[a]T. Saito

# Characterization of SiPM and development of test bench modules for the next-
generation cameras for Large-Sized Telescopes for Cherenkov Telescope Array

K. Hashiyama  H. Iwasaki  H. Kubo  M. Mizote  A. Okumura  H. Tajima  T.
Yamamoto

###### Abstract

The recent improvements in the performance of the silicon photomultipliers
(SiPMs) made them attractive options as photo sensors of imaging atmospheric
Cherenkov telescopes (IACTs). In fact, they are already adopted in some IACTs
such as FACT and the Small-Sized Telescopes of the Cherenkov Telescope Array
(CTA). However, the application to the Large-Sized Telescopes (LSTs) of CTA
requires additional studies. As the pixel size of LSTs is larger than the
nominal size of SiPMs, the signal from multiple sensors must be summed up.
Also, the high detection efficiency of the night sky background (NSB) photons
may degrade the telescope performance. To overcome this, the pulse width must
be as small as 3 ns and the detection efficiency for NSB photons must be
suppressed as much as possible. Heat generation and gain stabilization are
also issues. We studied different types of SiPMs from Hamamatsu photonics and
characterized them for the LST application, addressing the previous points.
Also, to prove the SiPM performance in LST, we are developing a SiPM module
which can be installed in the exisiting LST camera. Here we present the
results of this evaluation and the status of the test bench module
development.

##  1 Large-Size Telescopes for Cherenkov Telescope Array

The Cherenkov Telescope Array (CTA) is the next generation gamma-ray
observatory, covering the energy range from 20 GeV to 300 TeV. Such a wide
energy coverage can be achieved by using three different sizes of Imaging
Atmospheric Cherenkov Telescopes (IACTs), namely, Large-Sized Telescope (LST,
the reflector diameter $D_{r}=23$ m), Midium-Sized Telescope (MST, $D_{r}=\sim
10$ m)) , and Small-Sized Telescope (SST, $D_{r}=\sim 4$ m). The two arrays,
one in each hemisphere to survey the entire sky, will include a combination of
many SSTs, observing very high energy gamma-rays above 3 TeV, several MSTs
taking charge of the middle energy range between 300 GeV to 10 TeV and LSTs,
which are sensitive to the lowest energy range above 20 GeV.

Cherenkov light from an atmospheric air shower is dominated by optical photons
and IACTs are designed to catch them. As photo sensors, conventional
photomultipliers (PMTs) have been used in IACTs since the beginning of the
field. Semiconductor type photosensors, so-called Silicon Photomultiplers
(SiPMs), have also started to be used recently. The FACT telescope [1] was the
first IACT, that was equipped with SiPMs. It started operation in 2012. The
sensitivity of SiPM has been getting higher, with the optical cross talk
probability, which is non-desirable feature for most of applications, becoming
sufficiently low. The price per area has also decreased largely. As a
consequence, SiPM started to be used in other IACTs including some types of
cameras of SSTs and MSTs. We are considering to adopt SiPMs for a possible
upgrade of the LST camera.

In the case of LSTs, as the target energies of gamma rays are much lower than
other telescopes, the contamination of the night sky background (NSB) photons
to the air shower image is not negligible. Therefore, to evaluate new photo
sensors, the effect of night sky background (NSB) photons must be carefully
examined besides the Cherenkov photon detection efficiency. In this context,
we studied the characteristics of SiPMs from Hamamatsu photonics and we are
developing a 14 pixel SiPM module that can be used for LSTs.

##  2 Hamamatsu S13360-3075CN-UVE-1

Among different types of SiPM we tested, here we would like to report mainly
on Hamamatsu S13360-3075CN-UVE-1.

###  2.1 Sensor description

This sensor was developed by Hamamatsu photonics aiming for usage in IACTs.
Photodetection efficiency (PDE) is tuned for near-UV wavelengths, where
atmospheric Cherenkov light spectrum peaks. The most pronounced characteristic
of this is its pulse width, which will be discussed in the next sub-section.
The size of the sensitive area is 3 mm x 3 mm, which consists of 1600 Geiger
mode avalanche photo diode (g-APD) with a pitch of 75 um.

###  2.2 PulseShape

The pulse shape of this SiPM is shown in figure 1. There are a very fast
component and a very slow component as you can see in the middle and the left
panel of figure 1. The full width half maximum of the fast component is about
4 ns, while the slow component exhibits exponential decay with time constant
of about 2 $\mu$s.

Given that the typical duration of atmospheric Cherenkov flash from air
showers is a few to several ns, one can extract the charge only from the fast
component to record the shower image. However, it would be important to
understand the performance of the sensor if the next photon hit the g-APD
during the long tail. In section 2.4, we will study the recovery time.

Figure 1: Left: The overall picture of the pulse shape of Hamamatsu
S13360-3075CN-UVE-1. Middle: the pulse shape zoomed in the horizontal axis for
the fast component. The full width half maximum is about 4 ns. Right, the
pulse shape zoomed in the vertical axis for the slow component. It has an
exponential shape and the decay constant is about 2 $\mu$s.

###  2.3 Basic Performance

The basic performance such as charge resolution, the temperature dependence of
break down voltage, the dark count rate (DCR) as a function of over voltage
and temperature and the optical crosstalk probability as a function of the
over voltage and temperature were studied and results are shown in figure 2.
Temperature dependence of break down voltage is about 0.053 V/∘C. This
dependency may require a simple temperature compensation circuit when it is
used in IACT. DCR is highly temperature and over-voltage dependent, and it is
about 400 kHz at 25∘C with the over voltage of 3 volts. It is negligible
compared with NSB for LSTs (see section 2.5). Optical cross talk probably also
depends on the over voltage and about 4% at 3 V, which is also acceptable for
IACT application.

Figure 2: Basic performance of Hamamatsu S13360-3075CN-UVE-1. Top Left: Output
charge disbribution for weak input signals. Clearly separated distributions up
to 4 p.e. can be seen. Top right: Temperature dependence of the break down
voltage. It is about 0.053 V/∘C. Bottom left: Temperature and over-voltage
dependence of the dark count rate. Bottom right: Temperature and over-voltage
dependence of the optical cross talk probability.

###  2.4 Recovery Time

Given that this sensor has a 2 µs long tail in the pulse shape as shown in
section 2.2, we studied the recovery time of this sensor. The method is the
following. First we illuminate the sensor with a intense laser pulse such that
all 1600 g-APDs fire. Soon after that, the second laser flashes it again and
record the signal pulse amplitude. If the time difference between the two
flashes ($\Delta t$) is small, the amplitude is reduced because g-APDs are not
sufficiently charged up. The relation between the amplitude and $\Delta t$ is
shown in the figure 3. The relation can be expressed in $1-\rm{exp}(t/\tau)$
and $\tau$ is approximately 2 µs, which is similar to the time constant of the
slow component in the pulse shape.

Figure 3: Relative pulse amplitude as a function of the time since the
previous avalanche process. The curve can be fitted with $1-\rm{exp}(t/\tau)$
and $\tau$ is approximately 2 µs

###  2.5 Effect of NSB

The recovery time $\sim 2$ µs may not be a problem if NSB photon detection
rate is sufficiently low. However, if the NSB level is high, significant
fraction of g-APDs have a reduced over voltage at any moment. The left panel
of figure 4 shows the simulated over-voltage distribution over the 1600 g-APDs
at a given moment under different NSB conditions, based on numerical
simulations. As NSB rate goes higher, large fraction of g-APDs have lower
voltages.

The right panel of the figure 4 shows the drop of average PDE as a function of
the NSB level, based on numerical simulations. As the lower over-voltage leads
to lower PDE, the rate of NSB directly affects the PDE. The rate of NSB in per
3 mm $\times$ 3 mm area of this sensor in a dark night is expected to be about
15 MHz. The effect is negligible for dark nights, but if the rate is 10 times
larger the dark night, about 10% degradation is expected. If it is 100 times
larger, which is expected for the Moon night observations, the PDE will be
less than half of the dark condition.

Also, the variance of over voltages among g-APDs indicates the degradation of
charge resolution. It is expected by simulation and confirmed by the
measurements as shown in the figure 5. The resolution worsen by 50% under NSB
100 times stronger than the dark nights.

Figure 4: Left: Simulated distribution of over voltage over the 1600 g-APDs
under different NSB rates. Right: The mean and the standard deviation the PDE
distribution as a function the NSB rates. Three dashed vertical lines indicate
the NSB rate of the dark night, 10 time the dark night and 100 times the dark
night.

Figure 5: Charge resolution as a function of NSB rate. Measurements (left) and
numerical simulation (right). Three dashed vertical lines in the right panel
indicate the NSB rate of the dark night, 10 time the dark night and 100 times
the dark night.

###  2.6 Conclusion for Hamamatsu S13360-3075CN-UVE-1

The $\sim 4$ ns pulse width is very attractive for LSTs, as one can suppress
the contamination of NSB photons in the air shower images. However, this fast
pulse is accompanied with a very long tail and a slow recovery, which results
in the effective degradation of PDE and charge resolution under strong NSB.
One of the advantages of SiPM with respect to the conventional PMTs are
robustness of the sensor under strong illumination. For example, SiPM makes it
possible to operate IACTs under the strong Moon. But the loss of PDE and
charge resolution under strong background rates compromises this advantage.
The decision to adopt this sensor depends on the scientific objectives of LSTs
and their priorities. Further studies and discussions are needed.

