# Delamination from an adhesive sphere: Curvature–induced dewetting versus
buckling

Finn Boxa,b, Lucie Dominoa,c , Tiago Outerelo Corvoa, Mokhtar Adda-Bediad,
Vincent Démeryd,e, Dominic Vellaa and Benny Davidovitchf aMathematical
Institute, University of Oxford, Woodstock Rd, Oxford, OX2 6GG, UK
bDepartment of Physics and Astronomy, University of Manchester, Manchester M13
9PL, UK
cInstitute of Physics, Universiteit van Amsterdam, Science Park 904, 1098 XH,
Amsterdam, The Netherlands
dUniv Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, 69342 Lyon, France
eGulliver, CNRS, ESPCI Paris PSL, 10 rue Vauquelin, 75005 Paris, France
fDepartment of Physics, University of Massachusetts, Amherst, MA 01003, USA

###### Abstract

Everyday experience confirms the tendency of adhesive films to detach from
spheroidal regions of rigid substrates — what is a petty frustration when
placing a sticky bandage onto an elbow or knee is a more serious matter in the
coating and painting industries. Irrespective of their resistance to bending,
a key driver of such phenomena is Gauss’ Theorema Egregium, which implies that
naturally flat sheets cannot conform to doubly-curved surfaces without
developing a strain whose magnitude grows sharply with the curved area.
Previous attempts to characterize the onset of curvature-induced delamination,
and the complex patterns it gives rise to, assumed a dewetting-like mechanism
in which the propensity of two materials to form contact through interfacial
energy is modified by an elastic energy penalty. We show that this approach
may characterize moderately bendable adhesive sheets, but fails qualitatively
to describe the curvature-induced delamination of ultrathin films, whose
mechanics is governed by their propensity to buckle under minute levels of
compression. Combining mechanical and geometrical considerations, we introduce
a minimal model for curvature-induced delamination that accounts for two
elementary buckling motifs, shallow “rucks” and localized “folds”. We predict
nontrivial scaling rules for the onset of curvature-induced delamination and
various features of the emerging patterns, which compare well with
experimental observations. Beyond gaining control on the use of ultrathin
adhesives in cutting edge technologies such as stretchable electronics, our
analysis is a significant step towards quantifying the multiscale
morphological complexity that emerges upon imposing geometrical and mechanical
constraints on highly bendable solid objects.

##  I Introduction

The simplest way to form a composite material, simply sticking a layer of one
material to another, is encountered in everyday life from sticky notes to a
parent placing a band-aid on a child’s knee. Normally one does not consider
whether such an operation is at all possible. However, when at least one of
the two objects to be joined is curved, adhesion is no longer guaranteed. For
example, when a relatively thick, flat plate is adhered to a cylindrical
substrate the energetic penalty associated with detaching from the substrate
is small enough to be overcome by the elastic (bending) energy of the plate
that is released by detachment Neukirch2007  and leads to the failure of the
coating, or delamination. For sufficiently thin plates, this bending energy is
insignificant and adhesion proceeds as expected.

A qualitatively different picture emerges when the substrate is doubly-curved
(i.e. has two principal curvatures $\kappa_{1,2}\neq 0$, fig. 1A). While very
thin sheets are able to bend easily, stretching is much more difficult and so
Gauss’ _Theorema Egregium_ Wilson2008  limits them to maintaining their
initial Gaussian curvature, $K_{G}^{\mathrm{sheet}}$. If the Gaussian
curvature of the substrate $K_{G}^{\mathrm{subs}}=\kappa_{1}\kappa_{2}\neq
K_{G}^{\mathrm{sheet}}$ adhesion between the two frustrates the deformable
object (the sheet). Ultimately this frustration can be (partially) relieved by
delamination, which allows the sheet to ‘expel’ excess material and so conform
to the substrate Paulsen2019 ; Vella2019 . The central result of this study is
that this process may occur in two sharply distinct modes: a dewetting-like
instability (when the sheet is moderately bendable) or a buckling-like
instability (for highly bendable sheets) in which most of the sheet remains
adhered to the substrate and delamination occurs only at localized “rucks” or
“folds”. These distinctive modes of delamination make a significant difference
to the maximal size of sheet that can be smoothly adhered before delamination
occurs.

Figure 1: Illustrations of the geometric incompatibility of an intrinsically
flat sheet adhering to a doubly-curved object. A: A circular, naturally flat,
sheet of radius $W$ has Gaussian curvature $K_{G}^{\mathrm{sheet}}=0$ while a
spherical substrate of radius $R$ has Gaussian curvature
$K_{G}^{\mathrm{subs}}=\kappa_{1}\kappa_{2}=1/R^{2}$. B: Delamination blisters
spontaneously form when a (flat) band-aid is adhered to a child’s (curved)
knee. Panels C and D show more controlled realizations of this experiment in
sheets of large and small thicknesses, respectively. (Image in C taken from
Hure2011 .)

The geometrical frustration resulting from a difference in Gaussian curvature
occurs when a flat sheet is deposited on a sphere, as is shown schematically
in fig. 1A and in a practical scenario in fig. 1B. For the example shown in
fig. 1B, this incompatibility leads to failure in the form of delamination
blisters forming around the periphery. While this is a minor annoyance in the
example of a band-aid applied to a curved knee, in technological and
scientific applications conformability is key and so delamination is
problematic Xu2019 ; Yuk2020 ; Tringides2021 ; Yan2022 . As a result, a
variety of techniques have been developed to overcome geometric
incompatibility ranging from modifying the substrate geometry (as in the
Surface Force Apparatus Tabor1969 ; Israelachvili2015 ) to buffering the
excess area required to change the sheet’s shape Vella2019  by either removing
material Cho2014 ; Liu2021 ; Yan2022  or introducing sacrificial buckling
elements Wang2010 ; Jung2011 .

Despite the broad significance of geometric incompatibility for adhesion,
there seems to be little detailed understanding of _when_ and _how_ this
incompatibility is expected to lead to failure via delamination. The standard
picture of delamination induced by geometric incompatibility is due to Majidi
& Fearing Majidi2008 , who observed that if a flat sheet (of radius $W$) is
forced to adhere to a sphere of radius $R$, a strain of order $\varepsilon\sim
K_{G}^{\mathrm{subs}}W^{2}\sim W^{2}/R^{2}$ is induced. Perfect adhesion
therefore induces an elastic energy density $Y(W/R)^{4}$, where $Y=Et$ (with
$E$ the Young modulus and $t$ the sheet thickness) is the stretching modulus
of the sheet. Now, this elastic energy penalty can be avoided if the sheet
retains a planar shape by completely detaching from the substrate, at the
expense of paying an energetic penalty $\Gamma$ per unit area of lost contact.

Assuming the sheet is either fully adhered to the substrate or fully
delaminated from it, one may define a renormalized, curvature-dependent
adhesion energy density:

$\Gamma^{*}=\Gamma-c\cdot Y(W/R)^{4}\ ,$ (1)

where $c$ is a numerical constant that depends on the substrate geometry
($c=1/384$ for a spherical substrate Grason2013 ; Hohlfeld2015 ). Equation (1)
underlies an elementary description of delamination as a generalization of the
standard dewetting transition (which occurs as $\Gamma\to 0^{+}$), to
“curvature-induced dewetting” (which occurs as $\Gamma^{*}\to 0^{+}$). In this
generalization, the sheet’s elastic energy is simply viewed as additional to
the interfacial energies between the sheet, the substrate, and the surrounding
phase. In terms of the dimensionless parameters:

$\tilde{W}=W/R\ \ ;\ \ \beta=\Gamma/Y$ (2)

the curvature-induced dewetting scenario predicts that delamination occurs
when $\tilde{W}$ exceeds a critical value:

$\tilde{W}_{\mathrm{dewet}}\sim\beta^{1/4}\ .$ (3)

Although the curvature-induced dewetting mechanism does capture the basic
competition between adhesion and geometrical constraints, considering
$\Gamma^{*}$, defined in (1), as the single parameter that determines the
onset of delamination is problematic. Indeed, treating adhesion and elastic
energies as equivalent competitors obscures the fact that the former is
uniformly distributed while the latter is distributed in a nontrivial and
inhomogeneous manner: not only does the magnitude of strain vary significantly
with radial distance from the center, but the spatial structure of its
components is quite nontrivial. In particular, while the inner part of the
sheet is stretched both radially and azimuthally, the periphery is stretched
predominantly radially, and becomes azimuthally compressed when the sheet
exceeds a critical size Grason2013 ; Hohlfeld2015 :

$\tilde{W}_{\mathrm{comp}}\sim\beta^{1/2}\ .$ (4)

Notably, for $\beta\ll 1$, $\tilde{W}_{\mathrm{comp}}$ may be significantly
smaller than the delamination size $\tilde{W}_{\mathrm{dewet}}$ predicted by
the curvature-induced dewetting mechanism, Eq. (3). Since the thinner a sheet
is the less compression it can accommodate before buckling, one may expect
that the onset of curvature-induced delamination in sufficiently thin sheets
is not correctly described by the single parameter $\Gamma^{*}$, but requires
an explicit consideration of the strong nonuniformity and anisotropy, and the
consequent possibility of anisotropic instabilities such as radial wrinkling
Davidovitch2011 . The possible relevance of such a strongly anisotropic
response to delamination is clear in fig. 1C,D, which shows that delamination
occurs via the formation of radial blisters. In this paper, we show that an
anisotropic instability of this type is crucial in understanding the threshold
for delamination and that (3) is, at best, valid only for sheets with
moderate-to-large bendability.

As a first indication that the energetic picture is not the full story, we
present experimental results that interrogate the transition from smooth
adhesion to delamination for circular sheets of radius $W$ and thickness $t$
and a sphere of radius $R$. In these experiments (see Supplementary
Information), sheets (with thickness $t\in[100\mathrm{~{}nm},25\mathrm{~{}\mu
m}]$) are deposited from floating on a water bath onto the sphere. After the
system dries, the sheet is observed to either be smoothly adhered (represented
by a filled point in fig. 2) or be partially delaminated (an open point in
fig. 2). As expected, fig. 2 shows that above a critical radius ratio
$\tilde{W}>\tilde{W}_{c}$ the sheet delaminates (modulo some imperfection-
induced noise close to the transition).

Figure 2: A-D: Images showing typical delamination patterns for a range of
sphere sizes and sheet thicknesses. E: Initial presentation of experimental
results showing the regions of $(\beta,W/R=\tilde{W})$ parameter space for
which smooth adhesion (filled symbols) or localized delamination (open
symbols) were observed. Contrary to previous suggestions, the delamination
transition does _not_ occur when $\tilde{W}_{c}\sim\beta^{1/4}$. (Determining
the correct transition behavior is the focus of this paper.) Points are shown
for a variety of sheet thicknesses, indicated by point color (see colorbar to
the right), and material, indicated by shape: Polyimide (diamond),
Polycarbonate (triangles) and Polystyrene (circles).

Nevertheless, the experimental data shown in fig. 2 does not show the scaling
$\tilde{W}_{c}\sim\beta^{1/4}$ predicted by the curvature-induced dewetting
mechanism of (3): the only plausibly power-law behaviour that we observe
appears to be $\tilde{W}_{c}\sim\beta^{1/2}$, reminiscent of (4), and this
scaling is observed only in the very thinnest sheets. This suggests that the
simple theoretical picture of an energetic balance between stretching and
adhesion, as exemplified by (3), is not relevant for this simple experimental
system. In this paper, we will show that alternatives to the curvature-induced
dewetting picture laid out by Majidi & Fearing Majidi2008  exist; these do
take into consideration the onset of compressive azimuthal stress, (4) and may
actually be energetically favourable to the upper bound represented by (3).

##  II The onset of blistering

###  II.1 The importance of compression

We begin our re-examination of the theory by taking a step back to consider
the stress state within the elastic film as the sheet radius $W$ changes under
the assumption that the sheet remains perfectly attached to the sphere, and
hence axisymmetric. In this case, the vertical deformation of the sheet is

$\zeta=-r^{2}/(2R),$ (5)

(from the parabolic approximation to the sphere’s surface, valid when $W/R\ll
1$). Assuming that a tension $\Gamma$ acts at the edge of the sheet
(originating from the adhesion), the stress profile within the sheet can then
be readily calculated from this displacement, and has been presented in
related work Grason2013 ; Hohlfeld2015 . This calculation gives that the
radial and hoop stresses within the sheet induced by the deformation of (5)
are

$\displaystyle\sigma_{rr}$
$\displaystyle=\Gamma+\frac{Y}{16R^{2}}\left(W^{2}-r^{2}\right),$ (6)
$\displaystyle\sigma_{\theta\theta}$
$\displaystyle=\Gamma+\frac{Y}{16R^{2}}\left(W^{2}-3r^{2}\right).$ (7)

Since $r\leq W$, we see that $\sigma_{rr}>0$ throughout the sheet, but also
that the minimum value of the hoop stress,
$\sigma_{\theta\theta}^{\mathrm{min}}$ is

$\sigma_{\theta\theta}^{\mathrm{min}}=\sigma_{\theta\theta}(W)=\Gamma-\frac{YW^{2}}{8R^{2}}=Y\left(\beta-\frac{\tilde{W}^{2}}{8}\right),$
(8)

while the radial displacement of the sheet’s edge, $u_{r}^{\mathrm{axi}}(W)$,
satisfies

$\frac{u_{r}^{\mathrm{axi}}(W)}{W}=\frac{\sigma_{\theta\theta}(W)-\nu\sigma_{rr}(W)}{Y}=(1-\nu)\beta-\frac{1}{8}\tilde{W}^{2}.$
(9)

Crucially, (8) determines the numerical prefactor in the scaling (4), namely:

$\tilde{W}_{\mathrm{comp}}=2^{3/2}\beta^{1/2}\ ,$ (10)

so that for $\tilde{W}>\tilde{W}_{\mathrm{comp}}$ the hoop stress becomes
compressive at the edge of the sheet,
$\sigma_{\theta\theta}^{\mathrm{min}}<0$. As we discussed in the introduction,
this scaling matches the experimentally determined critical sheet size at the
onset of delamination for the thinnest elastic sheets (see fig. 2E).

