Conformal welding for critical Liouville quantum gravity
Nina Holden
ETH Z¨urich
Ellen Powell
ETH Z¨urich
Abstract
Consider two critical Liouville quantum gravity surfaces (i.e.,
γ
LQG for
γ
= 2), each with the
topology of
H
and with infinite boundary length. We prove that there a.s. exists a conformal welding
of the two surfaces, when the boundaries are identified according to quantum boundary length.
This results in a critical LQG surface decorated by an independent SLE
4
. Combined with the proof
of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018),
this shows that the welding operation is welldefined. Our result is a critical analogue of Sheffield’s
quantum gravity zipper theorem (2016), which shows that a similar conformal welding for subcritical
LQG (i.e.,
γ
LQG for
γ
∈
(0
,
2)) is welldefined.
1
Introduction
Let
D
1
and
D
2
be two copies of the unit disk
D
, and suppose that
φ
:
∂
D
1
→
∂
D
2
is a homeomorphism.
Then
φ
provides a way to identify the boundaries of
D
1
and
D
2
, and hence produce a topological sphere.
The classical
conformal welding
problem is to endow this topological sphere with a natural conformal
structure. When the sphere is uniformised (i.e., when it is conformally mapped to
S
2
) we get a simple
loop
η
on
S
2
, which is the image of the unit circle. Equivalently, the conformal welding problem consists of
finding a triple
{
η, ψ
1
, ψ
2
}
, where
η
is a simple loop on
S
2
, and
ψ
1
and
ψ
2
are conformal transformations
taking
D
1
and
D
2
, respectively, to the two components of
S
2
\
η
, such that
φ
=
ψ
1
◦
ψ

1
2
. If such a triple
exists and is uniquely determined by
φ
(up to M¨
obius transformations of the sphere) then one says that
the conformal welding (associated to
φ
) is welldefined.
The extension of this problem to the setting of
random
homeomorphisms has received much attention
in recent years; in particular, when the random curves and homeomorphisms are related to natural
conformally invariant objects such as Schramm–Loewner evolutions (SLE) and Liouville quantum gravity
(LQG). This will be the focus of the present paper. In particular, we consider the case of critical (
γ
= 2)
LQG, which is associated with SLE
4
.
Roughly speaking, LQG is a theory of random fractal surfaces obtained by distorting the Euclidean
metric by the exponential of a real parameter
γ
times a Gaussian free field (GFF) or a related kind
of distribution. Such random surfaces give rise to random conformal welding problems, for instance,
when the homeomorphism
φ
corresponds to gluing the boundaries of two discs according to their LQG
boundary lengths. Weldings of this type have been studied in several recent works [AJKS10, AJKS11,
She16, DMS18, MMQ18]. In particular, for a class of homeomorphisms defined in terms subcritical LQG
measures (
γ
LQG for
γ
∈
(0
,
2)) existence and uniqueness of the conformal welding was established by
Sheffield [She16], and the interface
η
was proven to have the law of an SLE
κ
with
κ
=
γ
2
∈
(0
,
4).