##  3 Light Guide

As can be seen in the previous section, the reduction of NSB detection
efficiency is very important if we use SiPMs in LSTs. For this purpose, a
special coating on the light guides are studied. The details are presented
here [2]. As you can see in the photo in figure 6, it reflects bluish light
well while red light is suppressed. The entrance and exit have a square shape.
For such a case, the parabola curve used in the so-called Winstone Cone [3] is
not the best option. The shape was optimized by using cubic Bezier curves
following the method described in [4]. Collection efficiency as a function of
incident angles estimated with the ray tracing simulation is shown in the
right panel of figure 6. The efficiency is perfect until $20-25$ degrees
(depending on the azimuthal angles) and then sharply falls. Only the photons
from the mirror dish would be collected onto the SiPM surface.

Figure 6: Left: The photo of LGs for the LST SiPM modules. Due to the coating,
the reflective surface look blue. Right: Simulated collection efficiency as a
function of the incident angle. Azimuthal angle 0 degree (parallel to one of
the edges) and 45 degrees are shown.

##  4 Design of the SiPM module

Applications of SiPM to a large size IACT are not well studied so far.
Especially, the effect of high and changing NSB rate together with temperature
dependence of the sensor performance could encompass hidden problems in SiPM
operation. To test them, we are developing a SiPM test bench module that can
be installed in the present LST camera. Figure 7 shows the completion
rendering of the module. It has 14 SiPM pixels with square shapes. The they
are aligned such that the module can be inserted without interfering with
existing PMT modules. Basically the same readout board can be used as the
current 7 PMT modules. The only change needed was a change in the electronics
to replace the 7 low-gain PMT readout channels with a second set of 7 high-
gain channels.

Figure 7: Left: The front view of the modules. 15 SiPM are aligned such that
they do not interfere with existing PMT pixels. Right: A 3D rendering of the
front part of the SiPM test bench module. The readout board will connected to
the right hand side of this picture.

##  5 Conclusion

Hamamatsu S13360-3075CN-UVE-1, which exhibits a fast pulse, is an attractive
option for LST cameras, but the slow recovery time needs to be carefully
examined since the performance degrades under strong NSB. The development of
test bench modules including square shaped light guides, that can be installed
in the existing LST camera, is progressing well.

## References

  * [1] H. Anderhub et al.  Nuclear Instruments and Methods in Physics Research Section A, 639(1):58–61, 2011.
  * [2] A. Okumura et al.  Proceedings of ICRC 2023, https://pos.sissa.it/444, 2023.
  * [3] R. Winston.  J. Opt. Soc. Amer., 60:245–247, 1970.
  * [4] Akira Okumura.  Astroparticle Physics, 38:18–24, 2012.

LST Acknowledgements

We gratefully acknowledge financial support from the following agencies and
organisations:

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação
de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Fundação de Amparo
à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Apoio à Ciência,
Tecnologia e Inovação do Paraná - Fundação Araucária, Ministry of Science,
Technology, Innovations and Communications (MCTIC), Brasil; Ministry of
Education and Science, National RI Roadmap Project DO1-153/28.08.2018,
Bulgaria; Croatian Science Foundation, Rudjer Boskovic Institute, University
of Osijek, University of Rijeka, University of Split, Faculty of Electrical
Engineering, Mechanical Engineering and Naval Architecture, University of
Zagreb, Faculty of Electrical Engineering and Computing, Croatia; Ministry of
Education, Youth and Sports, MEYS LM2015046, LM2018105, LTT17006, EU/MEYS
CZ.02.1.01/0.0/0.0/16_013/0001403, CZ.02.1.01/0.0/0.0/18_046/0016007 and
CZ.02.1.01/0.0/0.0/16_019/0000754, Czech Republic; CNRS-IN2P3, the French
Programme d’investissements d’avenir and the Enigmass Labex, This work has
been done thanks to the facilities offered by the Univ. Savoie Mont Blanc -
CNRS/IN2P3 MUST computing center, France; Max Planck Society, German
Bundesministerium für Bildung und Forschung (Verbundforschung / ErUM),
Deutsche Forschungsgemeinschaft (SFBs 876 and 1491), Germany; Istituto
Nazionale di Astrofisica (INAF), Istituto Nazionale di Fisica Nucleare (INFN),
Italian Ministry for University and Research (MUR); ICRR, University of Tokyo,
JSPS, MEXT, Japan; JST SPRING - JPMJSP2108; Narodowe Centrum Nauki, grant
number 2019/34/E/ST9/00224, Poland; The Spanish groups acknowledge the Spanish
Ministry of Science and Innovation and the Spanish Research State Agency (AEI)
through the government budget lines PGE2021/28.06.000X.411.01,
PGE2022/28.06.000X.411.01 and PGE2022/28.06.000X.711.04, and grants
PID2022-139117NB-C44, PID2019-104114RB-C31, PID2019-107847RB-C44,
PID2019-104114RB-C32, PID2019-105510GB-C31, PID2019-104114RB-C33,
PID2019-107847RB-C41, PID2019-107847RB-C43, PID2019-107847RB-C42,
PID2019-107988GB-C22, PID2021-124581OB-I00, PID2021-125331NB-I00; the “Centro
de Excelencia Severo Ochoa" program through grants no. CEX2019-000920-S,
CEX2020-001007-S, CEX2021-001131-S; the “Unidad de Excelencia María de Maeztu"
program through grants no. CEX2019-000918-M, CEX2020-001058-M; the “Ramón y
Cajal" program through grants RYC2021-032552-I, RYC2021-032991-I,
RYC2020-028639-I and RYC-2017-22665; the “Juan de la Cierva-Incorporación"
program through grants no. IJC2018-037195-I, IJC2019-040315-I. They also
acknowledge the “Atracción de Talento" program of Comunidad de Madrid through
grant no. 2019-T2/TIC-12900; the project “Tecnologiás avanzadas para la
exploracioń del universo y sus componentes" (PR47/21 TAU), funded by Comunidad
de Madrid, by the Recovery, Transformation and Resilience Plan from the
Spanish State, and by NextGenerationEU from the European Union through the
Recovery and Resilience Facility; the La Caixa Banking Foundation, grant no.
LCF/BQ/PI21/11830030; the “Programa Operativo" FEDER 2014-2020, Consejería de
Economía y Conocimiento de la Junta de Andalucía (Ref. 1257737), PAIDI 2020
(Ref. P18-FR-1580) and Universidad de Jaén; “Programa Operativo de Crecimiento
Inteligente" FEDER 2014-2020 (Ref. ESFRI-2017-IAC-12), Ministerio de Ciencia e
Innovación, 15% co-financed by Consejería de Economía, Industria, Comercio y
Conocimiento del Gobierno de Canarias; the “CERCA" program and the grant
2021SGR00426, both funded by the Generalitat de Catalunya; and the European
Union’s “Horizon 2020" GA:824064 and NextGenerationEU (PRTR-C17.I1). State
Secretariat for Education, Research and Innovation (SERI) and Swiss National
Science Foundation (SNSF), Switzerland; The research leading to these results
has received funding from the European Union’s Seventh Framework Programme
(FP7/2007-2013) under grant agreements No 262053 and No 317446; This project
is receiving funding from the European Union’s Horizon 2020 research and
innovation programs under agreement No 676134; ESCAPE - The European Science
Cluster of Astronomy & Particle Physics ESFRI Research Infrastructures has
received funding from the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement no. 824064.


# 3-circle Theorem for Willmore surface I

Yuxiang Li  Department of Mathematical Sciences, Tsinghua University
<EMAIL_ADDRESS>and  Hao Yin  School of Mathematical Sciences,
Unviersity of Science and Technology of China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the blow-up of Willmore surfaces. By using the
3-circle theorem, we prove a decay estimate of the second fundamental form
along the neck region. This estimate provides a new perspective and
streamlined proofs to a few key results in this field, such as the energy
identity(quantization), removable singularities and gap theorem.

Y.Li is supported by National Key R&D Program of China 2022YFA 1005400; H. Yin
is supported by NSFC 12141105.

##  1\. Introduction

In geometric analysis, especially in the study of variational problems, one
often encounters the phenomenon of bubble tree convergence. For example, the
harmonic maps from surfaces, the Yang-Mills fields in dimension four, the
Willmore surfaces in $\mathbb{R}^{n}$ and so on. It is important to understand
how various quantities behave after taking this limit.

The study of bubble tree convergence first appeared in the research on
harmonic maps from surfaces to manifolds and the study of harmonic heat flow.
The bubble tree phenomenon appears because of the $\epsilon$-regularity
theorem of the harmonic map equation. In this area, the two most natural
questions are: 1. whether the energy identity holds, that is, whether the
necks carry away energy; 2. whether the necks exist, and if so, what their
structure is. Many results have been obtained along this line. For example,
when the tension fields are $L^{p}$-bounded for $p>1$, there are no necks
between bubbles [26, 9, 30]; for a harmonic map sequence from a sequence of
Riemann surfaces, with diverging complex structures, Chen-Tian proved that the
necks converge to geodesics (see [6] and [7]).

In studying neck regions, typically, we use the Ding-Tian’s reduction [10] or
Parker’s reduction [25] to transform the problem into the proof of the
following decay estimate:

If $u_{k}$ is a sequence of harmonic maps defined on $[0,T_{k}]\times S^{1}$
with $T_{k}\rightarrow+\infty$, satisfying

$\lim_{T\rightarrow+\infty}\varlimsup_{k\rightarrow+\infty}\sup_{t\in[T,T_{k}-T]}\int_{[t,t+1]\times
S^{1}}|\nabla u_{k}|^{2}dtd\theta=0,$

then for some $q>0$,

$\int_{[t,t+1]\times S^{1}}|\nabla u_{k}|^{2}dtd\theta\leq
C(e^{-qt}+e^{-q(T_{k}-t)})\qquad\text{for}\quad t\in[0,T_{k}-1].$

With the above decay estimate, it is quite straightforward to derive

$\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}\int_{[T,T_{k}-T]\times
S^{1}}|\nabla
u_{k}|^{2}dtd\theta=0,\,\,\,\,\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}osc_{[T,T_{k}-T]\times
S^{1}}u_{k}=0,$

which implies the energy identity and the neckless theorem.