To understand how and when the appearance of compression affects the onset of
delamination, we begin by considering the case in which the sheet has very
little resistance to bending. In this case, a recent study of the one-
dimensional analogue problem Davidovitch2021  suggests that delamination
blisters in highly bendable sheets take the form of ‘folds’: the amplitude $A$
is large in comparison with the width of the blister $\lambda$ (as shown in
fig. 3) and, further, that $\lambda\sim\ell_{bc}$ where
$\ell_{bc}=(B/\Gamma)^{1/2}$ is the _bendocapillary length_ and $B\propto
Et^{3}$ is the bending stiffness of the sheet. In the axisymmetric case we
similarly expect folds to be energetically favorable when
$\tilde{W}>\tilde{W}_{\mathrm{comp}}$ and the bending stiffness is
sufficiently small. We therefore define a third dimensionless group, that
depends on the bending modulus

$\epsilon=\frac{B}{\Gamma R^{2}}=\left(\frac{\ell_{bc}}{R}\right)^{2}.$ (11)

Here $\epsilon$ is a dimensionless bending stiffness and $\epsilon^{-1}$
characterizes the degree of “bendability” of the adhesive film Davidovitch2011
.

Assuming that the blisters in a sheet delaminating from a sphere are radially
elongated and are hence locally one-dimensional, adopting a similar fold
shape, we now turn to understanding when such folds are expected and how many
of them should form. This will also allow us to perform a consistency check of
the fold ansatz _a posteriori_ , and to make the notion of ‘bendable’ more
precise.

Figure 3: Schematic of a thin sheet attached to a sphere highlighting the
adhered core region (blue), delamination blisters (red) and adhered ribbons,
or ‘stickons’, between the blisters (green). Depending on the relative
strength of bending and adhesion in this problem, the delamination blisters
may form either folds or rucks (shown below the main figure). The length,
$\ell$, of the blisters, as well as the width $w$ of the stickon region are
also shown.

###  II.2 The formation of folds

The key feature of a fold is that the bending energy is localized close to the
contact region and in the loop — the majority of the arc length of the
material is simply uncurved (albeit vertical) — and so the bending energy of a
single fold of extent $\ell$ (see fig. 3) in the radial direction is
$B\ell/\ell_{bc}=(B\Gamma)^{1/2}\ell$. As a result the bending energy in $n$
folds (all of radial length $\ell$) is

$U_{\mathrm{bend}}^{\mathrm{fold}}\sim n(B\Gamma)^{1/2}\ell.$ (12)

As should be expected, $U_{\mathrm{bend}}^{\mathrm{fold}}$ penalizes the
creation of more folds; to minimize this energy, the system should have as few
blisters as possible. However, forming a small number of blisters is expensive
in terms of strain energy because the portion of the sheet that remains
adhered still conforms to the sphere; if there are fewer blisters, the
attached ribbon-like elements between the blisters (or ‘stickons’, see fig. 3)
must be wider and so be more highly strained. The strain energy per unit area
of a stickon of width $w$ adhered to a sphere of radius of curvature $R$ is
$u_{\mathrm{rib}}\sim Yw^{4}/R^{4}$ Majidi2008 ; Meng2014 ; assuming that
stickons are wide compared to the blisters formed, $w\gg\lambda$, (i.e. most
of the sheet remains laminated to gain adhesion energy) we have that $w\sim
W_{c}/n$ close to onset. The total strain energy stored in these ribbons,
$U_{\mathrm{strain}}=u_{\mathrm{rib}}\times W_{c}\ell$, is

$U_{\mathrm{strain}}\sim\frac{YW_{c}^{4}}{n^{4}R^{4}}\times W_{c}\ell.$ (13)

(Note that there is still a strained, fully-laminated core region, but that
this does not play a role in the selection of the number of blisters that are
formed.)

As expected, the energy $U_{\mathrm{strain}}$ drives the system to have many
blisters, thereby competing with the bending energy
$U_{\mathrm{bend}}^{\mathrm{fold}}$ to determine the optimal number of folds

$n_{\mathrm{fold}}\sim\frac{W_{c}}{R}(\beta^{2}\epsilon)^{-1/10}\sim(\beta^{3}/\epsilon)^{1/10},$
(14)

where we use the assumption that folds form in tandem with the emergence of
compression, i.e.

$\tilde{W}_{c}\approx\tilde{W}_{\mathrm{comp}}\sim\beta^{1/2},$ (15)

in the last equality in (14).

To understand when this regime is expected experimentally, we note two
conditions on the formation of folds. First, folds are distinguished by being
much taller than their width, i.e. $A\gg\lambda$. To estimate the amplitude of
folds, we denote the sheet length to be wasted by these blisters by
$\Delta_{\mathrm{tot}}$, so that
$A\sim\Delta_{\mathrm{tot}}/n_{\mathrm{fold}}$. The fold ansatz is therefore
valid provided that $\Delta_{\mathrm{tot}}/n_{\mathrm{fold}}\gg\ell_{bc}$ or

$\frac{\Delta_{\mathrm{tot}}}{R}\gg(\epsilon^{4}\beta^{3})^{1/10}.$ (16)

Second, and simplifying further our analysis by considering the case $\nu=0$,
we note that since folds form close to the onset of compression in the sheet,
their total arclength, $\Delta_{\mathrm{tot}}$, may be estimated as
$\Delta_{\mathrm{tot}}\sim|u_{r}^{\mathrm{axi}}(W)|$. Recalling that the
underlying assumption in the fold regime is that
$\tilde{W}_{c}=\tilde{W}_{\mathrm{comp}}+\delta\tilde{W}$ with
$\delta\tilde{W}\ll\tilde{W}_{\mathrm{comp}}\sim\beta^{1/2}$, we can estimate
$|u_{r}^{\mathrm{axi}}(W)/W|\sim\tilde{W}_{\mathrm{comp}}\,\delta\tilde{W}$
from (9), so that the inequality (16) becomes
$\tilde{W}_{\mathrm{comp}}^{2}\,\delta\tilde{W}\gg(\epsilon^{4}\beta^{3})^{1/10}$
or

$(\epsilon^{4}\beta^{-7})^{1/10}\ll\delta\tilde{W}\ll\beta^{1/2}.$ (17)

(For other values of $\nu$, a detailed calculation, given in the Supplementary
Information, shows that (17) holds regardless.) The separation of scales in
(17) is only possible if

$\epsilon\ll\beta^{3}.$ (18)

(Note that to determine the conditions under which folds form it was not
necessary to evaluate $\delta\tilde{W}$ explicitly, and we hence have not
needed to calculate $\Delta_{\mathrm{tot}}$ at all!)

Equation (18) makes precise our earlier statements that folds are expected
when the bending stiffness of the sheet is sufficiently small: the appearance
of folds requires $\epsilon\ll\beta^{3}\lll 1$. We note also from (14) that
this high bendability regime corresponds automatically to a large number of
folds — just as a large number of wrinkles is associated with the small cost
of bending and, consequently, little resistance to compression, this is also
the case for folds.

When $\epsilon\gtrsim\beta^{3}$, the fold ansatz used above is no longer self-
consistent since it gives rise to folds for which the typical slope
$A/\lambda\lesssim 1$. If the slope of the delamination structures formed with
$\epsilon/\beta^{3}\gg 1$ were indeed small, they would be ridges Vella2009 ;
Pocivavsek2018 ; Guan2022  akin to rucks in rugs Vella2009prl ; Kolinski2009 ,
rather than the folds we have considered so far. We therefore turn to consider
rucks.

###  II.3 The formation of rucks

Unlike a fold, a small-slope ruck of height $A$ and width $\lambda\gg A$ has a
sinusoidal profile $\zeta(x)\approx\tfrac{1}{2}A\bigl{[}1+\cos(2\pi
x/\lambda)\bigr{]}$. As a result, a ruck ‘wastes’ a length
$\Delta=\int_{-\lambda/2}^{\lambda/2}\bigl{(}[1+\zeta^{\prime}(x)^{2}]^{1/2}-1\bigr{)}~{}\text{d}x\sim
A^{2}/\lambda$. However, for a ruck, $A$ and $\lambda$ cannot be chosen
independently: a local force balance Obreimoff1930  (or variational arguments
Majidi2007 ; Wagner2013 ) reveal that these two lengths are constrained by the
requirement that the radius of curvature of the blister at the delamination
point matches the bendo-capillary length Obreimoff1930 ; Majidi2007 ,
$\ell_{bc}$. This condition gives that $\lambda^{2}/A\sim\ell_{bc}$, which,
combined with the wasted length constraint, gives

$\lambda\sim\Delta^{1/3}\ell_{bc}^{2/3},\quad
A\sim\Delta^{2/3}\ell_{bc}^{1/3}.$ (19)

These scalings have been derived previously in the context of the ‘sticky
elastica’ problem Wagner2013  in which a blister with a given wasted length
$\Delta$ is formed and its dimensions measured. Unlike the sticky elastica
problem, however, the length to be wasted in each delamination blister is not
controlled here: while (9) implies a global constraint for the total amount of
length that must be wasted, $\Delta_{\mathrm{tot}}$, there is, as yet, no
constraint on the number of blisters that will form, each wasting a length
$\Delta=\Delta_{\mathrm{tot}}/n$.

Anticipating that the ruck ansatz will be the appropriate (i.e. self-
consistent) one for moderately bendable sheets (which, based on the fold case,
should correspond to $\epsilon/\beta^{3}\gg 1$) we also expect that the
threshold sheet size for delamination will be well beyond the critical size at
which a hoop compression first develops, i.e. that
$\tilde{W}_{c}\gg\beta^{1/2}$. From (9), we therefore have that
$|u_{r}(W_{c})|\sim R\tilde{W}_{c}^{3}$. Our working hypothesis is that, upon
delamination, this excess length is all wasted by buckling, so that

$\Delta_{\mathrm{tot}}\approx u_{r}(W_{c})\sim R\tilde{W}_{c}^{3}.$

We therefore repeat the energetic balance argument that allowed us to
determine the number of folds in the highly bendable limit: as before, bending
energy seeks to form as few rucks as possible, while the strain in the
laminated portions of the sheet drives it to form as many as possible. The
important difference with the earlier analysis of folds is that the bending
energy within a ruck is distributed all along its arc length; we therefore
have that the bending energy of all $n$ rucks is

$U_{\mathrm{bend}}^{\mathrm{ruck}}\sim n\times
B(A/\lambda^{2})^{2}\times\lambda\ell\sim\Gamma
n^{2/3}\Delta_{\mathrm{tot}}^{1/3}\ell_{bc}^{2/3}\ell.$ (20)

The strain energy stored in the laminated portions of the sheet, the stickons,
is (in scaling terms) independent of the form that the delamination blisters
take, so that (13) still holds. We can then determine that the optimal number
of rucks is

$n_{\mathrm{ruck}}\sim(\beta^{3}\epsilon)^{-1/14}\frac{W_{c}}{R}\left(\frac{\Delta_{\mathrm{tot}}}{W_{c}}\right)^{-1/14}\sim(\beta^{3}\epsilon)^{-1/14}\tilde{W}_{c}^{6/7}.$
(21)

At this stage, an important difference with the fold case emerges: we do not
_a priori_ know the value of $\tilde{W}$ at which ruck-like delamination
blisters will emerge, and so we cannot use (21) to determine
$n_{\mathrm{ruck}}$ in terms of $\epsilon$ and $\beta$. Instead, we determine
the threshold $\tilde{W}_{c}$ by the standard buckling criterion, i.e. by
equating the residual hoop compression in the delaminated state,
$\sigma_{\mathrm{res}}\sim B/\lambda^{2}$ with $\lambda$ given by (19) (see
ref. Paulsen2016 ) and the hoop compression of the axisymmetric (laminated)
state, $|\sigma_{\theta\theta}^{\mathrm{min}}|\sim Y\tilde{W}^{2}$, which we
evaluate from (8), recalling that $\tilde{W}_{c}\gg\beta^{1/2}$ for rucks.
This buckling criterion yields the delamination-into-rucks threshold:

$\tilde{W}_{c}\sim(\beta^{3}\epsilon)^{1/12}.$ (22)

We note that the threshold in (22) reproduces the fold scaling (15), i.e.
$\tilde{W}_{c}\sim\beta^{1/2}$, as $\epsilon/\beta^{3}\searrow 1$, while for
$\epsilon/\beta^{3}\gg 1$ the onset for delamination into rucks occurs at
$\tilde{W}_{c}\gg\beta^{1/2}$. Similarly, we note that when $\epsilon=O(1)$,
(22) recovers the prediction of the upper bound (3) that delamination is
favourable when $\tilde{W}\gtrsim\beta^{1/4}$.

As a final consistency check, we note that the aspect ratio of delamination
rucks at onset is

$\frac{A}{\lambda}\sim\left(\frac{\Delta}{\ell_{bc}}\right)^{1/3}\sim\left(\frac{\Delta_{\mathrm{tot}}/n}{\ell_{bc}}\right)^{1/3}\sim\left(\frac{\beta^{3}}{\epsilon}\right)^{1/12}.$
(23)

As expected, for $\epsilon\gg\beta^{3}$ the aspect ratio of the blisters
$A/\lambda\ll 1$ and our assumption of small-slope rucks (rather than large-
slope folds) is indeed self-consistent.

