would require a
systematic study with additional data.

There is an additional line (green data points of Fig. 1a.) that features a
$g$-factor of about 5.2, and is similar to the one reported by Strenzke et al.
in [10], resulting in $g\simeq$ 4.6, which remains an open question.

The similarity of the resonance lines measured by Morissette et al. and the
above mentioned publications points towards a common explanation, where
correlations related to MABG are excluded, as it has been measured in CVD
graphene. In addition, we stress that the lack of all these publications in
the references list of the manuscript by Morissette et al. is surprising as a
simple search on Google Scholar using the keywords “resistively detected
electron spin resonance graphene” yields an abundance of relevant content.

Finally, we enumerate a few deficiencies in the work by Morissette et al.: (i)
The validity of their theory should be confirmed with different sample sizes,
that would yield accordingly scaling zero-field splittings, however they only
report data on one sample. Existing literature, however, has shown little
variation of the SOC-induced splitting with sample size. (ii) The low-magnetic
field regime, labelled as regime ‘1’, lacks of experimental evidence.
Incidentally, previous measurements at low field have reported resonances
incompatible with their theory. (iii) The above mentioned resonance lines are
insensitive to carrier density, and thus, to filling of Moiré bands. (iv)
Morissette et al. claim to find a $g$-factor for the high energy line to be of
4.0, however, our fits indicate a value close to 5.2, which cannot be
explained within a 2-magnon picture that they invoke in their manuscript.

Conclusions: In summary, previous RD-ESR studies found identical resonant
features of zero-field splitting in monolayer and few-layer graphene. However,
these studies were completely disregarded by Morissette et al. in favor of an
inconsistent theory that highlights and hyperbolizes the correlations of MABG.
The astounding properties of MABG certainly endorse intensive studies,
however, the results need to be evaluated with greatest scientific care and
contrasted with existing literature.

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