There are two methods to obtain the decay. The first one is to derive various
differential inequalities that guarantee the convexity of the tangential
energy. This method was originated from Sack-Uhlenbeck [29], and it was
implemented independently by Ding [9] and by Lin-Wang [18]. The second is the
3-circle theorem introduced by Qing-Tian (see [26]). Note that the 3-circle
theorem has a long history and a much wider range of generalizations and
applications (for example, [8, 19, 31]).

Starting from 2001, Kuwert and Schätzle began using the blow-up analysis
methods to study Willmore surfaces and Willmore flows in $\mathbb{R}^{n}$,
obtaining $\epsilon$-regularity, gap theorems, and removability theorems for
singularities in codimension one [12, 13, 14]. In 2008, Rivière [27]
discovered that the Willmore surface equation could be written as a system of
two equations with Jacobian structures, allowing the use of Wente’s inequality
and harmonic analysis methods to study the Willmore equation. Using these
equations, Rivière also obtained $\epsilon$-regularity and removability
theorems for singularities in general codimensions.

In this paper, using the 3-circle lemma, we revisit the energy identity and
the neck structure problem of a sequence of closed Willmore surfaces with
bounded Willmore energy $W(f)$ and bounded genus, with the technical
assumption that the conformal structure remains bounded. Using the reduction
method due to Ding-Tian [10] and Parker [25] again, we can reduce the study of
the energy identity and the neck structure to the following problem.

Let $f_{k}:[0,T_{k}]\times S^{1}\rightarrow\mathbb{R}^{n}$ be a sequence of
conformal and Willmore immersion. We assume:

  * A.1)

$T_{k}\rightarrow+\infty$.

  * A.2)

for any $t_{k}\in(0,T_{k})$ with $t_{k}\rightarrow+\infty$ and
$T_{k}-t_{k}\rightarrow+\infty$, we have

$\int_{[t_{k},t_{k}+1]\times S^{1}}|A_{k}|^{2}dV_{g_{k}}\rightarrow 0.$

  * A.3)

if $e^{2u_{k}}(dt^{2}+d\theta^{2})=f_{k}^{*}(g_{\mathbb{R}^{n}})$, then
$\sup_{[0,T_{k}]\times S^{1}}\left|\nabla u_{k}\right|\leq\beta$.

  * A.4)

There is a Willmore immersion $\tilde{f}_{k}$ defined on $D$ such that
$f_{k}(t,\theta)=\tilde{f_{k}}(e^{-t},\theta)$ for all
$(t,\theta)\in[0,T_{k}]\times S^{1}$.

Q1: Do we have the energy identity

(1.1)
$\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}\int_{[T,T_{k}-T]\times
S^{1}}|H_{k}|^{2}dV_{g_{k}}=0?$

Q2: Let ${\bf n}_{k}$ be the Gauss map (into the Grassmannian $G(2,n)$). Do we
have

(1.2)
$\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}osc_{[T,T_{k}-T]\times
S^{1}}{\bf n}_{k}=0?$

Q3: If this is no true, what is the limit of the image of the Gauss map of
${\bf n}_{k}$?

######  Remark 1.1.

The assumption A.4) holds naturally when we consider a convergence of Willmore
immersions with the induced conformal structres bounded in the moduli space.

######  Remark 1.2.

The assumption A.3) does not follow from A.1), A.2) and A.4). To see this,
simply consider the sequence $f_{k}(t,\theta)=(e^{-kt}\cos k\theta,e^{-kt}\sin
k\theta,0)\in\mathbb{R}^{3}$. We do need it for the proof of our main results.
However, it arises naturally, when $f_{k}([0,T_{k}]\times S^{1})$ is a part of
closed surface $\Sigma^{\prime}_{k}$ with bounded energy and genus. We give a
proof of this claim in Lemma 5.1.

Based on the results in [27], Bernard and Rivière, in [3], gave an affirmative
answer to Q1 under the assumptions A.1)-A.4). In [22], Michelat and Rivière
proved some $L^{(2,1)}$ quantization result (Theorem C therein) and provided
the affirmative answer to Q2.

In this paper, we adapt the three circle argument to prove the decay estimate
(Theorem 1.3), which implies again the affirmative answers to Q1 and Q2.

Before stating our theorem, let’s first introduce two definitions. Given
$c\in\mathbb{R}^{n}$, $S\in\mathfrak{so}(\mathbb{R}^{n})$(the set of $n\times
n$ skew-symmetric matrices), and a conformal and Willmore immersion
$f:[0,T]\times S^{1}\rightarrow\mathbb{R}^{n}$, we define

$\displaystyle\tau_{1}(f,c)$ $\displaystyle=$
$\displaystyle\left(-2\int_{\left\\{t\right\\}\times
S^{1}}\partial_{t}Hd\theta-4\int_{\left\\{t\right\\}\times S^{1}}(H\cdot
A_{ti})g^{ij}(\partial_{j}f)d\theta+\int_{\left\\{t\right\\}\times
S^{1}}\left|H\right|^{2}\partial_{t}fd\theta\right)\cdot c$
$\displaystyle\tau_{2}(f,S)$ $\displaystyle=$ $\displaystyle
2\int_{\left\\{t\right\\}\times
S^{1}}(H\cdot\partial_{t}(Sf)-\partial_{t}H\cdot
Sf)d\theta-4\int_{\left\\{t\right\\}\times S^{1}}(H\cdot
A_{ti})g^{ij}(\partial_{j}f\cdot Sf)d\theta$
$\displaystyle+\int_{\left\\{t\right\\}\times
S^{1}}\left|H\right|^{2}\partial_{t}f\cdot Sfd\theta.$

It will be proved in Section 2 that $\tau_{1}$ and $\tau_{2}$ are independent
of $t$ in general and that if $A.4)$ is assumed, then $\tau_{1}(f,c)=0$ and
$\tau_{2}(f,S)=0$ for any $c\in\mathbb{R}^{n}$ and
$S\in\mathfrak{so}(\mathbb{R}^{n})$.

We need to point out that $\tau_{1}(f,c)=0$ corresponds to $d\omega=0$ in the
paper by Kuwert-Schätzle [13], and it is equivalent to the vanishing of the
first residue defined in [2]. While the second residue is defined in [2] to be
something else, both $\tau_{1}(f,c)=0$ and $\tau_{2}(f,S)=0$ were known as
conservation laws since [27] and were shown to be related to the Noether’s
theorem in [1].

Our main result is:

######  Theorem 1.3.

For any $q\in(0,2)$ and $m\in\mathbb{Z}_{+}$, there exists $\delta_{1}(m,q)>0$
such that if $f:[0,T]\times S^{1}\to\mathbb{R}^{n}$ is a conformal and
Willmore immersion with
$f^{*}(g_{\mathbb{R}^{n}})=e^{-2mt+2v}(dt^{2}+d\theta^{2})$, satisfying

  1. (a)

$\tau_{1}(f,S)=\tau_{2}(f,c)=0$ for any
$S\in\mathfrak{so}(\mathbb{R}^{n}),\,c\in\mathbb{R}^{n}$;

  2. (b)

$\sup_{[0,T]\times S^{1}}\left|\nabla v\right|<\delta_{1}$;

  3. (c)

$\tilde{\delta}:=\sup_{t\in[0,T-1]}\int_{[t,t+1]\times
S^{1}}\left|A\right|^{2}dV_{g}<\delta_{1}$,

then

(1.3) $\int_{[t,t+1]\times S^{1}}\left|A\right|^{2}dV_{g}\leq
C\tilde{\delta}\left(e^{-qt}+e^{-q(T-t)}\right)\qquad\text{for}\quad
t\in[0,T-1].$

######  Remark 1.4.

We can not expect a decay rate $q>2$ in general and we do not know if $q=2$ is
possible.

######  Remark 1.5.

We refer to Theorem 3.30 in the recent paper [23] of Michelat and Rivière for
a similar but different decay estimate.

The proof of this theorem uses the 3-circle argument twice. First, we apply it
to the equation of mean curvature $H$,

$2\Delta H+4{\rm div}(H\cdot A_{pq}g^{ip}\partial_{i}f)-{\rm
div}(|H|^{2}\nabla f)=0.$

Under the assumption (c), this equation implies that $H$ is almost a (vector-
valued) harmonic function on $[0,T]\times S^{1}$. In order to apply the
3-circle lemma for harmonic functions, we divide the cylinder $[0,T]\times
S^{1}$ into cylinders of fixed length $Q_{i}$. We would like to show some
convexity property for the quantity $\int_{Q_{i}}\left|H\right|^{2}dV_{g}$
(see Section 3 for details). This is not true even for an exact harmonic
function as indicated by the special case $e^{mt}\cos m\theta$, $e^{mt}\sin
m\theta$. It turns out that these two are the only obstructions. Fortunately,
thanks to the assumption (a), we are able to show that they do not appear in
the limit when we apply the 3-circle to (scaled) $H$. Therefore, we obtain the
decay of Willmore energy (Theorem 3.1).