##  III Experimental measurements of onset

Combining the two predictions (15) and (22), we have that the critical radius
at the onset of delamination scales with $\epsilon$ and $\beta$ as

$\tilde{W}_{c}\sim\begin{cases}\beta^{1/2}&\epsilon\ll\beta^{3},\\\
(\epsilon\beta^{3})^{1/12}&\beta^{3}\ll\epsilon\ll 1\ .\end{cases}$ (24)

We note that these two results may alternatively be written:

$\frac{\tilde{W}_{c}}{\beta^{1/2}}\sim\begin{cases}1&\epsilon/\beta^{3}\ll
1,\\\ (\epsilon/\beta^{3})^{1/12}&\epsilon/\beta^{3}\gg 1.\end{cases}$ (25)

Since this form presents different results in terms of the effective
bendability of the sheets, $\epsilon/\beta^{3}$, it is a useful one for
reconsidering the experimental data presented in fig. 2, to which we now turn.

As a first comparison between the predictions of the theoretical picture
presented above and the experiments already described, we reconsider the data
for the state of the system (laminated or delaminated) as a function of sheet
radius, presented in fig. 2. Figure 4A shows the data of fig. 2 plotted in the
way suggested by (25); this plot shows that these data collapse well when
plotted in this way and, further that the boundary between delaminated and
adhered states is consistent with the asymptotic forms predicted in (25).
(Note that the numerical value of the threshold $\tilde{W}_{c}$ has been fixed
to be the value expected in the limit $\epsilon/\beta^{3}\to 0$,
$\tilde{W}_{c}=2\sqrt{2}$, by our procedure for determining the adhesion
energy $\Gamma$, see Supplementary Information. Nevertheless, the existence of
a plateau in this regime is clear, as is the scaling for
$\epsilon/\beta^{3}\gg 1$.) We also note that the plateau in $\tilde{W}_{c}$
is observed with $\epsilon/\beta^{3}$ finite but large ($\lesssim 100$),
rather than strictly $\epsilon/\beta^{3}\ll 1$; this indicates the presence of
a large numerical prefactor that cannot be determined by our scaling analysis.

Figure 4: Comparison of theory and experiment. A: The regions of parameter
space in which smooth adhesion (filled points) and partial delamination (open
points) are observed. Here, data points are those from fig. 2 but now plotted
as suggested by (25); this shows a good collapse of the data as well as the
two distinct regimes predicted by theory. (The scaling prediction for rucks,
(22), is shown by the dash-dotted line; the prediction for folds, (15), is
shown by the dashed line and corresponds to the dashed line in fig. 2E.) B:
The experimentally measured aspect ratio, showing that experiments are
consistent with the scaling for aspect ratio of rucks given in (23) and shown
by the dash-dotted line. Here aspect ratios are determined from optical
interference (open points) and profilometry (closed point); symbols are used
as in fig. 2 to show sheet type and thickness.

The features of the experimental data as plotted in fig. 4A are non-trivial
tests of the presented theoretical picture. However, another useful comparison
with experiments comes from the aspect ratio of the delamination blisters,
measured near the edge of the sheet, in the ruck regime. (This was measured in
the case of rucks using an optical interference technique, described in the
Supplementary Information, that cannot resolve the large slopes of folds.)
These measurements are presented in fig. 4B and are also consistent with (23),
and hence the theoretical picture as a whole. Note, however, that these
experiments were not performed ‘at’ threshold, but at approximately a constant
distance beyond it. It is therefore natural to consider the problem of what
happens beyond the initial threshold a little further.

##  IV Beyond threshold: Spatial structure of folds

The argument so far has focussed on the behaviour at the onset of the
delamination instability. However, similar arguments can be used to understand
what the desired spatial structure of the fold and ruck pattern might be
beyond threshold. Provided that the radial position $r$ is large compared to
the blister width, $r\gg\lambda$, the same arguments used to derive the number
of blisters at onset can be repeated with $W_{c}$ replaced by $r$. In this way
we find from (14) that

$n_{\mathrm{fold}}\sim\frac{r}{R}(\beta^{2}\epsilon)^{-1/10},$ (26)

while in the ruck case (21) gives

$n_{\mathrm{ruck}}(r)\sim(\epsilon\beta^{3})^{-1/14}\left(\frac{r}{R}\right)^{6/7}.$
(27)

We see that the evolution of mode number of instability with radial position,
$r$, depends on whether that instability takes place via folds or rucks —
compare the linear scaling with radial position $r$ of (26) and the sub-linear
scaling with $r$ of (27). These two scalings can be written in terms of common
variables as:

$n\left(\frac{\beta^{3}}{\epsilon}\right)^{1/10}\sim\begin{cases}\frac{r}{R}\left(\frac{\beta}{\epsilon^{2}}\right)^{1/10}&\epsilon\ll\beta^{3},\\\
\left[\frac{r}{R}\left(\frac{\beta}{\epsilon^{2}}\right)^{1/10}\right]^{6/7}&\epsilon\gg\beta^{3}.\end{cases}$
(28)

The predictions of (28) are compared with experimental results in fig. 5:
points with larger values of $\epsilon/\beta^{3}$ (paler/yellower points) are
consistent with the sub-linear scaling expected for rucks, while those with
smaller values of $\epsilon/\beta^{3}$ (darker/bluer points) are consistent
with the linear scaling expected for folds. The collapse of our experimental
data shown in the main portion of fig. 5 shows two further noteworthy
features: firstly, while the exponents of the two behaviours are ostensibly
close ($6/7$ versus unity), the associated prefactors seem to be very
different, separating the ruck and fold behaviours; secondly, the steep
transition region between the two behaviours suggests that the central regions
of highly bendable sheets may form rucks.

Figure 5: The spatial distribution of the fold number, $n$. Here experiments
are performed with PS sheets (filled symbols) and PC sheets (open symbols)
with the colour bar encoding the value of $\epsilon/\beta^{3}$. According to
the theory, see (28), when plotted in this way bluer/darker points should lie
closer to the linear scaling expected for folds while yellower/lighter points
should lie closer to the $6/7$ exponent expected for rucks. The experimental
results are largely consistent with the theory, and suggest that the exponent
of spatial variation is larger for smaller $\epsilon/\beta^{3}$. Moreover, the
rescaled data show a reasonable collapse when compared with the raw data
(inset).

##  V Conclusions

We have shown that the threshold size $\tilde{W}_{c}$ above which a bendable
sheet delaminates from an adhesive spherical substrate is not determined
merely by the adhesion strength and stretching modulus, as has been supposed
previously Majidi2008 ; Hure2011 ; HUre2013 ; Bense2020 . Instead, there is a
strong effect of the bending modulus underlying the nontrivial dependence of
$\tilde{W}_{c}$ on the two dimensionless groups $\beta$ and $\epsilon$ that is
given in (24).

In particular, we have shown that delamination allows the sheet to expel
excess area (allowing conformation) in two different ways. When
$\epsilon/\beta^{3}\ll 1$ (or, equivalently, $\Gamma\gg Y\sqrt{t/R}$), the
sheet delaminates from the sphere by forming many large-slope folds, while for
$\epsilon/\beta^{3}\gg 1$ (i.e. $\Gamma\ll Y\sqrt{t/R}$), delamination occurs
via small-slope rucks. Despite the differences in the morphology of the
delamination pattern, in both cases the onset of instability occurs for sheets
that are significantly smaller than the previously presented ‘upper bound’
$\tilde{W}_{\mathrm{dewet}}\sim\beta^{1/4}$, which was obtained by
incorporating the total elastic energy into the standard dewetting criterion
for two phases Majidi2008 .

The key difference between the previously identified upper bound on the size
of sheets that can conform perfectly to a sphere and our results arises from
the fact that $\tilde{W}_{\mathrm{dewet}}$ does not take into consideration
the inhomogeneous, anisotropic distribution of strain in the laminated state,
and the consequent presence of compression even for rather small sheet sizes
$\tilde{W}_{\mathrm{comp}}<\tilde{W}\ll\tilde{W}_{\mathrm{dewet}}$, where
$\tilde{W}_{\mathrm{comp}}\sim\beta^{1/2}$, (15). Indeed, the emergence of
fold and ruck patterns follows directly from the tendency of a thin sheet to
maximize adhesion by “trading” an energetically-expensive strain with
energetically-cheap bending. This principle is analogous to the one underlying
wrinkle patterns in sheets confined to curved deformable substrates Hure2012 ;
King2012 ; Paulsen2016 ; Davidovitch2019 ; the morphological complexity of
delamination patterns in comparison to their wrinkle counterparts follows from
the binary (non-analytic) nature of the adhesion energy, which penalizes only
the lateral extent of blisters and is insensitive to the deflection amplitude.

We have seen that sheets do not conform perfectly to a spherical substrate
once their lateral size increases beyond a limit. In applications that involve
a sheet (approximately) conforming to a doubly-curved substrate, it is
nevertheless of interest to determine the extent to which some delamination is
minor: does the sheet remain largely conformed to the substrate? We therefore
define the ‘conformability’ of the sheet to the sphere, $\mathcal{C}(r)$, as
the proportion of the infinitesimal annulus $[r,r+\delta r]$ that is covered
by the sheet, i.e.

$\mathcal{C}(r)=1-\frac{n(r)\lambda(r)}{2\pi r}.$ (29)

Using the results already given we have that the delaminated proportion,
${\cal D}(r)=1-\mathcal{C}(r)$ is

${\cal
D}(r)\sim\begin{cases}(\epsilon^{2}/\beta)^{1/5}&\epsilon\ll\beta^{3}\ll 1,\\\
(\epsilon^{2}/\beta)^{1/7}(r/R)^{4/7}&\beta^{3}\ll\epsilon\ll 1.\end{cases}$
(30)

The above expressions show three desirable features of the emergence of folds
in highly bendable sheets for uniform conformation: ${\cal C}\to 1$ in the
(singular) infinite bendability limit $\epsilon\to 0$ (i.e. conformation
becomes asymptotically perfect in this limit), ${\cal D}(r)$ is independent of
$r$, and, throughout this regime, ${\cal D}\lesssim\beta$. In contrast, the
conformation obtained by rucks is nonuniform and less effective in comparison
to folds.

We now briefly consider the adhesion of graphene to a spherical substrate, as
might be desired in a Surface Force Apparatus, for example. Taking values
typical for graphene of $Y\sim 400\mathrm{~{}N\,m^{-1}}$ Lee2008  and bending
energy $B\sim 1\mathrm{~{}eV}\sim 10^{-19}\mathrm{~{}J}$ Lu2009 , together
with a radius of curvature typical of the Surface Force Apparatus, $R\sim
1\mathrm{~{}cm}$, and an adhesive energy $\Gamma\sim 0.1\mathrm{~{}N\,m^{-1}}$
we find that $\beta\sim 2\times 10^{-5}$, $\epsilon\sim 10^{-14}$ so that
$\epsilon/\beta^{3}\sim 1$, which is sufficient for graphene to lie within the
high bendability regime. In particular, we expect that
$\tilde{W}_{c}\sim\beta^{1/2}\sim 5\times 10^{-3}$: sheets of graphene adhered
to a hemisphere of radius of curvature $1\mathrm{~{}cm}$ will only adhere
smoothly if their radius $W\lesssim 50\mathrm{~{}\mu m}$. This is a
significantly more stringent constraint on the adhesion of graphene to doubly-
curved surfaces than would have been the case in the picture presented by the
upper bound Majidi2008 , $\tilde{W}_{\mathrm{dewet}}\simeq 700~{}\mathrm{\mu
m}$. It may also explain why applications in which graphene spontaneously
adheres to curved substrates has been observed to form delamination blisters
Xu2019  with a morphology similar to the rucks and folds studied here.

The theory and experiments presented in this paper show that the anisotropic
and inhomogeneous response of thin elastic sheets to compression manifests
itself in both the macroscopic and the microscopic behaviour of such sheets:
both large scale features such as the threshold between delamination and
smooth adhesion (not to mention conformability) _and_ small scale features
like the number of blisters can only be understood through a proper
understanding of these different responses. Moreover, this understanding
points the way to better control of conformability in scenarios ranging from
the humble band-aid to industrial coatings and beyond.

###### Acknowledgements.

The research leading to these results has received funding from the European
Research Council under the European Union’s Horizon 2020 Programme / ERC Grant
Agreement no. 637334 (DV), the Royal Society through URF/R1/211730 (FB), the
Leverhulme Trust (DV), the National Science Foundation under grant DMR 1822439
(BD), and a visiting professor fellowship from ENS Lyon (June 2018), where
this work was commenced. We are grateful to Daniel Bonn for the use of a
profilometer.  Supplementary information for “ _Delamination from an adhesive
sphere: Curvature–induced dewetting versus buckling_ ” This Supplementary
Information file contains further details of the theoretical picture presented
in the main paper in §S1, including the detailed calculation of the length
available to waste in the formation of folds. §S2 provides details of the
experimental methods.

##  S1 Further theory details

###  S1.1 General calculation of length wasted in folds

To show that folds can only occur when $\epsilon\ll\beta^{3}$ (eqn [17] of the
main text) it was assumed that the Poisson ratio $\nu=0$. This simplified the
calculation of the length that must be wasted by fold formation,
$\Delta_{\mathrm{tot}}$, but the result actually holds for all values of
$\nu$. To show this, we consider here how the argument is modified for non-
zero Poisson ratios.