Next, we apply the 3-circle lemma to the equation of the Gauss map ${\bf n}$,

$\Delta{\bf n}+A_{G(2,{n})}(d{\bf n},d{\bf n})=\nabla^{\perp}H$

where $A_{G(2,n)}$ is the second fundamental form of the Grassmannian
$G(2,n)$. Since $A=\nabla{\bf n}$, it suffcies to prove the decay for the
Dirichlet energy of ${\bf n}$ along the cylinder. For that purpose, we compare
the size of $H$ and $A$. For those parts of the cylinder on which $A$ is
smaller than a multiple of $H$, the decay of $A$ follows from that of $H$,
which is known. For those parts that $H$ is significantly smaller than $A$,
the above equation shows that ${\bf n}$ is “almost” a harmonic map. By now, it
is standard to apply the 3-circle lemma to study the energy decay of a
harmonic map from cylinder and a key ingredient of the proof is some Pohozaev
identity which implies that the tangential energy on each
$\left\\{t\right\\}\times S^{1}$ is the same as the radial energy. In our
case, we are able to show that this is “almost” true when $H$ is much smaller
than $A$ (see Lemma 4.2 for exact statement). Note that ${\bf n}$ is only
defined on a cylinder (that is not simply connected) and in general we do not
expect the Pohozaev identity even for harmonic maps defined on cylinders.

The details of these two steps appear in Section 3 and Section 4 respectively.
Together they prove Theorem 1.3. In Section 5, we apply Theorem 1.3 to give
positive answers to Q1 and Q2 under the assumption of A.1)-A.4). The main task
is to verify the assumption (b). This follows from A.2) and A.3) via a proof
by contradiction.

In Section 6, we discuss further applications of Theorem 1.3. It is natural
that a decay estimate as (4.1) has applications such as removable singularity
theorem and gap theorem. A problem is that for these applications, it is not
natural to assume A.4), hence we will have to prove the assumption (a) before
we can use Theorem 1.3. In Section 6.1, we reprove the removable singularity
theorem due to Kuwert and Schätzle [13, 14] and Rivière [27] using our decay
estimate. In Section 6.2, we prove a version of gap theorem for conformal and
Willmore immersions defined on $\mathbb{R}\times S^{1}$.

To conclude the introduction, we discuss briefly the results on Willmore
immersions with the assumptions A.1)-A.3), but not A.4). Under conditions
A.1)-A.3), Laurain and Rivière studied problem Q1 in [16] and they provide
some sufficient condition for the validity of the energy identity (1.1). Very
recently, Dorian Martino proved that energy identity is true when $n=3$ and
the index is bounded [20]. He also proved that the conformal Gauss map
converges to a geodesic. In a subsequent paper, we will apply the 3-circle
theorem again to show that for a sequence of conformal and Willmore immersions
satisfying A.1)-A.3), but not A.4), the image of the Gauss map of $f_{k}$
converges to a geodesic in $G(2,n)$.

Acknowledgement. The authors would like to thank Professor Ernst Kuwert for
his insights and helpful discussions.

##  2\. A Pohozaev type identity

As in the proof of the energy identity and no neck theorem in the harmonic map
case (see [10, 18, 26] for example), it is important to have some Pohozaev
identity. In the case of a harmonic map $u:D\to N$, where $D$ is the unit disk
in $\mathbb{R}^{2}$ and $N$ is any target manifold, the Pohozaev identity
implies that the tangential part of the Dirichet energy is exactly the same as
the radial part. In the Willmore case, it is not immediately clear what are
the tangential part and the radial part of the energy. Instead, we find that
the following result will serve a similar purpose in the proof of energy decay
in Theorem 1.3.

######  Theorem 2.1.

Let $f:D\to\mathbb{R}^{n}$ be a smooth conformal Willmore immersion. Let
$(r,\theta)$ be the polar coordinate and $t=-\log r$. For any $t>0$, any
$c\in\mathbb{R}^{n}$ and any $S\in\mathfrak{so}(\mathbb{R}^{n})$, we have

(2.1) $\begin{split}0=&2\int_{\left\\{t\right\\}\times
S^{1}}(H\cdot\partial_{t}(Sf)-\partial_{t}H\cdot(Sf+c))d\theta-4\int_{\left\\{t\right\\}\times
S^{1}}(H\cdot A_{ti})g^{ij}(\partial_{j}f\cdot(Sf+c))d\theta\\\
&+\int_{\left\\{t\right\\}\times
S^{1}}\left|H\right|^{2}\partial_{t}f\cdot(Sf+c)d\theta.\end{split}$

Or equivalently,

$\tau_{1}(f,c)=\tau_{2}(f,S)=0.$

Here $g$ is the pullback metric $f^{*}(g_{\mathbb{R}^{n}})$ and $A_{ti}$ is
$A(\partial_{t}f,\partial_{i}f)$ with $A$ the second fundamental form.

######  Remark 2.2.

This result is not new. For a clear presentation of how such identities can be
obtained via Noether’s theorem, we refer to Bernard’s paper [1]. Here we have
put it in a form convenient for us and include a proof for completeness.

###### Proof.

Let $f_{s}:D\rightarrow\mathbb{R}^{n}(s\in(-\epsilon,\epsilon))$ be a smooth
variation with $f_{0}=f$ and $\dot{f_{s}}|_{s=0}=\varphi$. Denote the pullback
metric $f_{s}^{*}(g_{\mathbb{R}^{n}})$ by $g_{s}$ and the mean curvature
vector $\Delta_{g_{s}}f_{s}$ by $H_{s}$. We omit the subscript $s$ when $s=0$.

For any subdomain $\Omega\subset D$, direct computation shows

$\displaystyle\frac{d}{ds}W(f_{s},\Omega)|_{s=0}$ $\displaystyle=$
$\displaystyle\frac{d}{ds}\int_{\Omega}(g_{s}^{ij}(f_{s})_{ij}-\Gamma_{s,ij}^{k}\partial_{k}f_{s})^{2}\sqrt{|g_{s}|}dx|_{s=0}$
$\displaystyle=$ $\displaystyle
2\int_{\Omega}H\cdot\Delta_{g}\varphi\sqrt{\left|g\right|}dx+2\int_{\Omega}H\cdot(\dot{g}_{s}^{ij}\partial^{2}_{ij}f-\dot{\Gamma}_{s,ij}^{k}\partial_{k}f)|_{s=0}\sqrt{\left|g\right|}dx+\int_{\Omega}|H|^{2}\frac{d}{ds}\sqrt{\left|g_{s}\right|}|_{s=0}dx$
$\displaystyle=$ $\displaystyle 2\int_{\Omega}H\cdot\Delta_{g}\varphi
dV_{g}-2\int_{\Omega}g^{ip}(\dot{g}_{s,pq}|_{s=0})g^{jq}(\partial^{2}_{ij}f\cdot
H)dV_{g}+\int_{\Omega}|H|^{2}\nabla_{g}f\cdot\nabla_{g}\varphi dV_{g}$
$\displaystyle=$ $\displaystyle 2\int_{\Omega}H\cdot\Delta_{g}\varphi
dV_{g}-4\int_{\Omega}H\cdot
A_{ij}g^{ip}g^{jq}(\partial_{p}f\cdot\partial_{q}\varphi)dV_{g}+\int_{\Omega}|H|^{2}\nabla_{g}f\cdot\nabla_{g}\varphi
dV_{g}$ $\displaystyle=$ $\displaystyle 2\int_{\Omega}H\cdot\Delta\varphi
dx-4\int_{\Omega}H\cdot
A_{ij}g^{ip}\delta^{jq}(\partial_{p}f\cdot\partial_{q}\varphi)dx+\int_{\Omega}|H|^{2}\nabla
f\cdot\nabla\varphi dx.$

Here in the last line, we have used the conformal invariance and $\Delta$ and
$\nabla$ are the Laplace and the gradient with respect to the Euclidean metric
on $D$.

If $\Omega=D$ and $\varphi$ has compact support in $D$, then integration by
parts shows that $f$ being a Willmore immersion satisfies the Euler-Lagrange
equation

(2.2) $2\Delta H+4{\rm div}(H\cdot A_{pq}g^{ip}\partial_{i}f)-{\rm
div}(|H|^{2}\nabla f)=0.$

Now, for any $S\in\mathfrak{so}(\mathbb{R}^{n})$ and $c\in\mathbb{R}^{n}$, let
$e^{sS}$ be the family of rotations generated by $S$ and consider a special
variation given by

$f_{s}=e^{sS}f+sc.$

Hence,

$\varphi=\dot{f_{s}}|_{s=0}=Sf+{c}$

and

$W(f_{s},\Omega)=W(f,\Omega),\qquad\text{for}\quad s\in(-\epsilon,\epsilon),$

which implies that $\frac{d}{ds}W(f_{s},\Omega)|_{s=0}=0$. Since $f$ satisfies
(2.2), the boundary term that is involved in the integration by parts when
deriving (2.2) must vanish, in spite of the fact that $\varphi$ is not
supported in $\Omega$.

If we take $\Omega=D_{r}$ for $0<r<1$, we obtain

$2\int_{\partial
D_{r}}(H\cdot\partial_{r}\varphi-\partial_{r}H\cdot\varphi)d\sigma-4\int_{\partial
D_{r}}(H\cdot A_{rq})g^{iq}(\partial_{i}f\cdot\varphi)d\sigma+\int_{\partial
D_{r}}|H|^{2}\partial_{r}f\cdot\varphi d\sigma=0$

where $A_{rq}=A(\partial_{r}f,\partial_{q}f)$.

The proof is done by substituting $\varphi=Sf+c$ and translating $(r,\theta)$
coordinate to $(t,\theta)$ coordinate. $\hfill\Box$

######  Remark 2.3.

When $f$ is a conformal and Willmore immersion from $[0,T]\times S^{1}$, we
regard $f$ as a map from $D\setminus D_{e^{-T}}$. Taking
$\Omega=D_{r_{1}}\setminus D_{r_{2}}$, we learn from the same computations
above that the definition of $\tau_{1}$ and $\tau_{2}$ are independent of $t$.

##  3\. 3-circle lemma for Willmore energy

For some $L>0$, we define for any $i\in\mathbb{Z}$

$Q_{i}=[(i-1)L,iL]\times S^{1}.$

Usually, for each $Q_{i}$, we attach some quantity $\Phi_{i}\geq 0$ (which is
some integral over $Q_{i}$). The 3-circle lemma aims at showing that for each
$q\in(0,2)$, there is $L_{0}(q)$ and if $L>L_{0}$, we have

(3.1) $\Phi_{i}\leq e^{-qL}(\Phi_{i-1}+\Phi_{i+1})$

holds for $i$ in a certain range.

The advantage of having (3.1) is that we can show for each
$q^{\prime}\in(0,q)$, the existence of $L_{1}(q,q^{\prime})$ such that if
$L>L_{1}$, we have

$\text{either}\quad\Phi_{i-1}\geq
e^{q^{\prime}L}\Phi_{i}\qquad\text{or}\quad\Phi_{i+1}\geq
e^{q^{\prime}L}\Phi_{i}\\\ $

and

$\displaystyle\Phi_{i}\geq e^{q^{\prime}L}\Phi_{i-1}$
$\displaystyle\Longrightarrow$ $\displaystyle\Phi_{i+1}\geq
e^{q^{\prime}L}\Phi_{i}$ $\displaystyle\Phi_{i}\geq e^{q^{\prime}L}\Phi_{i+1}$
$\displaystyle\Longrightarrow$ $\displaystyle\Phi_{i-1}\geq
e^{q^{\prime}L}\Phi_{i}.$

Hence (3.1) is very useful in proving exponential decay of certain quantity
along a cylinder and $q^{\prime}$ represents the speed of decay.

In this section, we will study the Willmore energy of a conformal Willmore
immersion $f$ defined on

$Q=Q_{1}\cup Q_{2}\cup Q_{3}.$

The main result of this section is a 3-circle type lemma for the Willmore
energy. More precisely,

######  Theorem 3.1.

For any $m\in\mathbb{Z}_{+}$ and $q\in(0,2)$, there exist $L_{0}(m,q)>0$ and
$\epsilon_{0}(m,q)>0$. If $L>L_{0}$, the following holds. Let
$f:Q\rightarrow\mathbb{R}^{n}$ be a conformal and Willmore immersion with
$g=f^{*}(g_{\mathbb{R}^{n}})=e^{-2mt+2v}(dt^{2}+d\theta^{2})$ for some smooth
function $v$. If

(3.2) $\tau_{1}(f,c)=\tau_{2}(f,S)=0,\qquad\text{for any}\quad
S\in\mathfrak{so}(\mathbb{R}^{n}),\,\,\,\,c\in\mathbb{R}^{n}$

and

(3.3) $\int_{Q_{i}}|A|^{2}+\|\nabla
v\|_{C^{0}(Q_{i})}<\epsilon_{0},\,\,\,\,{i=1,2,3},$

then

(3.4) $\int_{Q_{2}}\left|H\right|^{2}dV_{g}\leq
e^{-qL}\left(\int_{Q_{1}}\left|H\right|^{2}dV_{g}+\int_{Q_{3}}\left|H\right|^{2}dV_{g}\right).$

###  3.1. Harmonic functions

The assumption (3.3) implies that the mean curvature is approximately a
harmonic function. We need the following 3-circle property of a harmonic
function.

######  Lemma 3.2.

For each $m\in\mathbb{Z}_{+}$ and $q\in(0,2)$, there is $L_{0}(m,q)>0$ such
that if $L>L_{0}(m,q)$ then the following holds. Assume that $u\neq 0$ is a
harmonic function defined on $Q$. The following expansion is well known

(3.5) $u=a+bt+\sum_{k=1}^{\infty}\left((a_{k}e^{-kt}+b_{k}e^{kt})\cos
k\theta+(a_{k}^{\prime}e^{-kt}+b_{k}^{\prime}e^{kt})\sin k\theta\right).$

If $b_{m}=b_{m}^{\prime}=0$, then

(3.6)
$\int_{Q_{2}}|u|^{2}e^{-2mt}dtd\theta<e^{-qL}\left(\int_{Q_{1}}\left|u\right|^{2}e^{-2mt}dtd\theta+\int_{Q_{3}}\left|u\right|^{2}e^{-2mt}dtd\theta\right).$

###### Proof.

By a direct calculation, we have (for $i=1,2,3$)

$\displaystyle\|ue^{-mt}\|^{2}_{L^{2}(Q_{i})}$ $\displaystyle=$ $\displaystyle
2\pi\displaystyle{\int}_{(i-1)L}^{iL}|a+bt|^{2}e^{-2mt}dt$
$\displaystyle+\pi\int_{(i-1)L}^{iL}\sum_{k=1}^{+\infty}|a_{k}e^{-kt}+b_{k}e^{kt}|^{2}e^{-2mt}dt+\pi\int_{(i-1)L}^{iL}\sum_{k=1}^{+\infty}|a_{k}^{\prime}e^{-kt}+b_{k}^{\prime}e^{kt}|^{2}e^{-2mt}dt$
$\displaystyle=$ $\displaystyle 2\pi\int_{(i-1)L}^{iL}|a+bt|^{2}e^{-2mt}dt$
$\displaystyle+\frac{\pi}{2}\sum_{k=1}^{\infty}{\Big{(}}|a_{k}|^{2}\frac{1-e^{-2(k+m)L}}{(k+m)}e^{-2(k+m)(i-1)L}+|b_{k}|^{2}\frac{e^{2(k-m)L}-1}{k-m}e^{2(k-m)(i-1)L}+$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\frac{a_{k}b_{k}}{m}(1-e^{-2mL})e^{-2m(i-1)L}{\Big{)}}$
$\displaystyle+\frac{\pi}{2}\sum\limits_{k=1}^{\infty}{\Big{(}}{|a_{k}^{\prime}|}^{2}\frac{1-e^{-2(k+m)L}}{(k+m)}e^{-2(k+m)(i-1)L}+{|b_{k}^{\prime}|}^{2}\frac{e^{2(k-m)L}-1}{k-m}e^{(k-m)(i-1)L}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\frac{a_{k}^{\prime}b_{k}^{\prime}}{m}(1-e^{-2mL})e^{-2m(i-1)L}{\Big{)}}.$

In the above computation, when $k=m$, the expression
$\frac{e^{2(k-m)L}-1}{k-m}$ does not make sense. However, since we have
assumed $b_{m}=b^{\prime}_{m}=0$, this is not a problem.

By setting $C_{k},D_{k},E_{k}$ and
$C^{\prime}_{k},D^{\prime}_{k},E^{\prime}_{k}$ properly, we can rewrite the
last line in the above equation

$\displaystyle\left\|ue^{-mt}\right\|_{L^{2}(Q_{i})}^{2}$ $\displaystyle:=$
$\displaystyle
A_{i}+\sum_{k=1}\left(C_{k}e^{-2(k+m)(i-1)L}+D_{k}e^{2(k-m)(i-1)L}+E_{k}e^{-2m(i-1)L}\right)$
$\displaystyle+\sum_{k=1}\left(C^{\prime}_{k}e^{-2(k+m)(i-1)L}+D^{\prime}_{k}e^{2(k-m)(i-1)L}+E^{\prime}_{k}e^{-2m(i-1)L}\right)$
$\displaystyle:=$ $\displaystyle
A_{i}+\sum_{k=1}^{\infty}B_{i;k}+\sum_{k=1}^{\infty}B^{\prime}_{i;k}$
$\displaystyle:=$ $\displaystyle A_{i}+B_{i}+B_{i}^{\prime}.$

Step 1.  Claim: there exists $C(m)>0$ such that

(3.7) $B_{2;k}\leq C(m)e^{-2L}(B_{1;k}+B_{3;k}).$

Summing over $k$, we obtain

(3.8) $B_{2}\leq C(m)e^{-2L}(B_{1}+B_{3}).$

The proof of the claim is divided into several cases.

Case 1: $k<m$. By $k\geq 1$ and the Schwarz inequality,

$\displaystyle\left|\frac{2a_{k}b_{k}}{m}\right|$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{m^{2}-1}}{m}\left|2\frac{a_{k}}{\sqrt{m+k}}\frac{b_{k}}{\sqrt{m-k}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{m^{2}-1}}{m}\left(\frac{\left|a_{k}\right|^{2}}{m+k}+\frac{\left|b_{k}\right|^{2}}{m-k}\right).$

Then

$(1-e^{-2mL})\left|\frac{2a_{k}b_{k}}{m}\right|\leq\frac{\sqrt{m^{2}-1}}{m}\left((1-e^{-2(k+m)L})\frac{\left|a_{k}\right|^{2}}{m+k}+\frac{1-e^{-2mL}}{1-e^{2(k-m)L}}(1-e^{2(k-m)L})\frac{\left|b_{k}\right|^{2}}{m-k}\right).$

By the definition of $C_{k},D_{k}$ and $E_{k}$, for $L$ sufficiently large, we
have $c(m)\in(0,1)$ such that

$\left|E_{k}\right|\leq(1-c(m))(C_{k}+D_{k}),$

which implies

$C_{k}+D_{k}+E_{k}=c(m)(C_{k}+D_{k})+(1-c(m))(C_{k}+D_{k})+E_{k}\geq
c(m)(C_{k}+D_{k}).$

Hence,

$\displaystyle B_{2;k}$ $\displaystyle=$ $\displaystyle
C_{k}e^{-2(k+m)L}+D_{k}e^{2(k-m)L}+E_{k}e^{-2mL}$ $\displaystyle\leq$
$\displaystyle e^{-2L}(C_{k}+D_{k})+(1-c(m))e^{-2L}(C_{k}+D_{k})$
$\displaystyle\leq$ $\displaystyle 2e^{-2L}(C_{k}+D_{k})$ $\displaystyle\leq$
$\displaystyle C(m)e^{-2L}(C_{k}+D_{k}+E_{k})$ $\displaystyle=$ $\displaystyle
C(m)e^{-2L}B_{1;k}.$

Here $C(m)$ is a constant depending only on $m$ that may vary from line to
line.