The key observation is that for highly bendable sheets, the formation of
delamination folds relaxes the compressive stress, i.e.
$\sigma_{\theta\theta}\approx 0$ Davidovitch2011 . (More precisely, the
maximal level of the compressive stress, $|\min({\sigma_{\theta\theta}})|$,
accommodated by the axisymmetric, unbuckled state, vanishes as $\epsilon\to
0$.) As such, the linear constitutive relation (the tensor equation equivalent
of Hooke’s law) gives the azimuthal (hoop) strain at the edge of the sheet at
threshold as:

$\left.\epsilon_{\theta\theta}\right|_{r=W}=\frac{\sigma_{\theta\theta}-\nu\sigma_{rr}}{Y}\approx-\nu\frac{\sigma_{rr}(W)}{Y}=-\nu\beta.$
(S1)

The azimuthal strain may also be expressed in terms of the radial and normal
displacements ($u_{r}$ and $\zeta$, respectively) as:

$\epsilon_{\theta\theta}=\frac{u_{r}}{r}+\frac{1}{2r^{2}}\left(\frac{\partial\zeta}{\partial\theta}\right)^{2}.$
(S2)

We therefore combine Eqns [S1] and [S2] with the assumption that in the fold
(high bendability) limit, the radial displacement $u_{r}(W)$ does not change
significantly from its value in the axisymmetric setup (i.e. that sufficiently
close to threshold, $u_{r}(W)\approx
u_{r}^{\mathrm{axi}}(W)\cdot\bigl{[}1+O(\epsilon^{a})\bigr{]}$ for some $a>0$)
to calculate the length to be wasted $\Delta_{\mathrm{tot}}$ by folds,
$\Delta_{\mathrm{tot}}=\int_{0}^{2\pi}\tfrac{1}{2r^{2}}\left(\tfrac{\partial\zeta}{\partial\theta}\right)^{2}~{}W\mathrm{d}\theta$.
We find that

$-\nu\beta=\left.\epsilon_{\theta\theta}\right|_{r=W}=\frac{u_{r}(W)}{W}+\frac{\Delta_{\mathrm{tot}}}{W}\approx\frac{u_{r}^{\mathrm{axi}}(W)}{W}+\frac{\Delta_{\mathrm{tot}}}{W}$
(S3)

and hence

$\frac{\Delta_{\mathrm{tot}}}{W}\approx-\frac{u_{r}^{\mathrm{axi}}(W)}{W}-\nu\beta=\left|\frac{u_{r}^{\mathrm{axi}}(W)}{W}+\nu\beta\right|.$
(S4)

Recalling that in the fold regime we anticipate that the critical size for the
fold instability is close to that at which compression first occurs, i.e.
$W_{c}=W_{\mathrm{comp}}+\delta W$ with $\delta W/W_{\mathrm{comp}}\ll 1$, and
using Eq. [9] of the main text we then have

$\frac{\Delta_{\mathrm{tot}}}{W}\approx\left|\frac{u_{r}^{\mathrm{axi}}(W)}{W}+\nu\beta\right|=\left|(1-\nu)\beta-\frac{1}{8}\tilde{W}^{2}+\nu\beta\right|=\left|\beta-\frac{1}{8}\left(2\sqrt{2}\beta^{1/2}+\delta\tilde{W}\right)^{2}\right|\approx\beta^{1/2}\delta\tilde{W}/\sqrt{2}.$
(S5)

(S5) is identical to the scaling relationship used to derive eqn [17] of the
main text under the assumption that $\nu=0$; as a result the main conclusion,
eqn [17] of the main text, holds irrespective of Poisson ratio.

###  S1.2 Leading order correction of $w_{c}/R$ when $\epsilon/\beta^{3}\ll 1$

The above paragraph highlighted the central role of the compressive stress in
the analysis of the highly bendable regime. Namely, it is the hoop component
of the axisymmetric stress, $\sigma_{\theta\theta}<0$, that is being
suppressed upon delamination, rather than the corresponding strain
$\epsilon_{\theta\theta}$ (which may retain a finite value
$\epsilon_{\theta\theta}\approx-\nu\beta$, even after delamination). Pushing
this observation further, we can improve our characterization of the threshold
value to delamination folds by equating the maximal compression in the
axisymmetric state, $\sigma^{\mathrm{axi}}_{\theta\theta}(r=W)$, with the
residual compressive stress in the folded state,
$\sigma^{\mathrm{fold}}_{\theta\theta}$. The former is expressed using Eqs (8)
and (10) of the main text together with
$(\tilde{W}_{c}/\tilde{W}_{\mathrm{comp}})-1\ll 1$ to give

$\sigma^{\mathrm{axi}}_{\theta\theta}(W)\sim-Y\tilde{W}_{\mathrm{comp}}\cdot(\tilde{W}_{c}-\tilde{W}_{\mathrm{comp}}).$
(S6)

The latter, $\sigma_{\theta\theta}^{\mathrm{fold}}$, is evaluated via the
residual (unavoidable) compression in a “stickon”:
$\sigma_{\theta\theta}^{\mathrm{fold}}\sim-Y(w/R)^{2}\sim-Yn(W)^{-2}\beta$,
where we have used that the width of the delaminated zone is negligible, i.e.
$\lambda(r)\ll r$, to estimate the width of each stickon as $w\sim W/n$. Using
Eq. [14] of the main text to estimate $n$, we therefore find that:

$\sigma^{\mathrm{fold}}_{\theta\theta}\sim-Y(\epsilon\beta^{2})^{1/5}.$ (S7)

Combining Eqs. [S6] and (S7) we obtain:

$\tilde{W}_{c}(\beta,\epsilon)\sim\tilde{W}_{\mathrm{comp}}\cdot\left[1+c(\epsilon/\beta^{3})^{1/5}\right],$
(S8)

for some constant $c$. This determines, at a scaling level, the correction to
Eq. [15] of the main text. We shall see below that this prediction is
consistent with experimental results performed on the thinnest PS sheets.

##  S2 Experimental Methods

###  S2.1 Sheet fabrication and description

Polystyrene (PS) sheets were fabricated by spin-coating PS (Goodfellow,
Cambridge) dissolved in toluene onto glass slides. The thickness of the
resulting solid PS sheets was measured using interferometry (Film Metrics
F20), and circles of known radius were then cut from the sheet and floated on
a bath of tap water. Here, results are presented for sheet thicknesses
$120\mathrm{~{}nm}\leq t\leq 1.55\mathrm{~{}\mu m}$, with a typical variation
of $20\mathrm{~{}nm}$ within each film.

To reach larger thicknesses (lower bendability), pre-fabricated sheets of
other materials were purchased from Goodfellow (Cambridge, UK). In particular,
sheets of Polycarbonate (PC) with $t=2\mathrm{~{}\mu m},6\mathrm{~{}\mu m}$,
and Polyimide (PI) with $t=13\mathrm{~{}\mu m},25\mathrm{~{}\mu m}$ were used.

The geometrical properties of the system (sheet radius $W$, thickness $t$ and
sphere radius of curvature $R$) were all known or measured as previously
described; similarly the Young’s modulus of the sheets is used as reported
from the literature, i.e. $E=3.4\mathrm{~{}GPa}$ (PS) from Huang2007 , and
$E=2.35\mathrm{~{}GPa}$ (Polycarbonate) and $E=2.9~{}\mathrm{GPa}$ (Polyimide)
from Goodfellow data sheets.

The only unknown in our system is the dry adhesion energy between the sheet
and the sphere. This cannot be measured directly using standard blister tests
because of the extremely thin sheets used. Instead, we use the behaviour with
the very smallest values of $\epsilon/\beta^{3}$ (where the plateau in
$\tilde{W}_{c}$ is predicted to occur) to fit $\Gamma\approx
0.45~{}\mathrm{~{}Nm^{-1}}$; more details of this fitting procedure are given
in §1.S2.3. (It is only possible to measure $\Gamma$ directly for the thinnest
sheets, PS, since these reach the smallest values of $\epsilon/\beta^{3}$; we
therefore use the same value of $\Gamma$ for all materials.)

With these parameter values the dimensionless bending stiffness of the sheet,
$\epsilon=B/(\Gamma R^{2})$, varies in the interval $[2\times 10^{-10},0.02]$.

###  S2.2 Experimental protocol

All sheets were floated on the surface of water either as part of the
fabrication process (for PS sheets) or manually (for other materials).
Commercially available polycarbonate spherical caps were then placed below the
floating sheet and the liquid level gradually reduced until the sheet was
deposited on the surface of the sphere. The system was then allowed to dry for
at least ten minutes before the sphere was inspected to determine whether any
delamination had occurred during the drying process. (In some instances,
especially close to the transition, sheets delaminated in a portion even
though experiments that are nearby in parameter space remained perfectly
adhered. Such experiments are believed to be anomalous, caused by deposition
not occurring axisymmetrically on the sphere, for example, but are
nevertheless indicated by an open point in figures 2 and 4 of the main text
for completeness.)

###  S2.3 Adhesion strength

Figure S1: Experimental data obtained with the thinnest PS sheets show that
the linear relationship of eqn [S10] is observed experimentally. The
uncertainty in the value of the intercept is shown by the two dashed lines
surrounding the solid line (which corresponds to
$\Gamma=0.45\mathrm{~{}Nm^{-1}}$.

The more detailed version of the theory presented in §S1.2 gives more
information about how the critical size for delamination is expected to behave
with $\epsilon/\beta^{3}\ll 1$; this information can be used in the estimation
of the adhesion strength $\Gamma$, as we now discuss. In particular, we expect
from (S8) that the critical sheet size for delamination with
$\epsilon/\beta^{3}\ll 1$ will be:

$\frac{w_{c}}{R}=2\sqrt{2}\beta^{1/2}\left[1+c\left(\frac{\epsilon}{\beta^{3}}\right)^{1/5}\right]$
(S9)

with $c$ undetermined. Now denoting $\ell_{ec}=\Gamma/E$ and recalling that
$\beta=\ell_{ec}/t$ we note that this may be rewritten:

$\frac{w_{c}}{R}t^{1/2}=2\sqrt{2}\ell_{ec}^{1/2}+c^{\prime}t^{6/5}R^{-2/5}/\ell_{ec}^{3/10}.$
(S10)

We therefore expect that plotting $w_{c}\times t^{1/2}/R$ versus
$t^{6/5}R^{-2/5}$ will give approximately a straight line with intercept
$2\sqrt{2}\ell_{ec}^{1/2}$. Plotting the PS data in this way yields fig. S1,
which shows the linear relationship expected. Fitting this linear
relationship, we find that $\ell_{ec}\approx 0.13\pm 0.02\mathrm{~{}nm}$ for
PS (with the constant $c\approx 0.08$). This corresponds to $\Gamma\approx
0.45\pm 0.07\mathrm{~{}Nm^{-1}}$ and so we use the value $\Gamma\approx
0.45\mathrm{~{}Nm^{-1}}$ in all of the plots presented in the main text.

###  S2.4 Blister shape

The theory presented throughout the main text is based on the observation that
extremely bendable sheets should form folds (rather than rucks) at the onset
of the delamination instability. Profiles of delamination blisters around the
periphery of the sheet (all located close to the edge of the sheet with
$r/W=0.95$) were measured using a custom made 3D scanning platform that
combines a confocal distance sensor (Microepsilon IF2405) with precision
linear stages (Physik Instrument), giving (in principle) horizontal and
vertical resolutions of $0.5\mathrm{~{}\mu m}$ and $1\mathrm{~{}\mu m}$,
respectively. We also used a profilometer (Keyence: VK-X1000), reconstructing
the profile from the interference pattern observed with light of wavelength
$440\mathrm{~{}nm}$.

While the quantities of interest are the blister width and height ($\lambda$
and $A$ respectively) at an experimental level, it is difficult to measure
$\lambda$ directly. Instead we replace $\lambda$ with the blister’s full-width
at half maximum, $\ell_{1/2}$. Noting that $\Delta/W\sim W^{2}/R^{2}$, we then
find that in the ruck regime, we should expect:

$\frac{A}{\lambda}\sim\left(\frac{\beta}{\epsilon^{2}}\right)^{1/14}\left(\frac{W}{R}\right)^{5/7}.$
(S11)

Experimental estimates of the typical aspect ratio of the blister,
$A/\ell_{1/2}$, measured at the edge of the sheet are plotted in fig. 4B of
the main text. These results are consistent with the prediction for the ruck
regime (S11). (Here, experiments were performed with a range of values of
$\epsilon$, $\beta$ and $\tilde{\Delta}\sim W^{2}/R^{2}$. Most experiments
were performed with $W/(R\beta^{1/2})=12\pm 1$, corresponding to a horizontal
line in the regime diagram of fig. 4A of the main text. However, for the
thickest sheets, this fixed value of $W/(R\beta^{1/2})$ would not lead to
delamination; experiments were therefore performed with sufficiently large
$W/(R\beta^{1/2})$ that well-developed delamination occurred.)

Unfortunately, the interference techniques used did not allow for the
measurement of blister profiles in the profiles for experiments with the most
bendable sheets (those with the smallest values of $\epsilon/\beta^{3}$).
Nevertheless, the results shown in fig. 4 of the main text are overall
consistent with the prediction that more bendable sheets should lead to
delamination blisters with larger aspect ratio.

###  S2.5 Spatial distribution of blisters

Figure S2: Image detection of blisters at three different radial positions for
a PS sheet with $\beta=2.5\times 10^{-4}$, $\epsilon=8.4\times 10^{-9}$ and
$W/R=0.21$. Left: circles are drawn at various radii (indicated by dashed
curves) and the image intensity mapped along this curve. Right: The image
intensity along each curve (plotted as a function of arc length) shows a
series of peaks; these peaks are counted to give the number of blisters at a
given radial position. (Note that the plots in the right hand figure have been
offset vertically for clarity.)

For regimes of parameter space in which delamination occurred, a second series
of experiments was conducted in which the spatial distribution of the
delamination blisters was measured. To do this, images were captured with
different focal planes, and a composite image created. (An example of such a
composite image is shown in fig. 1D of the main text.) From these images, the
spatial distribution of the number of blisters could be determined by counting
peaks in the image intensity in circles of increasing radius — such intensity
peaks correspond to blisters. This process was automated via image processing
techniques developed in Matlab: typically, peaks are detected when the peak
intensity is more than two standard deviations from the mean of the intensity
signal. For some (noisy) images, this threshold was modified to give results
that accord with manual counting of peaks. An example of the composite image
and the resulting intensity plots is shown in fig. S2.

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# Generic conformally flat hypersurfaces in $\mathbb{R}^{4}$

Xiu Ji, TongZhu Li
Department of Mathematics, Beijing institute of technology,
Beijing,100081,China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>The work is supported by the grant
No. 11571037 and No.11471021 of NSFC.

###### Abstract

In this paper, we study generic conformally flat hypersurfaces in the
Euclidean $4$-space $\mathbb{R}^{4}$ using the framework of Möbius geometry.
First, we classify locally the generic conformally flat hypersurfaces with
closed Möbius form under the Möbius transformation group of $\mathbb{R}^{4}$.
Such examples come from cones, cylinders, or rotational hypersurfaces over the
surfaces with constant Gaussian curvature in $3$-spheres, Euclidean
$3$-spaces, or hyperbolic $3$-spaces, respectively. Second, we investigate the
global behavior of the generic conformally flat hypersurface and give some
integral formulas about these hypersurfaces.

2000 Mathematics Subject Classification: 53A30; 51B10.

Key words: generic conformally flat hypersurface, Möbius metric, Möbius form,
Möbius transformation.

##  1 Introduction

A Riemannian manifold $(M^{n},g)$ is conformally flat, if every point has a
neighborhood which is conformal to an open set in the Euclidean space
$\mathbb{R}^{n}$. A hypersurface of the Euclidean space $\mathbb{R}^{n+1}$ is
said to be conformally flat if so it is with respect to the induced metric.
The dimension of the hypersurface seems to play an important role in the study
of conformally flat hypersurfaces. For $n\geq 4$, the immersed hypersurface
$f:M^{n}\to\mathbb{R}^{n+1}$ is conformally flat if and only if at least $n-1$
of the principal curvatures coincide at each point by the result of Cartan-
Schouten ([1],[10]). Cartan-Schouten’s result is no longer true for three
dimensional hypersurfaces. Lancaster ([6]) gave some examples of conformally
flat hypersurfaces in $\mathbb{R}^{4}$ having three different principal
curvatures. For $n=2$, the existence of isothermal coordinates means that any
Riemannian surface is conformally flat.

A conformally flat hypersurface $f:M^{3}\to\mathbb{R}^{4}$ in $\mathbb{R}^{4}$
is said to be generic, if the second fundamental form has three distinct
eigenvalues everywhere on $M^{3}$. Standard example of generic conformally
flat hypersurface comes from cone, cylinder, or rotational hypersurface over a
surface with constant Gaussian curvature in $3$-sphere $\mathbb{S}^{3}$,
Euclidean $3$-space $\mathbb{R}^{3}$, or hyperbolic $3$-space
$\mathbb{H}^{3}$, respectively. The (local) classification of these
hypersurfaces is far from complete. However, several partial classification
results of generic conformally flat hypersurfaces were given in [2], [3],
[4],[5], [7],[11],[12] and [13].

It is known that the conformal transformation group of $\mathbb{R}^{n}$ is
isomorphic to its Möbius transformation group if $n\geq 3$. As conformal
invariant objects, generic conformally flat hypersurfaces are investigated in
this paper using the framework of Möbius geometry. If an immersed hypersurface
in $\mathbb{R}^{n+1}$ has not any umbilical point, then we can define the so-
called Möbius metric on the hypersurface, which is invariant under Möbius
transformations [14]. Together with another quadratic form (called the Möbius
second fundamental form) they form a complete system of invariants for
hypersurfaces $(dim\geq 3)$ in Möbius geometry [14]. Other important Möbius
invariants of the hypersurface are the Möbius form and the Blaschke tensor.
First, we find that the standard examples of generic conformally flat
hypersurface has closed Möbius form, and vice versa.

######  Theorem 1.1.

Let $f:M^{3}\rightarrow\mathbb{R}^{4}$ be a generic conformally flat
hypersurface. The Möbius form is closed if and only if the hypersurface $f$ is
locally Möbius equivalent to one of the following hypersurfaces in
$\mathbb{R}^{4}$:
$(1)$ a cylinder over a surface in $\mathbb{R}^{3}$ with constant Gaussian
curvature,
$(2)$ a cone over a surface in $\mathbb{S}^{3}$ with constant Gaussian
curvature,
$(3)$ a rotational hypersurface over a surface in $\mathbb{H}^{3}$ with
constant Gaussian curvature.

Second, we investigate the global behavior of compact generic conformally flat
hypersurfaces by the Möbius invariants. Let $(M^{n},g)$ be a Riemannian
manifold. $K(P)$ denotes the sectional curvature of sectional plane
$P(\in\wedge^{2}TM^{n})$. We call the sectional curvature $K(P)$ have sign if
$K(P)\geq 0$ for all $P\in\wedge^{2}TM^{n}$, or $K(P)\leq 0$ for all
$P\in\wedge^{2}TM^{n}$.

######  Theorem 1.2.

Let $f:M^{3}\rightarrow\mathbb{R}^{4}$ be a generic conformally flat
hypersurface. If the hypersurface $M^{3}$ is compact, then the sectional
curvature of the Möbius metric can not have sign.

######  Theorem 1.3.

Let $f:M^{3}\rightarrow\mathbb{R}^{4}$ be a generic conformally flat
hypersurface. If the hypersurface $M^{3}$ is compact, then

$\int_{M^{3}}\\{|\tilde{A}|^{2}+\frac{1}{3}R^{2}-|Ric|^{2}-\frac{2}{27}\\}dv_{g}=0,$

where $\tilde{A}:=A-\frac{1}{3}tr(A)g$ denotes the trace-free Blaschke tensor,
$|Ric|$ denotes the norm of the Ricci curvature of $g$, and $R$ denotes the
scalar curvature of $g$.

######  Corollary 1.1.

Let $f:M^{3}\rightarrow\mathbb{R}^{4}$ be a generic conformally flat
hypersurface. If the hypersurface $M^{3}$ is compact, then

$\int_{M^{3}}\\{|\tilde{A}|^{2}-\frac{2}{27}\\}dv_{g}>0,$

where $\tilde{A}:=A-\frac{1}{3}tr(A)g$ denotes the trace-free Blaschke tensor.

The paper is organized as follows. In section 2, we review the elementary
facts about Möbius geometry of hypersurfaces in $\mathbb{R}^{n+1}$. In section
3, we investigate local behavior of generic conformally flat hypersurfaces in
$\mathbb{R}^{4}$ and prove Theorem 1.1. In section 4, we investigate global
behavior of generic conformally flat hypersurfaces in $\mathbb{R}^{4}$ and
prove Theorem 1.2 and Theorem 1.3.

##  2 Möbius invariants of hypersurfaces in $\mathbb{R}^{n+1}$

In [14], Wang has defined Möbius invariants of submanifolds in
$\mathbb{S}^{n+1}$ and given a congruent theorem of hypersurfaces in
$\mathbb{S}^{n+1}$. In this section, we define Möbius invariants and give a
congruent theorem of hypersurfaces in $\mathbb{R}^{n+1}$ in the same way in
[14]. For details we refer to
$\cite[cite]{[\@@bibref{}{liu}{}{}]},\cite[cite]{[\@@bibref{}{w}{}{}]}$.

Let $\mathbb{R}^{n+3}_{1}$ be the Lorentz space, i.e., $\mathbb{R}^{n+3}$ with
inner product $<\cdot,\cdot>$ defined by

$<x,y>=-x_{0}y_{0}+x_{1}y_{1}+\cdots+x_{n+2}y_{n+2},$

for
$x=(x_{0},x_{1},\cdots,x_{n+2}),y=(y_{0},y_{1},\cdots,y_{n+2})\in\mathbb{R}^{n+3}$.

Let $f:M^{n}\rightarrow\mathbb{R}^{n+1}$ be a hypersurface without umbilical
points and assume that $\\{e_{i}\\}$ is an orthonormal basis with respect to
the induced metric $I=df\cdot df$ with $\\{\theta_{i}\\}$ the dual basis. Let
$II=\sum_{ij}h_{ij}\theta_{i}\theta_{j}$ and $H=\sum_{i}\frac{h_{ii}}{n}$ be
the second fundamental form and the mean curvature of $f$, respectively. We
define the Möbius position vector $Y:M^{n}\rightarrow\mathbb{R}^{n+3}_{1}$ of
$f$ by

$Y=\rho\left(\frac{1+|f|^{2}}{2},\frac{1-|f|^{2}}{2},f\right)~{},~{}~{}\rho^{2}=\frac{n}{n-1}(|II|^{2}-nH^{2}).$

######  Theorem 2.1.

[14] Two hypersurfaces $f,\bar{f}:M^{n}\rightarrow\mathbb{R}^{n+1}$ are Möbius
equivalent if and only if there exists $T$ in the Lorentz group $O(n+2,1)$
such that $\bar{Y}=YT.$

It follows immediately from Theorem 2.1 that

$g=<dY,dY>=\rho^{2}df\cdot df$

is a Möbius invariant, called the Möbius metric of $f$.

Let $\Delta$ be the Laplacian with respect to $g$. Define

$N=-\frac{1}{n}\Delta Y-\frac{1}{2n^{2}}<\Delta Y,\Delta Y>Y,$

which satisfies $<Y,Y>=0=<N,N>,~{}~{}<N,Y>=1.$

Let $\\{E_{1},\cdots,E_{n}\\}$ be a local orthonormal basis for $(M^{n},g)$
with dual basis $\\{\omega_{1},\cdots,\omega_{n}\\}$. Write $Y_{i}=E_{i}(Y)$.
Then we have

$<Y_{i},Y>=<Y_{i},N>=0,~{}<Y_{i},Y_{j}>=\delta_{ij},~{}~{}1\leq i,j\leq n.$

Let $\xi$ be the mean curvature sphere of $f$ written as

$\xi=\left(\frac{1+|f|^{2}}{2}H+f\cdot e_{n+1},\frac{1-|f|^{2}}{2}H-f\cdot
e_{n+1},Hf+e_{n+1}\right)~{},$

where $e_{n+1}$ is the unit normal vector field of $f$ in $\mathbb{R}^{n+1}$.
Thus $\\{Y,N,Y_{1},\cdots,Y_{n},\xi\\}$ forms a moving frame in
$\mathbb{R}^{n+3}_{1}$ along $M^{n}$. We will use the following range of
indices in this section: $1\leq i,j,k\leq n$. We can write the structure
equations as following:

$\displaystyle dY=\sum_{i}Y_{i}\omega_{i},$ $\displaystyle
dN=\sum_{ij}A_{ij}\omega_{i}Y_{j}+\sum_{i}C_{i}\omega_{i}\xi,$ $\displaystyle
dY_{i}=-\sum_{j}A_{ij}\omega_{j}Y-\omega_{i}N+\sum_{j}\omega_{ij}Y_{j}+\sum_{j}B_{ij}\omega_{j}\xi,$
$\displaystyle d\xi=-\sum_{i}C_{i}\omega_{i}Y-\sum_{ij}\omega_{i}B_{ij}Y_{j},$

where $\omega_{ij}$ is the connection form of the Möbius metric $g$ and
$\omega_{ij}+\omega_{ji}=0$. The tensors

${\bf A}=\sum_{ij}A_{ij}\omega_{i}\otimes\omega_{j},~{}~{}{\bf
B}=\sum_{ij}B_{ij}\omega_{i}\otimes\omega_{j},~{}~{}{\bf
C}=\sum_{i}C_{i}\omega_{i}$

are called the Blaschke tensor, the Möbius second fundamental form and the
Möbius form of $f$, respectively. The covariant derivative of
$C_{i},A_{ij},B_{ij}$ are defined by

$\displaystyle\sum_{j}C_{i,j}\omega_{j}=dC_{i}+\sum_{j}C_{j}\omega_{ji},$
$\displaystyle\sum_{k}A_{ij,k}\omega_{k}=dA_{ij}+\sum_{k}A_{ik}\omega_{kj}+\sum_{k}A_{kj}\omega_{ki},$
$\displaystyle\sum_{k}B_{ij,k}\omega_{k}=dB_{ij}+\sum_{k}B_{ik}\omega_{kj}+\sum_{k}B_{kj}\omega_{ki}.$

The integrability conditions for the structure equations are given by

(2.1) $\displaystyle A_{ij,k}-A_{ik,j}=B_{ik}C_{j}-B_{ij}C_{k},$ (2.2)
$\displaystyle C_{i,j}-C_{j,i}=\sum_{k}(B_{ik}A_{kj}-B_{jk}A_{ki}),$ (2.3)
$\displaystyle B_{ij,k}-B_{ik,j}=\delta_{ij}C_{k}-\delta_{ik}C_{j},$ (2.4)
$\displaystyle
R_{ijkl}=B_{ik}B_{jl}-B_{il}B_{jk}+\delta_{ik}A_{jl}+\delta_{jl}A_{ik}-\delta_{il}A_{jk}-\delta_{jk}A_{il},$
(2.5) $\displaystyle R_{ij}:=\sum_{k}R_{ikjk}=-\sum_{k}B_{ik}B_{kj}+(tr{\bf
A})\delta_{ij}+(n-2)A_{ij},$ (2.6)
$\displaystyle\sum_{i}B_{ii}=0,~{}~{}\sum_{ij}(B_{ij})^{2}=\frac{n-1}{n},~{}~{}tr{\bf
A}=\sum_{i}A_{ii}=\frac{1}{2n}(1+\frac{n}{n-1}R),$

Here $R_{ijkl}$ denotes the curvature tensor of $g$, and $R=\sum_{ij}R_{ijij}$
is the Möbius scalar curvature. We know that all coefficients in the structure
equations are determined by $\\{g,{\bf B}\\}$ when $n\geq 3$. Thus we have

######  Theorem 2.2.

[14] Two hypersurfaces $f:M^{n}\rightarrow\mathbb{R}^{n+1}$ and
$\bar{f}:M^{n}\rightarrow\mathbb{R}^{n+1}(n\geq 3)$ are Möbius equivalent if
and only if there exists a diffeomorphism $\varphi:M^{n}\rightarrow M^{n}$
which preserves the Möbius metric and the Möbius second fundamental form.