Case 2: $k=m$.

Note that $D_{k}=E_{k}=0$. We have

$B_{2;k}=e^{-2(k+m)L}C_{k}\leq e^{-2L}C_{k}=e^{-2L}B_{1;k}.$

Case 3: $k>m$ and $E_{k}\geq 0$.

When $E_{k}\geq 0$, we know $D_{k}e^{2(k-m)(2L)}\leq B_{3;k}$, which implies
that

$\displaystyle B_{2;k}$ $\displaystyle=$ $\displaystyle
C_{k}e^{-2(k+m)L}+D_{k}e^{2(k-m)L}+E_{k}e^{-2mL}$ $\displaystyle=$
$\displaystyle C_{k}e^{-2(k+m)L}+e^{-2(k-m)L}D_{k}e^{4(k-m)L}+E_{k}e^{-2mL}$
$\displaystyle\leq$ $\displaystyle
e^{-2L}(C_{k}+E_{k})+e^{-2L}(D_{k}e^{2(k-m)L\cdot 2})$ $\displaystyle\leq$
$\displaystyle e^{-2L}(B_{1}^{k}+B_{3}^{k}).$

Case 4: $k>m$ and $E_{k}<0$.

Choose $L_{0}$ such that for $L>L_{0}$,

$2e^{-2(k+m)L}\leq 2e^{-2L-2mL}<e^{-2mL}\quad\text{and}\quad
2e^{-2(k-m)L-4mL}<e^{-2mL}.$

Together with $E_{k}<0$, we obtain

$(e^{-2(k+m)L}+e^{-2(k-m)L-4mL})E_{k}>e^{-2mL}E_{k},$

which allows us to estimate

$\displaystyle B_{2;k}$ $\displaystyle=$ $\displaystyle
C_{k}e^{-2(k+m)L}+D_{k}e^{2(k-m)L}+E_{k}e^{-2mL}$ $\displaystyle\leq$
$\displaystyle e^{-2(k+m)L}(C_{k}+E_{k})+e^{-2(k-m)L}(D_{k}e^{2(k-m)\cdot
2L}+E_{k}e^{-2m\cdot 2L})$ $\displaystyle\leq$ $\displaystyle
e^{-2(k+m)L}(C_{k}+D_{k}+E_{k})+e^{-2(k-m)L}(C_{k}e^{-(k+m)\cdot
2L}+D_{k}e^{2(k-m)\cdot 2L}+E_{k}e^{-2m\cdot 2L})$ $\displaystyle=$
$\displaystyle e^{-2(k+m)L}B_{1;k}+e^{-2(k-m)L}B_{3;k}$ $\displaystyle\leq$
$\displaystyle e^{-2L}(B_{1;k}+B_{3;k}).$

Here in the last line above we used the fact that $B_{i;k}$ is nonnegative,
which is obvious if we go back to its definition.

Step 2. We show that for any $q<2$, we can choose $L_{0}$, such that for any
$L>L_{0}$, there holds

(3.9) $A_{2}\leq e^{-qL}(A_{1}+A_{3}).$

Setting

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle|a|^{2}+\frac{ab}{m}+\frac{|b|^{2}}{2m^{2}}=|a+\frac{b}{2m}|^{2}+\frac{|b|^{2}}{4m^{2}}$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle\frac{|b|^{2}}{m}+2ab$
$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle|b|^{2},$

and

$F(t)=\frac{e^{-2mt}}{2m}(c_{1}+c_{2}t+c_{3}t^{2}),$

we verify that $-F(t)$ is a primitive function of $(a+bt)^{2}e^{-2mt}$ such
that

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle F(0)-F(L)$
$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle F(L)-F(2L)$
$\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle F(2L)-F(3L).$

It is easy to check the following easy facts

  1. (1)

$c_{1}+c_{2}t+c_{3}t^{2}\geq 0$ for any $t\in\mathbb{R}$, hence $F(t)\geq 0$
for all $t\in\mathbb{R}$;

  2. (2)

$c_{1}\geq 0$;

  3. (3)

$c_{2}L\leq 2mc_{1}L$;

  4. (4)

$c_{3}L^{2}\leq 4m^{2}c_{1}L^{2}$.

Using these properties, we have

$\displaystyle F(L)$ $\displaystyle=$
$\displaystyle\frac{e^{-2mL}}{2m}(c_{1}+c_{2}L+c_{3}L^{2})\leq
C(m)e^{-2mL}L^{2}c_{1}\leq C(m)L^{2}e^{-2L}F(0)$ $\displaystyle F(3L)$
$\displaystyle=$
$\displaystyle\frac{e^{-6mL}}{2m}(c_{1}+3c_{2}L+9c_{3}L^{2})\leq
C(m)e^{-2mL}L^{2}c_{1}\leq C(m)L^{2}e^{-2L}F(0).$

Hence for a fixed $q<2$, we may choose $L_{0}$ such that for any $L>L_{0}$, we
have

$(1+e^{-qL})F(L)\leq\frac{e^{-qL}}{2}F(0),\,\,\,\,e^{-qL}F(3L)\leq\frac{e^{-qL}}{2}F(0).$

Taking the sum of the above inequalities and noticing that $F(2L)\geq 0$, we
get

$(1+e^{-qL})F(L)+e^{-qL}F(3L)\leq
e^{-qL}F(0)\leq(1+e^{-qL})F(2L)+e^{-qL}F(0),$

which implies that

$F(L)-F(2L)\leq e^{-qL}(F(0)-F(L)+F(2L)-F(3L)).$

The proof for (3.9) is done.

Finally, by the same argument as in the Step 1, we also have

$B_{2}^{\prime}\leq C(m)e^{-2L}(B_{1}^{\prime}+B_{3}^{\prime}).$

Together with (3.8), we can choose $L_{0}$ such that if $L>L_{0}$, we have

$B_{2}+B_{2}^{\prime}\leq e^{-qL}(B_{1}+B_{1}^{\prime}+B_{3}+B_{3}^{\prime}).$

and the inequality can be made strict as long as the right hand side is not
zero. For fixed $q\in(0,2)$, we use Step 2 for $q^{\prime}$ (satisfying
$q<q^{\prime}<2$) to see

$A_{2}\leq e^{-qL}(A_{1}+A_{3})$

and again the inequality can be made strict as long as $A_{1}+A_{3}$ is not
zero. Since we have assumed that $u$ is not identically zero, at least one of
the above inequalities are strict. The proof of (3.6) is done by adding them
up. $\hfill\Box$

###  3.2. Proof of Theorem 3.1

###### Proof.

Given $m$ and $q$, let $L_{0}$ be determined by Lemma 3.2. Argue by
contradiction. Assume that there exist $L>L_{0}$ and a sequence of $f_{k}$
with $g_{k}=f_{k}^{*}(g_{\mathbb{R}^{n}})=e^{-2mt+2v_{k}}(dt^{2}+d\theta^{2})$
satisfying

(3.10) $\int_{Q}\left|A_{k}\right|^{2}dV_{g_{k}}+\left\|\nabla
v_{k}\right\|_{C^{0}(Q)}\to 0$

but

$\int_{Q_{2}}\left|H_{k}\right|^{2}dV_{g_{k}}>e^{-qL}\left(\int_{Q_{1}}\left|H_{k}\right|^{2}dV_{g_{k}}+\int_{Q_{3}}\left|H_{k}\right|^{2}dV_{g_{k}}\right),$

that is,

(3.11)
$\int_{Q_{2}}|H_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta>e^{-qL}\left(\int_{Q_{1}}|H_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta+\int_{Q_{3}}|H_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta\right)$

for $k$ sufficiently large.

After a scaling for each $f_{k}$, by (3.10), we assume that $v_{k}$ converges
to zero uniformly on $Q$. We still denote the scaled maps by $f_{k}$. This
will not cause any problem because every assumption and the conclusion of the
theorem is invariant after the scaling.

The boundedness of $v_{k}$ on $Q$ allows us to use the $\epsilon$-regularity
estimate ([3, Theorem I.1]) to control any derivative of $f_{k}$ on any
compact set of $Q$. Hence, up to translation and taking subsequence, we may
assume that $f_{k}$ converges locally smoothly to a conformal Willmore
immersion $f_{\infty}$. By (3.10), the second fundamental form of $f_{\infty}$
vanishes and the induced metric $g_{\infty}=e^{-2mt}(dt^{2}+d\theta^{2})$. The
vanishing of the second fundamental form implies that the image of
$f_{\infty}$ lies in a plane and after a rotation of $\mathbb{R}^{n}$ (which
may flip the orientation if necessary), we may assume that
$f_{\infty}(t,\theta)=(u(t,\theta),v(t,\theta),0,...,0)$. Since $f_{\infty}$
is conformal, the explict formula of $g_{\infty}$ implies that

$\left|\partial_{t}u\right|^{2}+\left|\partial_{t}v\right|^{2}=\left|\partial_{\theta}u\right|^{2}+\left|\partial_{\theta}v\right|^{2}=e^{-2mt}\quad\text{and}\quad\partial_{t}u\cdot\partial_{\theta}u+\partial_{t}v\cdot\partial_{\theta}v=0.$

It follows from elementary complex analysis that after a rotation of the
domain (i.e. $\theta\mapsto\theta+\theta_{0}$) if necessary and a translation
in $\mathbb{R}^{n}$, $f_{\infty}$ is uniquely determined

$f_{\infty}(t,\theta)=\frac{e^{-mt}}{m}(\cos m\theta,\sin
m\theta,0,\cdots,0).$

We observe that all assumptions and the conclusion of Theorem 3.1 is invariant
under scaling, translation and rotation. Hence, we may assume that $f_{k}$
converges to $f_{\infty}$ locally smoothly in $Q$.

######  Remark 3.3.

It is important to note that $\tau_{1}(f,c)=0$ (for all $c\in\mathbb{R}^{n}$)
is invariant under scaling, translation and rotation, while $\tau_{2}(f,S)=0$
(for all $S\in\mathfrak{so}(\mathbb{R}^{n})$) is invariant under scaling and
rotation, but not translation. However, if $\tau_{1}(f,c)=0$ is assumed, then
$\tau_{2}(f,S)=0$ becomes invariant under translation. These facts can be
verified directly using the definitions. It will be clear that for the rest of
the proof only $\tau_{2}(f,S)=0$ is used. However, we still need to assume
$\tau_{1}(f,c)=0$ in the theorem to ensure that $\tau_{2}(f,S)=0$ is invariant
under translation.

Set

$\phi_{k}=H_{k}/\|H_{k}e^{-mt}\|_{L^{2}(Q_{2})}.$

Here the $L^{2}$ norm is taken with respect to the flat metric
$dt^{2}+d\theta^{2}$. By (3.11), we have

(3.12)
$\int_{Q_{2}}|\phi_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta>e^{-qL}\left(\int_{Q_{1}}|\phi_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta+\int_{Q_{3}}|\phi_{k}|^{2}e^{-2mt+2v_{k}}dtd\theta\right)$

which implies

(3.13) $\|\phi_{k}e^{-mt}\|_{L^{2}(Q)}<C.$

If we rewrite the equation of $H_{k}$ (see (2.2)) in the form of a linear
equation (whose coefficients are allowed to depend on $H_{k}$, $A_{k}$ and
their derivatives), we obtain the following equation for $\phi_{k}$

(3.14) $\Delta\phi_{k}+\alpha_{k}\cdot\nabla\phi_{k}+\beta_{k}\phi_{k}=0.$

Thanks to (3.10), we have

$\|\alpha_{k}\|_{C^{l}(K)}+\|\beta_{k}\|_{C^{l}(K)}\rightarrow 0$

for any $l\in\mathbb{N}$ and any compact subset $K\subset Q$. The equation
(3.14) and the $L^{2}$ bound (3.13) imply any $C^{l}$ estimates of $\phi_{k}$
on $K$. By taking subsequence, we assume $\phi_{k}$ converges in
$C^{\infty}_{loc}(Q)$ to some harmonic function denoted by $\phi$. By our
definition of $\phi_{k}$,

$\|\phi e^{-mt}\|_{L^{2}(Q_{2})}=1.$

By the Fatou’s lemma and (3.10), (3.12), we have

(3.15) $\int_{Q_{2}}|\phi|^{2}e^{-2mt}dtd\theta\geq
e^{-qL}\left(\int_{Q_{1}}|\phi|^{2}e^{-2mt}dtd\theta+\int_{Q_{3}}|\phi|^{2}e^{-2mt}dtd\theta\right).$

In order to get a contradiction to the three circle property given by Lemma
3.2, it suffices to show that $b_{m}=b^{\prime}_{m}=0$ in the following
expansion of harmonic function $\phi$

$\phi=a+bt+\sum_{k=1}^{\infty}\left((a_{k}e^{-kt}+b_{k}e^{kt})\cos
k\theta+(a_{k}^{\prime}e^{-kt}+b_{k}^{\prime}e^{kt})\sin k\theta\right).$

Since $H_{k}$ is a normal vector,

$\phi\cdot\partial_{t}(f_{\infty})=\lim_{k\rightarrow+\infty}\frac{{H}_{k}\cdot(\partial_{t}f_{k})}{\|H_{k}e^{-mt}\|_{L^{2}(Q_{2})}}=0.$

Similarly, we have $\phi\cdot\partial_{\theta}(f_{\infty})=0$. Hence, we may
assume

$\phi=(0,0,\phi_{3},\cdots,\phi_{n}).$

Taking the limit $k\to\infty$, we get from $\tau_{2}(f_{k},S)=0$ that

(3.16) $0=\int_{\\{t\\}\times
S^{1}}\left(\phi\cdot\partial_{t}(Sf_{\infty})-\partial_{t}\phi\cdot(Sf_{\infty})\right)d\theta.$

Letting $S=(s_{ij})$ with

$s_{ij}=\left\\{\begin{array}[]{ll}1&i=1,j=n\\\ -1&i=n,j=1\\\
0&others,\end{array}\right.$

we obtain from (3.16) that

$0=\int_{\left\\{t\right\\}\times S^{1}}\phi_{n}(me^{-mt}\cos
m\theta)-\partial_{t}\phi_{n}(-e^{-mt}\cos m\theta)d\theta.$

This implies that $b_{m}=0$ for the expansion of $\phi_{n}$. By choosing
different $S$, the same argument shows that $b_{m}=b^{\prime}_{m}=0$ for the
expansion of $\phi_{3},\cdots,\phi_{n}$. This concludes the proof of Theorem
3.1. $\hfill\Box$

Consequently, following the discussion at the beginning of Section 3, we have
the following

######  Corollary 3.4.

For any $q\in(0,2)$ and $m\in\mathbb{Z}_{+}$, there exist
$L_{0}(m,q),\delta_{0}(m,q)>0$, such that if $L>L_{0}$ then the following
holds. Let $f:[0,kL]\times S^{1}\rightarrow\mathbb{R}^{n}$ be a conformal and
Willmore immersion with
$f^{*}(g_{\mathbb{R}^{n}})=e^{-2mt+2v}(dt^{2}+d\theta^{2})$. Assume
$\tau_{1}(f,S)=0$ and $\tau_{2}(f,c)=0$ for any
$S\in\mathfrak{so}(\mathbb{R}^{n})$ and $c\in\mathbb{R}$. If

$\|\nabla
v\|_{L^{\infty}(Q_{i})}+\int_{Q_{i}}|A|^{2}dV_{g}<\delta_{0},\qquad\forall\,\,\,\,1\leq
i\leq k,$

then we have

(3.17)
$\int_{Q_{i}}|H|^{2}dV_{g}<C(e^{-iqL}\int_{Q_{1}}|H|^{2}dV_{g}+e^{-(k-i)qL}\int_{Q_{k}}|H|^{2}dV_{g}).$

##  4\. 3-circle lemma for the second fundamental form

In this section, we prove Theorem 1.3. It follows directly from the following
theorem.

######  Theorem 4.1.

For any $q\in(0,2)$ and $m\in\mathbb{Z}_{+}$, there exist
$L_{2}(m,q),\delta_{2}(m,q)>0$, such that if $L>L_{2}$ then the following
holds. Let $f:[0,kL]\times S^{1}\rightarrow\mathbb{R}^{n}$ be a conformal and
Willmore immersion with
$f^{*}(g_{\mathbb{R}^{n}})=e^{-2mt+2v}(dt^{2}+d\theta^{2})$. Assume
$\tau_{1}(f,S)=0$ and $\tau_{2}(f,c)=0$ for any
$S\in\mathfrak{so}(\mathbb{R}^{n})$ and $c\in\mathbb{R}$. If

$\|\nabla
v\|_{L^{\infty}(Q_{i})}+\int_{Q_{i}}|A|^{2}dV_{g}<\delta_{2},\qquad\forall\,\,\,\,1\leq
i\leq k,$

then we have the following decay estimate for the second fundamental form $A$

(4.1) $\int_{Q_{i}}|A|^{2}dV_{g}<C\left(e^{-iqL}\int_{Q_{1}\cup Q_{2}\cup
Q_{3}}|A|^{2}dV_{g}+e^{-(k-i)qL}\int_{Q_{k-2}\cup Q_{k-1}\cup
Q_{k}}|A|^{2}dV_{g}\right),$

where $C$ depends on $m$, $q$, $L$.

Assuming Theorem 4.1, for $q,m$ as above, take $\delta_{2}$ and $L_{2}$ as
given in Theorem 4.1 and fix some $L>L_{2}\geq 1$ such that $T=kL$ for some
$k\in\mathbb{Z}_{+}$. Then (a)-(c) in Theorem 1.3 implies that all assumptions
of Theorem 4.1 hold with $\delta_{1}=\delta_{2}/(2L)$. Since

$\int_{Q_{1}\cup Q_{2}\cup Q_{3}}\left|A\right|^{2}dV_{g},\int_{Q_{k-2}\cup
Q_{k-1}\cup Q_{k}}\left|A\right|^{2}dV_{g}\leq C(L)\tilde{\delta},$

(1.3) is a consequence of (4.1). This proves Theorem 1.3.

The proof of Theorem 4.1 consists of two parts. The first part shows that when
$H$ is much smaller than $A$, the 3-circle holds for the $L^{2}$ norm of $A$
(Theorem 4.3). The second part combines this with Corollary 3.4 to prove
Theorem 4.1.