By equation (2.2), we have

(2.7) $dC=0\Leftrightarrow\sum_{k}(B_{ik}A_{kj}-B_{jk}A_{ki})=0.$

For the second covariant derivative of $B_{ij}$ defined by

$dB_{ij,k}+\sum_{m}B_{mj,k}\omega_{mi}+\sum_{m}B_{im,k}\omega_{mj}+\sum_{m}B_{ij,m}\omega_{mk}=\sum_{m}B_{ij,km}\omega_{m},$

we have the following Ricci identities

$B_{ij,kl}-B_{ij,lk}=\sum_{m}B_{mj}R_{mikl}+\sum_{m}B_{im}R_{mjkl}.$

We call eigenvalues of $(B_{ij})$ as _Möbius principal curvatures_ of $f$.
Clearly the number of distinct Möbius principal curvatures is the same as that
of its distinct Euclidean principal curvatures.

Let $k_{1},\cdots,k_{n}$ be the principal curvatures of $f$, and
$\\{\lambda_{1},\cdots,\lambda_{n}\\}$ the corresponding Möbius principal
curvatures, then the curvature sphere of principal curvature $k_{i}$ is

$\xi_{i}=\lambda_{i}Y+\xi=\left(\frac{1+|f|^{2}}{2}k_{i}+f\cdot
e_{n+1},\frac{1-|f|^{2}}{2}k_{i}-f\cdot e_{n+1},k_{i}f+e_{n+1}\right)~{}.$

Note that $k_{i}=0$ if and only if,

$<\xi_{i},(1,-1,0,\cdots,0)>=0.$

This means that the curvature sphere of principal curvature $k_{i}$ is a
hyperplane in $\mathbb{R}^{n+1}$.

##  3 Generic conformally flat hypersurfaces in $\mathbb{R}^{4}$

In this section, we give some local properties of the Möbius invariants of
generic conformally flat hypersurfaces in $\mathbb{R}^{4}$.

Let $(M^{n},g)$ be an n-dimensional Riemannian manifold, and $\\{e_{1},\cdots
e_{n}\\}$ be a local orthonormal frame field on $(M^{n},g)$, and
$\\{\omega_{1},\cdots,\omega_{n}\\}$ its dual coframe field. The Weyl
conformal tensor
$W=\sum_{ijkl}W_{ijkl}\omega_{i}\otimes\omega_{j}\otimes\omega_{k}\otimes\omega_{l}$
and the Schouten tensor $S=\sum_{ij}S_{ij}\omega_{i}\otimes\omega_{j}$ of
$(M^{n},g)$ are defined by, respectively,

$\begin{split}&W_{ijkl}=R_{ijkl}-\frac{1}{n-2}\\{R_{ik}\delta_{jl}-R_{jk}\delta_{il}+\delta_{ik}R_{jl}-\delta_{jk}R_{il}-\frac{R}{(n-1)}(\delta_{ik}\delta_{jl}-\delta_{jk}\delta_{il})\\},\\\
&S_{ij}=R_{ij}-\frac{R}{2(n-1)}\delta_{ij},\end{split}$

where $R_{ij}$ denotes the Ricci curvature and $R$ the scalar curvature of
$(M^{n},g)$.

A result of Weyl states that a Riemannian manifold $(M^{n},g)$ of dimension
$n(\geq 4)$ is conformally flat if and only if the Weyl conformal tensor
vanishes, and a Riemannian manifold $(M^{n},g)$ of dimension $3$ is
conformally flat if and only if the Schouten tensor is a Codazzi tensor (i.e.,
$S_{ij,k}=S_{ik,j}$). Using the Weyl’s result, we can prove the following
lemma (or see [15]),

######  Lemma 3.1.

[15] A Riemannian product $(M_{1},g_{1})\times(M_{2},g_{2})=(M_{1}\times
M_{2},g_{1}+g_{2})$ is conformally flat if and only if either

(1) $(M_{i},g_{i})$ is one dimensional curve, and $(M_{j},g_{j}),(i\neq j)$ is
a space form, or

(2) $(M_{1},g_{1})$ and $(M_{2},g_{2})$ are space forms of dimension at least
two, with non-zero opposite curvatures.

For hypersurfaces in $\mathbb{R}^{n+1}$, when $n\geq 4$, it is well-known from
the Cartan-Schouten that a hypersurface $f:M^{n}\to\mathbb{R}^{n+1}$ is
conformally flat if and only if at least $n-1$ of the principal curvatures
coincide at each point. But Cartan-Schouten’s result is no longer true in
dimension $3$, since there exist generic conformally flat hypersurfaces.

Let $f:M^{3}\to\mathbb{R}^{4}$ be a generic hypersurface. We choose an
orthonormal basis $\\{E_{1},E_{2},E_{3}\\}$ with respect to the Möbius metric
$g$ such that

(3.8) $(B_{ij})=diag\\{b_{1},b_{2},b_{3}\\},~{}~{}~{}b_{1}<b_{2}<b_{3}.$

Let $\\{\omega_{1},\omega_{2},\omega_{3}\\}$ be the dual of
$\\{E_{1},E_{2},E_{3}\\}$. The conformal fundamental forms of $f$ are defined
by

$\Theta_{1}=\sqrt{(b_{3}-b_{1})(b_{2}-b_{1})}\omega_{1},~{}\Theta_{2}=\sqrt{(b_{3}-b_{2})(b_{2}-b_{1})}\omega_{2},~{}\Theta_{3}=\sqrt{(b_{3}-b_{1})(b_{3}-b_{2})}\omega_{3}.$

Using the equation (2.5) and (2.6), the Schouten tensor of $f$ is

$S=\sum_{ij}(-\sum_{l}B_{il}B_{lj}+A_{ij}+\frac{1}{6}\delta_{ij})\omega_{i}\wedge\omega_{j}.$

Thus

(3.9) $S_{ij,k}=-\sum_{l}(B_{il,k}B_{lj}+B_{il}B_{lj,k})+A_{ij,k}.$

If the hypersurface $f$ is conformally flat, then $S_{ij,k}=S_{ik,j}$.
Combining the equations (2.1) and (3.8), we obtain the following equation

(3.10) $b_{k}B_{ik,j}-b_{j}B_{ij,k}=2(B_{ij}C_{k}-B_{ik}C_{j}).$

Using the equation (2.3), we have the following equations,

(3.11) $\begin{split}&B_{12,3}=B_{13,2}=0,\\\
&B_{ij,i}=\frac{3b_{i}}{b_{j}-b_{i}}C_{j},~{}~{}B_{ii,j}=\frac{b_{i}-b_{k}}{b_{j}-b_{i}}C_{j},~{}~{}i\neq
j,j\neq k,i\neq k.\end{split}$

Using
$dB_{ij}+\sum_{k}B_{kj}\omega_{ki}+\sum_{k}B_{ik}\omega_{kj}=\sum_{k}B_{ij,k}\omega_{k}$
and (3.8), we get

(3.12)
$\omega_{ij}=\sum_{k}\frac{B_{ij,k}}{b_{i}-b_{j}}\omega_{k}=\frac{B_{ij,i}}{b_{i}-b_{j}}\omega_{i}+\frac{B_{ij,j}}{b_{i}-b_{j}}\omega_{j}.$

The following lemma is trivial by the equation (3.11) and (3.12), (or see
[3],[13]).

######  Lemma 3.2.

Let $M^{3}\to\mathbb{R}^{4}$ be a generic hypersurface. The following are
equivalent: (1), the hypersurface is conformally flat;
(2), the schouten tensor is a Codazzi tensor;
(3), the conformal fundamental forms $\Theta_{1},\Theta_{2},\Theta_{3}$ are
closed.

Next, we give the standard examples of generic conformally flat hypersurfaces
in $\mathbb{R}^{4}$.

######  Example 3.1.

Let $u:M^{2}\longrightarrow\mathbb{R}^{3}$ be an immersed surface. We define
_the cylinder over $u$_ in $\mathbb{R}^{4}$ as

$f=(id,u):\mathbb{R}^{1}\times
M^{2}\longrightarrow\mathbb{R}^{1}\times\mathbb{R}^{3}=\mathbb{R}^{4},~{}~{}~{}~{}f(t,y)=(t,u(x)),$

where $id:\mathbb{R}^{1}\longrightarrow\mathbb{R}^{1}$ is the identity map.

The first fundamental form $I$ and the second fundamental form $II$ of the
cylinder $f$ are, respectively,

$I=I_{\mathbb{R}^{1}}+I_{u},\;\;II=II_{u},$

where $I_{u},II_{u}$ are the first and second fundamental forms of $u$,
respectively, and $I_{\mathbb{R}^{1}}$ denotes the standard metric of
$\mathbb{R}^{1}$. Let $\\{k_{1},k_{2}\\}$ be principal curvatures of surface
$u$. Obviously the principal curvatures of hypersurface $f$ are
$\\{0,k_{1},k_{2}\\}.$ The Möbius metric $g$ of hypersurface $f$ is

(3.13)
$g=\rho^{2}I=\frac{n}{n-1}(|II|^{2}-nH^{2})I=\left(4H_{u}^{2}-3K_{u}\right)(I_{\mathbb{R}^{1}}+I_{u}),$

where $H_{u},K_{u}$ are the mean curvature of $u$ and Gaussian curvature of
$u$, respectively. Therefore combining Lemma 3.1 we have the following result.

######  Proposition 3.1.

Let $f:M^{3}\to\mathbb{R}^{4}$ be a cylinder over a surface
$u:M^{2}\to\mathbb{R}^{3}$, then the cylinder $f$ is conformally flat if and
only if the surface $u$ is of constant Gaussian curvature.

######  Example 3.2.

Let $u:M^{2}\longrightarrow\mathbb{S}^{3}\subset\mathbb{R}^{4}$ be an immersed
surface. We define _the cone over $u$_ in $\mathbb{R}^{4}$ as

$f:\mathbb{R}^{+}\times
M^{2}\longrightarrow\mathbb{R}^{4},~{}~{}~{}~{}f(t,x)=tu(x).$

The first and second fundamental forms of the cone $f$ are, respectively,

$I=I_{\mathbb{R}^{1}}+t^{2}I_{u},\;\;II=t~{}II_{u},$

where $I_{u},II_{u},I_{\mathbb{R}^{1}}$ are understood as before. Let
$\\{k_{1},k_{2}\\}$ be principal curvatures of surface $u$. The principal
curvatures of hypersurface $f$ are
$\\{0,\frac{1}{t}k_{1},\frac{1}{t}k_{2}\\}.$ The Möbius metric $g$ of
hypersurface $f$ is

(3.14)
$g=\frac{1}{t^{2}}\left[4H_{u}^{2}-3(K_{u}-1)\right](I_{\mathbb{R}^{1}}+t^{2}I_{u})=\left[4H_{u}^{2}-3(K_{u}-1)\right](I_{\mathbb{H}^{1}}+I_{u}),$

where $H_{u},K_{u}$ are the mean curvature and Gaussian curvature of $u$,
respectively, $I_{\mathbb{H}^{1}}=\frac{dt^{2}}{t^{2}}$. Therefore combining
Lemma 3.1 we have the following result.

######  Proposition 3.2.

Let $f:M^{3}\to\mathbb{R}^{4}$ be a cone over a surface
$u:M^{2}\to\mathbb{S}^{3}$, then the cone $f$ is conformally flat if and only
if the surface $u$ is of constant Gaussian curvature.

######  Example 3.3.

Let $\mathbb{R}^{3}_{+}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}|x_{3}>0\\}$ be
the upper half-space endowed with the standard hyperbolic metric

$ds^{2}=\frac{1}{x_{3}^{2}}[dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}]~{}.$

Let $u=(x_{1},x_{2},x_{3}):M^{2}\longrightarrow\mathbb{R}^{3}_{+}$ be an
immersed surface. We define _rotational hypersurface over $u$_ in
$\mathbb{R}^{4}$ as

$f:\mathbb{S}^{1}\times
M^{2}\longrightarrow\mathbb{R}^{4},~{}~{}~{}f(\phi,x_{1},x_{2},x_{3})=(x_{1},x_{2},x_{3}\phi),$

where $\phi:\mathbb{S}^{1}\longrightarrow\mathbb{S}^{1}$ is the unit circle.

The first fundamental form and the second fundamental form of $u$ is,
respectively,

$\begin{split}&I_{u}=\frac{1}{x_{3}^{2}}(dx_{1}\cdot dx_{1}+dx_{2}\cdot
dx_{2}+dx_{3}\cdot dx_{3}),\\\ &II_{u}=\frac{1}{x_{3}^{2}}(dx_{1}\cdot
d\eta_{1}+dx_{2}\cdot d\eta_{2}+dx_{3}\cdot
d\eta_{3})-\frac{\eta_{3}}{x_{3}}I_{u}.\end{split}$

The first and the second fundamental forms of $f$ is, respectively,

$I=dx\cdot
dx=x_{3}^{2}(I_{\mathbb{S}^{1}}+I_{u}),~{}~{}II=x_{3}II_{u}-\eta_{3}I_{u}-\eta_{3}I_{\mathbb{S}^{1}}.$

Let $\\{k_{1},k_{2}\\}$ be principal curvatures of $u$. Then principal
curvatures of hypersurface $f$ are
$\\{\frac{-\eta_{3}}{x_{3}^{2}},\frac{k_{1}}{x_{3}}-\frac{\eta_{3}}{x_{3}^{2}}~{},~{}\frac{k_{2}}{x_{3}}-\frac{\eta_{3}}{x_{3}^{2}}\\}.$
Thus the Möbius metric of the rotational hypersurface $f$ is

(3.15)
$g=\rho^{2}I=\left[4H_{u}^{2}-3(K_{u}+1)\right](I_{\mathbb{S}^{1}}+I_{u}),$

where $H_{u},K_{u}$ are the mean curvature and Gaussian curvature of $u$,
respectively. Therefore combining Lemma 3.1 we have the following result,

######  Proposition 3.3.

Let $f:M^{3}\to\mathbb{R}^{4}$ be a rotational hypersurface over a surface
$u:M^{2}\to\mathbb{R}^{3}_{+}$, then the hypersurface $f$ is conformally flat
if and only if the surface $u$ is of constant Gaussian curvature.

######  Proposition 3.4.

Let $f:M^{3}\to\mathbb{R}^{4}$ be one of generic conformally flat
hypersurfaces given by above three examples (3.1) (3.2) (3.3). Then the Möbius
form is closed $(i.e.,dC=0)$.

###### Proof.

The Möbius metric $g$ in above three examples (3.1) (3.2) (3.3) can be unified
in a single formula:

$g=\left[4H_{u}^{2}-3(K_{u}-\epsilon)\right](ds^{2}+I_{u})=\rho^{2}(ds^{2}+I_{u}),$

where $I_{u}$, $K_{u}$, and $H_{u}$ are the induced metric, Gaussian
curvature, and mean curvature of $u:M^{2}\rightarrow N^{3}(\epsilon)$,
respectively.

Let $\\{e_{2},e_{3}\\}$ be a local orthonormal basis on $TM^{2}$ with respect
to $I_{u}$, consisting of unit principal vectors of $u$ and
$e_{1}=\frac{\partial}{\partial s}$. Then $\\{e_{1},e_{2},e_{3}\\}$ is an
orthonormal basis for $T(I\times M^{2})$ with respect to $ds^{2}+I_{u}$. Let
$\tilde{R}_{ijkl}$ denote the curvature tensor of the metric $ds^{2}+I_{u}$,
and $R_{ijkl}$ the curvature tensor for $g=\rho^{2}[ds^{2}+I_{u}]$. From Yau’s
paper [15], we have

(3.16)
$\begin{split}&R_{ijij}=\rho^{2}\tilde{R}_{ijij}+\rho\rho_{ii}+\rho\rho_{jj}-|\nabla\rho|^{2},i\neq
j\\\
&R_{ijik}=\rho^{2}\tilde{R}_{ijik}+\rho\rho_{jk},~{}~{}when~{}\\{i,j,k\\}~{}are~{}distinct,\\\
\end{split}$

which implies that $(B_{ij})=diag(b_{1},b_{2},b_{3})$ and
$(A_{ij})=diag(a_{1},a_{2},a_{3})$ under the local orthonormal basis
$\\{\rho^{-1}e_{1},\rho^{-1}e_{2},\rho^{-1}e_{3}\\}$ by the equation (2.4).
Thus $dC=0$ by the equation (2.7). ∎

Next, we prove Theorem 1.1. From Proposition 3.4, we prove another hand of
Theorem 1.1 and we assume $dC=0$. From the equation (2.7), under the
orthonormal basis $\\{E_{1},E_{2},E_{3}\\}$ in (3.8) we find

(3.17) $(A_{ij})=diag\\{a_{1},a_{2},a_{3}\\}.$

The equations (3.8) and (3.17) imply that

$R_{ijik}=A_{jk}=0,~{}~{}j\neq k,$

by the equation (2.4).

From the definition of $B_{ij,kl}$ and (2.3), (3.11) and (3.12), we have

(3.18)
$\begin{split}&B_{23,31}=\frac{3b_{2}B_{33,1}-3b_{3}B_{22,1}}{(b_{3}-b_{2})^{2}}C_{2}+\frac{3b_{3}}{b_{2}-b_{3}}[C_{2,1}-\frac{B_{12,1}}{b_{1}-b_{2}}C_{1}]+B_{31,3}\frac{B_{12,1}}{b_{1}-b_{2}},\\\
&B_{23,13}=(B_{22,1}-B_{33,1})\frac{B_{23,3}}{b_{2}-b_{3}}+(B_{11,2}-B_{33,2})\frac{B_{13,3}}{b_{1}-b_{3}},\end{split}$

Using Ricci identity $B_{23,31}-B_{23,13}=(b_{3}-b_{2})R_{2313}=0$ and (3.18),
we have

$b_{3}C_{1,2}=\frac{2b_{1}^{2}+b_{2}b_{3}}{(b_{2}-b_{1})(b_{1}-b_{3})}C_{1}C_{2}=-C_{1}C_{2}.$

Similarly we have

$b_{1}C_{2,3}=-C_{2}C_{3},~{}~{}b_{2}C_{1,3}=-C_{1}C_{3}.$

Therefore

(3.19) $b_{k}C_{i,j}=-C_{i}C_{j},i\neq j,i\neq k,k\neq j.$

Now we define $\\{C_{i,jk}\\}$ given by

$dC_{i,j}+\sum_{m}C_{m,j}\omega_{mi}+\sum_{m}C_{i,m}\omega_{mj}=\sum_{m}C_{i,jm}\omega_{m}.$

Let $\\{i,j,k\\}$ be distinct. Taking derivative for (3.19) along $E_{k}$ and
invoking (3.11) and (3.12), we get

(3.20)
$\begin{split}&B_{kk,k}C_{i,j}+b_{k}[C_{i,jk}-C_{k,j}\frac{B_{ki,k}}{b_{k}-b_{i}}-C_{i,k}\frac{B_{kj,k}}{b_{k}-b_{j}}]\\\
&=-C_{i}[C_{j,k}-C_{k}\frac{B_{jk,k}}{b_{k}-b_{j}}]-C_{j}[C_{i,k}-C_{k}\frac{B_{ik,k}}{b_{k}-b_{i}}].\end{split}$

If $b_{1}b_{2}b_{3}=0$, we can assume that $b_{1}=0$, which implies that
$b_{2}=-b_{3}=\sqrt{\frac{1}{3}}$ by the equation (2.6). Using (3.11), we have
$C_{1}=C_{2}=C_{3}=0$ and $B_{ij,k}=0$.

Next we assume that $b_{1}b_{2}b_{3}\neq 0$. Because
$b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=\frac{2}{3},B_{ij,j}=B_{jj,i}-C_{i}$, from
(3.19) and (3.20) we have

(3.21)
$b_{k}C_{i,jk}=-\frac{4}{3}\frac{C_{i}C_{j}C_{k}}{b_{i}b_{j}b_{k}}=-\frac{4}{3}\frac{C_{1}C_{2}C_{3}}{b_{1}b_{2}b_{3}}.$

Since $C_{i,jk}=C_{j,ik}=C_{k,ij}$ and $b_{i}\neq b_{j},i\neq j$, from (3.21)
we get

$C_{i,jk}=C_{j,ik}=C_{k,ij}=0,~{}~{}C_{1}C_{2}C_{3}=0.$

We can assume that $C_{1}=0$, then

(3.22)
$\omega_{12}=\frac{B_{12,1}}{b_{1}-b_{2}}\omega_{1},~{}\omega_{13}=\frac{B_{13,1}}{b_{1}-b_{3}}\omega_{1},~{}\omega_{23}=\frac{B_{23,2}}{b_{2}-b_{3}}\omega_{2}+\frac{B_{23,3}}{b_{2}-b_{3}}\omega_{3}.$

From (3.22), combining
$d\omega_{12}-\omega_{13}\wedge\omega_{32}=\frac{-1}{2}\sum_{kl}R_{12kl}\omega_{k}\wedge\omega_{l},$
we obtain

$\begin{split}&\frac{-1}{2}\sum_{kl}R_{12kl}\omega_{k}\wedge\omega_{l}=d(\frac{B_{12,1}}{b_{1}-b_{2}})\wedge\omega_{1}\\\
&+[(\frac{B_{12,1}}{b_{1}-b_{2}})^{2}+\frac{B_{13,1}B_{23,2}}{(b_{1}-b_{3})(b_{2}-b_{3})}]\omega_{1}\wedge\omega_{2}+\frac{B_{13,1}}{b_{1}-b_{3}}[\frac{B_{12,1}}{b_{1}-b_{2}}+\frac{B_{23,3}}{b_{2}-b_{3}}]\omega_{1}\wedge\omega_{3},\end{split}$

which implies that

(3.23)
$\begin{split}&E_{3}(\frac{B_{12,1}}{b_{1}-b_{2}})-\frac{B_{13,1}}{b_{1}-b_{3}}[\frac{B_{12,1}}{b_{1}-b_{2}}+\frac{B_{23,3}}{b_{2}-b_{3}}]=0,\\\
&E_{2}(\frac{B_{12,1}}{b_{1}-b_{2}})-[(\frac{B_{12,1}}{b_{1}-b_{2}})^{2}+\frac{B_{13,1}B_{23,2}}{(b_{1}-b_{3})(b_{2}-b_{3})}]=R_{1212}=b_{1}b_{2}+a_{1}+a_{2}.\end{split}$

Similarly we have

(3.24)
$\begin{split}&E_{2}(\frac{B_{13,1}}{b_{1}-b_{3}})-\frac{B_{12,1}}{b_{1}-b_{2}}[\frac{B_{13,1}}{b_{1}-b_{3}}-\frac{B_{23,2}}{b_{2}-b_{3}}]=0,\\\
&E_{3}(\frac{B_{13,1}}{b_{1}-b_{3}})-[(\frac{B_{13,1}}{b_{1}-b_{3}})^{2}-\frac{B_{12,1}B_{23,3}}{(b_{1}-b_{2})(b_{2}-b_{3})}]=R_{1313}=b_{1}b_{3}+a_{1}+a_{3}.\end{split}$

Under the local basis above, $\\{Y,N,Y_{1},Y_{2},Y_{3},\xi\\}$ forms a moving
frame in $\mathbb{R}^{6}_{1}$ along $M^{3}$. We define

(3.25)
$\begin{split}&F=b_{1}Y+\xi,\;\;X_{2}=\frac{B_{12,1}}{b_{1}-b_{2}}Y+Y_{2},\;\;X_{3}=\frac{B_{13,1}}{b_{1}-b_{3}}Y+Y_{3},\\\
&T=a_{1}Y+N-\frac{B_{12,1}}{b_{1}-b_{2}}Y_{2}-\frac{B_{13,1}}{b_{1}-b_{3}}Y_{3}-b_{1}\xi,\\\
&Q=2a_{1}+b_{1}^{2}+(\frac{B_{12,1}}{b_{1}-b_{2}})^{2}+(\frac{B_{13,1}}{b_{1}-b_{3}})^{2}.\end{split}$

Clearly $F$ is the curvature sphere of principal curvature $k_{1}$. And

$\begin{split}&<F,X_{2}>=<F,X_{3}>=<F,T>=<F,Y_{1}>=0,\\\
&<T,X_{2}>=<T,X_{3}>=<T,Y_{1}>=<X_{2},X_{3}>=0,\\\
&<F,F>=<X_{2},X_{2}>=<X_{3},X_{3}>=1,~{}<T,T>=Q.\end{split}$

From structure equation of the hypersurface and (3.22), we get

(3.26)
$E_{1}(F)=0,~{}~{}E_{2}(F)=(b_{1}-b_{2})X_{2},~{}~{}E_{3}(F)=(b_{1}-b_{3})X_{3}.$

Thus curvature sphere $F$ induces a surface $\tilde{M}=M^{3}/L$ in the de-
Sitter space $\mathbb{S}^{5}_{1}$

$F:\tilde{M}=M^{3}/L\longrightarrow\mathbb{S}^{5}_{1},$

where fibers $L$ are integral submanifolds of distribution
$D=span\\{E_{1}\\}$. We define $V=span\\{T,Y_{1}\\}.$ Clearly we have

$F\bot V.$

Using (3.22), (3.23) and (3.24), we can get that

(3.27)
$\begin{split}&E_{1}(Y_{1})=-T,~{}~{}E_{2}(Y_{1})=0,~{}~{}E_{3}(Y_{1})=0,\\\
&E_{1}(T)=QY_{1},~{}~{}E_{2}(T)=\frac{B_{12,1}}{b_{1}-b_{2}}T,~{}~{}E_{3}(T)=\frac{B_{13,1}}{b_{1}-b_{3}}T.\end{split}$

This implies that subspace $V$ is parallel along $M^{3}$. Similarly we have

(3.28)
$E_{1}(Q)=0,E_{2}(Q)=2\frac{B_{12,1}}{b_{1}-b_{2}}Q,E_{3}(Q)=2\frac{B_{13,1}}{b_{1}-b_{3}}Q.$

Regarding (3.28) as a linear first-order differential equation for $Q$, we see
that $Q\equiv 0$ or $Q\neq 0$ on an connected manifold $M^{n}$. Therefore
there are three possibilities for the induced metric on the fixed subspace
$V\subset\mathbb{R}^{6}_{1}$.

Case 1, $Q=0$, then $<T,T>=Q=0$, therefore $V$ is endowed with a degenerate
inner product.

By (3.27), $T$ determines a fixed light-like direction in
$\mathbb{R}^{6}_{1}$. Up to a Möbius transformation, we may take to be

$T=\lambda(1,-1,0,0,0,0),~{}~{}\lambda\in C^{\infty}(M^{3}).$

Since $V$ is a fixed degenerate subspace in $\mathbb{R}^{6}_{1}$, we can find
a space-like vector $v$ such that $V=Span\\{e=(1,-1,0,0,0,0),v\\}$ and
$<e,F>=<v,F>=0$. We interpret the geometry of the hypersurface
$f:M^{3}\rightarrow\mathbb{R}^{4}$ as below:

1) $v$ determines a fixed hyperplane $\Sigma$ in $\mathbb{R}^{4}$ because of
$<T,v>=0$.

2) $F$ is a two parameter family of hyperplanes orthogonal to the fixed
hyperplane $\Sigma$ in $\mathbb{R}^{4}$.

Therefore $f(M^{3})$, as the envelope of this family of hyperplanes $F$, is
clearly a cylinder over a surface $\tilde{M}\subset\mathbb{R}^{3}$.

Case 2, $Q<0$, then $<T,T>$ is negative, and $V$ is a Lorentz subspace in
$\mathbb{R}^{6}_{1}$. Up to a Möbius transformation, we can assume that

$V=Span\\{T,Y_{1}\\}=Span\\{p_{0}=(1,1,0,0,0,0),p_{1}=(1,-1,0,0,0,0)\\}.$

Using the stereographic projection, $p_{0},p_{1}$ correspond to the origin $O$
and the point at infinity $\infty$ of $\mathbb{R}^{4}$, respectively. Since
$F\perp V$, $F$ is a two parameter family of hyperplanes (passing $O$ and
$\infty$). Therefore $f(M^{3})$, as the envelope of this family of hyperplanes
$F$, is clearly a cone (with vertex $O$) over a surface
$\tilde{M}\subset\mathbb{S}^{3}$.

Case 3, $Q>0$, then $<T,T>$ is positive, and $V$ is a space-like subspace in
$\mathbb{R}^{6}_{1}$. Up to a Möbius transformation, we can assume that

$V=Span\\{T,Y_{1}\\}=Span\\{(0,0,1,0,0,0),(0,0,0,1,0,0)\\}=\mathbb{R}^{2}.$

Thus $V$ is a fixed two dimensional plane
$\mathbb{R}^{2}\subset\mathbb{R}^{4}$, and $F$ is a two parameter family of
hyper-sphere orthogonal to this fixed plane $\mathbb{R}^{2}$ with centers
locating on it. Thus $F$ envelopes a rotational hypersurface $f(M^{3})$ (over
a surface $\tilde{M}\subset\mathbb{R}^{3}_{+}$).

From Case 1, Case 2, Case 3, we prove that if $dC=0$, then the hypersurface is
Möbius equivalent to one of the standard examples of generic conformally flat
hypersurface. thus we complete the proof of Theorem 1.1.

##  4 Global behavior of the generic conformally flat hypersurface

Let $f:M^{3}\to\mathbb{R}^{4}$ be a generic conformally flat hypersurface. We
say that the pair $(U,\omega)$ is admissible if
(1), $U$ is an open subset of $M^{3}$,
(2), $\omega=(\omega_{1},\omega_{2},\omega_{3})$ is a orthonormal co-frame
field on $U$ with respect to the Möbius metric $g$,
(3), $\omega_{1}\wedge\omega_{2}\wedge\omega_{3}=dv_{g}$,
(4), $B=\sum_{i}b_{i}\omega_{i}\otimes\omega_{i}$.

Denote by $F=(E_{1},E_{2},E_{3})$ the dual frame field of $\omega$. Then it is
easily-seen that, $(U,\omega)$ is admissible if and only if $E_{i}$ is an unit
principal vector associated to $b_{i}$ for each $1\leq i\leq 3$, and
$\\{E_{1},E_{2},E_{3}\\}$ is an oriented basis associated to the orientation
of $M^{3}$. Denote by $\\{\omega_{ij}\\}$ the connection form with respect to
$(U,\omega_{1},\omega_{2},\omega_{3})$. Thus under the admissible frame field
$\\{E_{1},E_{2},E_{3}\\}$,

$(B_{ij})=diag\\{b_{1},b_{2},b_{3}\\}.$

Now we introduce two $2$-forms on $M^{3}$: for every admissible co-frame field
$(U,\omega)$, set

$\begin{split}&\Phi=\omega_{12}\wedge\omega_{3}+\omega_{23}\wedge\omega_{1}+\omega_{31}\wedge\omega_{2},\\\
&\Psi=(b_{1}-b_{2})^{2}\omega_{12}\wedge\omega_{3}+(b_{2}-b_{3})^{2}\omega_{23}\wedge\omega_{1}+(b_{1}-b_{3})^{2}\omega_{31}\wedge\omega_{2}.\end{split}$

If $(U,\omega)$ and $(\tilde{U},\tilde{\omega})$ are both admissible co-frame
fields with $U\cap\tilde{U}\neq\emptyset$. Then on $U\cap\tilde{U}$,
$\omega_{i}=\epsilon_{i}\tilde{\omega}_{i},~{}~{}\omega_{ij}=\epsilon_{i}\epsilon_{j}\tilde{\omega}_{ij}$
for every $1\leq i\leq 3$, where $\epsilon_{i}=1$ or $-1$ and
$\epsilon_{1}\epsilon_{2}\epsilon_{3}=1$. Thus

$\omega_{12}\wedge\omega_{3}=\tilde{\omega}_{12}\wedge\tilde{\omega}_{3},~{}~{}\omega_{23}\wedge\omega_{1}=\tilde{\omega}_{23}\wedge\tilde{\omega}_{1},~{}~{}\omega_{31}\wedge\omega_{2}=\tilde{\omega}_{31}\wedge\tilde{\omega}_{2}.$

Therefore the $2$-forms $\Phi,\Psi$ are well-defined on $M^{3}$. Combining
$d\omega_{ij}-\sum_{k}\omega_{ik}\wedge\omega_{kj}=\frac{-1}{2}\sum_{kl}R_{ijkl}\omega_{k}\wedge\omega_{l}$,
$d\omega_{i}=\sum_{k}\omega_{ik}\wedge\omega_{k}$ and the equation (3.11),
(3.12), we get

$\begin{split}&d(\omega_{12}\wedge\omega_{3})=-R_{1212}\omega_{1}\wedge\omega_{2}\wedge\omega_{3}\\\
&+[\frac{-9b_{1}b_{2}C_{3}^{2}}{(b_{1}-b_{3})^{2}(b_{2}-b_{3})^{2}}+\frac{9b_{2}b_{3}C_{1}^{2}}{(b_{1}-b_{2})^{2}(b_{1}-b_{3})^{2}}+\frac{9b_{1}b_{3}C_{2}^{2}}{(b_{1}-b_{2})^{2}(b_{2}-b_{3})^{2}}]\omega_{1}\wedge\omega_{2}\wedge\omega_{3}.\end{split}$

Similarly we can compute $d(\omega_{23}\wedge\omega_{1})$ and
$d(\omega_{31}\wedge\omega_{2})$. Therefore we have

(4.29)
$\begin{split}&d\Phi=[\frac{9b_{1}b_{2}C_{3}^{2}}{(b_{1}-b_{3})^{2}(b_{2}-b_{3})^{2}}-R_{1212}-R_{1313}-R_{2323}]\omega_{1}\wedge\omega_{2}\wedge\omega_{3}\\\
&+[\frac{9b_{2}b_{3}C_{1}^{2}}{(b_{1}-b_{2})^{2}(b_{1}-b_{3})^{2}}+\frac{9b_{1}b_{3}C_{2}^{2}}{(b_{1}-b_{2})^{2}(b_{2}-b_{3})^{2}}]\omega_{1}\wedge\omega_{2}\wedge\omega_{3}.\end{split}$

Using $db_{i}=\sum_{k}B_{ii,k}\omega_{k}$ and the same computation as $d\Phi$,
we can obtain

(4.30)
$\begin{split}&d\Psi=-[(b_{1}-b_{2})^{2}R_{1212}+(b_{1}-b_{3})^{2}R_{1313}+(b_{2}-b_{3})^{2}R_{2323}]dv_{g}\\\
&+[\frac{18b_{1}b_{2}C_{3}^{2}}{(b_{1}-b_{3})^{2}(b_{2}-b_{3})^{2}}+\frac{18b_{2}b_{3}C_{1}^{2}}{(b_{1}-b_{2})^{2}(b_{1}-b_{3})^{2}}+\frac{18b_{1}b_{3}C_{2}^{2}}{(b_{1}-b_{2})^{2}(b_{2}-b_{3})^{2}}]dv_{g},\end{split}$

where $dv_{g}=\omega_{1}\wedge\omega_{2}\wedge\omega_{3}$. Combining the
equation (4.29) and the equation (4.30), we have

(4.31)
$2d\Phi-d\Psi=\\{[(b_{1}-b_{2})^{2}-2]R_{1212}+[(b_{2}-b_{3})^{2}-2]R_{2323}+[(b_{1}-b_{3})^{2}-2]R_{1313}\\}dv_{g}.$

If $M^{3}$ is compact, the equation (4.31) implies that

(4.32)
$\int_{M^{3}}\\{[(b_{1}-b_{2})^{2}-2]R_{1212}+[(b_{2}-b_{3})^{2}-2]R_{2323}+[(b_{1}-b_{3})^{2}-2]R_{1313}\\}dv_{g}=0.$

From $b_{1}+b_{2}+b_{3}=0$ and $b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=\frac{2}{3}$, we
can derive that

$(b_{1}-b_{2})^{2}+(b_{2}-b_{3})^{2}+(b_{1}-b_{3})^{2}=2.$

which implies that

(4.33)
$(b_{1}-b_{2})^{2}-2<0,~{}~{}(b_{2}-b_{3})^{2}-2<0,~{}~{}(b_{1}-b_{3})^{2}-2<0.$

Now we assume that the sectional curvature of $M^{3}$ with respect to the
Möbius metric have sign, for example, the sectional curvature is nonnegative.
The equation (4.32) and (4.33) imply that the sectional curvature vanishes,
i.e.,

$R_{1212}=R_{2323}=R_{1313}=0.$

In [8], authors classify the hypersurfaces $f:M^{3}\to\mathbb{R}^{4}$ with
constant Möbius sectional curvature, which are non-compact. This is a
contradiction, thus we finish the proof of Theorem 1.2.

Using the equation (2.4) and the equation (2.6), we have

(4.34)
$\begin{split}&[(b_{1}-b_{2})^{2}-2]R_{1212}+[(b_{2}-b_{3})^{2}-2]R_{2323}+[(b_{1}-b_{3})^{2}-2]R_{1313}\\\
&=\frac{2}{9}-\frac{10}{3}tr(A)+3[b_{1}^{2}a_{1}+b_{2}^{2}a_{2}+b_{3}^{2}a_{3}].\end{split}$

On the other hand, the equation (2.5) implies that

(4.35)
$|Ric|^{2}=\frac{2}{9}+5tr(A)^{2}+|A|^{2}-\frac{4}{3}tr(A)-2[b_{1}^{2}a_{1}+b_{2}^{2}a_{2}+b_{3}^{2}a_{3}],$

where $|Ric|$ denote the norm of the Ricci curvature. Combining the equation
(4.34) and the equation (4.35), we can derive that

(4.36)
$\begin{split}&[(b_{1}-b_{2})^{2}-2]R_{1212}+[(b_{2}-b_{3})^{2}-2]R_{2323}+[(b_{1}-b_{3})^{2}-2]R_{1313}\\\
&=\frac{5}{9}-\frac{16}{3}tr(A)+\frac{15}{2}tr(A)^{2}+\frac{3}{2}|A|^{2}-\frac{3}{2}|Ric|^{2}.\end{split}$

Let $\tilde{A}:=A-\frac{1}{3}tr(A)g$ denote the trace-free Blaschke tensor,
then $|\tilde{A}|^{2}=|A|^{2}-\frac{1}{3}tr(A)^{2}$. Thus from the equation
(4.36), we have

(4.37)
$\begin{split}&[(b_{1}-b_{2})^{2}-2]R_{1212}+[(b_{2}-b_{3})^{2}-2]R_{2323}+[(b_{1}-b_{3})^{2}-2]R_{1313}\\\
&=\frac{3}{2}|\tilde{A}|^{2}-\frac{3}{2}[|Ric|^{2}-\frac{1}{3}R^{2}]-\frac{1}{9}.\end{split}$

Now if the hypersurface $M^{3}$ is compact, then

(4.38)
$\int_{M^{3}}\\{|\tilde{A}|^{2}+\frac{1}{3}R^{2}-|Ric|^{2}-\frac{2}{27}\\}dv_{g}=0.$

Therefore we finish the proof of Theorem 1.3.

Since $R=tr(Ric)$, we have $\frac{1}{3}R^{2}-|Ric|^{2}\leq 0$ on $M^{3}$. if
$\frac{1}{3}R^{2}-|Ric|^{2}\equiv 0$, then the sectional curvature $K=0$, and
there is a contradiction by the results in [8]. Thus Corollary 1.1 is proved.

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