###  4.1. When $H$ is small.

The Gauss map ${\bf n}$ takes value in the Grassmannian $G(2,n)$. There is a
natural embedding of $G(2,n)$ into $\Lambda^{2}(\mathbb{R}^{n})$, the $2$-nd
exterior power of $\mathbb{R}^{n}$. The induced metric from this embedding
agrees with the metric as the symmetric space $O(n)/O(2)\times O(n-2)$. We
denote the second fundamental form of $G(2,n)$ by $A_{G(2,n)}$. The main
result in [28] shows

(4.2) $\tau({\bf n}):=\Delta{\bf n}+A_{G(2,n)}(d{\bf n},d{\bf
n})=\nabla^{\perp}H,$

where $H$ is the mean curvature and $\nabla^{\perp}H$ is the covariant
derivative of the normal bundle. The equation (4.2) is to be understood with
an isomorphism (see Section 2 of [28]) that identifies $\nabla^{\perp}H$ with
a section of ${\mathbf{n}}^{-1}(TG(2,n))$.

We regard (4.2) as a perturbed harmonic map equation. In the neck analysis of
harmonic maps, it is important to have some Pohozaev identity that relates the
tangential energy of $\mathbf{n}$ with the radial energy. In our setting, this
is achieved by the following lemma.

######  Lemma 4.2.

Let $f:(t_{1},t_{2})\times S^{1}\to\mathbb{R}^{n}$ be any conformal immersion
with $f^{*}(g_{\mathbb{R}^{n}})=e^{2u}(dt^{2}+d\theta^{2})$. Let
${\mathbf{n}},H,A$ be the Gauss map, mean curvature and second fundamental
form of $f$ respectively. Then for some universal constant $C$,

(4.3) $\left|\left|\partial_{t}{\bf n}\right|^{2}-\left|\partial_{\theta}{\bf
n}\right|^{2}\right|\leq
Ce^{2u}|A||H|,\qquad\forall(t,\theta)\in(t_{1},t_{2})\times S^{1}.$

###### Proof.

The proof is by direct computation.

We set $e_{1}=e_{t}=e^{-u}\partial_{t}f$,
$e_{2}=e_{\theta}=e^{-u}\partial_{\theta}f$, and select $e_{3}$, $\cdots$,
$e_{n}$ (depending smoothly on $(t,\theta)$) such that $\\{e_{i}\\}$ is an
orthonormal basis of $\mathbb{R}^{n}$. The Gauss map is given by

${\bf n}(t,\theta)=e_{t}\wedge e_{\theta}.$

By the definition of the second fundamental form $A$ and the Christoffel
symbol $\Gamma$ of the induced connection, we have

$\displaystyle e^{-u}\frac{\partial e_{t}}{\partial t}$ $\displaystyle=$
$\displaystyle\Gamma_{tt}^{\theta}e_{\theta}+e^{-2u}A_{tt}^{\alpha}e_{\alpha}$
$\displaystyle e^{-u}\frac{\partial e_{\theta}}{\partial t}$ $\displaystyle=$
$\displaystyle\Gamma_{\theta t}^{t}e_{t}+e^{-2u}A_{\theta
t}^{\alpha}e_{\alpha}$ $\displaystyle e^{-u}\frac{\partial
e_{t}}{\partial\theta}$ $\displaystyle=$
$\displaystyle\Gamma_{t\theta}^{\theta}e_{\theta}+e^{-2u}A_{t\theta}^{\alpha}e_{\alpha}$
$\displaystyle e^{-u}\frac{\partial e_{\theta}}{\partial\theta}$
$\displaystyle=$
$\displaystyle\Gamma_{\theta\theta}^{t}e_{t}+e^{-2u}A_{\theta\theta}^{\alpha}e_{\alpha},$

where $\alpha=3,\cdots,n$ and $A_{tt}$ is short for $A(f_{t},f_{t})$, etc. By
definition, we have

$H=e^{-2u}(A_{tt}^{\alpha}+A_{\theta\theta}^{\alpha})e_{\alpha},\,\,\,\,|A|^{2}=e^{-4u}\sum_{\alpha}(|A_{tt}^{\alpha}|^{2}+2|A_{t\theta}^{\alpha}|^{2}+|A_{\theta\theta}^{\alpha}|^{2}).$

By a direct calculation, we have

$\frac{\partial(e_{t}\wedge e_{\theta})}{\partial
t}=e^{-u}\left(A_{tt}^{\alpha}e_{\alpha}\wedge e_{\theta}+A_{\theta
t}^{\alpha}e_{t}\wedge e_{\alpha}\right)$

and

$\frac{\partial(e_{t}\wedge
e_{\theta})}{\partial\theta}=e^{-u}\left(A_{t\theta}^{\alpha}e_{\alpha}\wedge
e_{\theta}+A_{\theta\theta}^{\alpha}e_{t}\wedge e_{\alpha}\right).$

Hence,

$\left|\partial_{t}{\bf n}\right|^{2}-\left|\partial_{\theta}{\bf
n}\right|^{2}=e^{-2u}\sum_{\alpha}|A_{tt}^{\alpha}|^{2}-|A_{\theta\theta}^{\alpha}|^{2}=e^{2u}\sum_{\alpha}e^{-2u}(A_{tt}^{\alpha}+A_{\theta\theta}^{\alpha})e^{-2u}(A_{tt}^{\alpha}-A_{\theta\theta}^{\alpha}),$

from which (4.3) follows. $\hfill\Box$

Next, we use the 3-circle argument to prove

######  Theorem 4.3.

For any $q\in(0,2)$ and $\beta>0$, there exists $L_{1}(q,\beta)$ such that if
$L>L_{1}$, the following holds. There exists $\epsilon(q,\beta)>0$ and
$\delta(q,\beta)>0$ such that if $f$ is a conformal and Willmore immersion
defined on $Q=Q_{1}\cup Q_{2}\cup Q_{3}$ satisfying

  1. (1)

$\left|\nabla u\right|\leq\beta$ where
$f^{*}(g_{\mathbb{R}^{n}})=e^{2u}(dt^{2}+d\theta^{2})$;

  2. (2)

$\int_{Q}\left|H\right|^{2}dV_{g}\leq\delta\int_{Q}\left|A\right|^{2}dV_{g}$;

  3. (3)

$\left\|A\right\|_{L^{2}(Q)}<\epsilon$;

then

$\int_{Q_{2}}\left|A\right|^{2}dV_{g}\leq
e^{-qL}\left(\int_{Q_{1}}\left|A\right|^{2}dV_{g}+\int_{Q_{3}}\left|A\right|^{2}dV_{g}\right).$

###### Proof.

If otherwise, then we can find $f_{k}$ satisfying (1), such that

(4.4)
$\lim_{k\to\infty}\frac{1}{\|A_{k}\|^{2}_{L^{2}(Q)}}\int_{Q}\left|H_{k}\right|^{2}dV_{g_{k}}=\lim_{k\to\infty}\left\|A_{k}\right\|_{L^{2}(Q)}=0$

and

(4.5) $\int_{Q_{2}}|A_{k}|^{2}dV_{g_{k}}\geq
e^{-qL}\left(\int_{Q_{1}}|A_{k}|^{2}dV_{g_{k}}+\int_{Q_{3}}|A_{k}|^{2}dV_{g_{k}}\right).$

Similar to the proof of Theorem 3.1, by a scaling of $f_{k}$ and (1), we first
assume that $u_{k}$’s are bounded. Then by (4.4), we use the
$\epsilon$-regularity of Willmore surface to show that $f_{k}$ (after
translation) converges smoothly locally in $Q$. Moreover, we know $A_{k}$
converges to $0$ and $u_{k}$ converges to some smooth function $u$.

We multiply both sides of (4.2) by $\frac{1}{\left\|\nabla{\bf
n}_{k}\right\|_{L^{2}(Q_{2})}}$ and set $\phi_{k}=\frac{{\bf
n}_{k}-c_{k}}{\left\|\nabla{\bf n}_{k}\right\|_{L^{2}(Q_{2})}}$ where $c_{k}$
is the average of ${\bf n}_{k}$ over $Q_{2}$, to get

(4.6) $\Delta\phi_{k}+A_{G(2,n)}(d{\bf
n}_{k},d\phi_{k})=\frac{\nabla^{\perp}H_{k}}{\left\|\nabla{\bf
n}_{k}\right\|_{L^{2}(Q_{2})}}.$

Observe that $H_{k}$ satisfies (2.2) and we have control over any $C^{l}$ norm
of $H_{k}$, $f_{k}$ and $A_{k}$ so that we can rewrite (2.2) as a linear
equation of $\phi_{k}$ with bounded coefficients. The elliptic interior
estimate then implies that

$\sup_{[\eta,3L-\eta]\times S^{1}}\left|\nabla^{\perp}H_{k}\right|\leq
C(\eta)\left\|H_{k}\right\|_{L^{2}(Q)}.$

Thanks to (4.4) and (4.5), by taking $k\to\infty$, we obtain a (vector-valued)
harmonic function $\phi$ as the limit of $\phi_{k}$ defined on $Q$. By (4.5)
and the choice of scaling constant, we have

$\int_{Q_{2}}\left|\nabla\phi\right|^{2}dtd\theta=1,\quad\int_{Q}\left|\nabla\phi\right|^{2}dtd\theta<C.$

It follows from (4.3) and (4.4) that

$\int_{Q_{2}}\left|\partial_{t}\phi\right|^{2}-\left|\partial_{\theta}\phi\right|^{2}dtd\theta=0.$

For such harmonic functions, there is a version of three circle lemma which
implies the existence of $L_{1}$ such that if $L>L_{1}$

(4.7)
$\int_{Q_{2}}\left|\nabla\phi\right|^{2}dt\theta<e^{-qL}\left(\int_{Q_{1}}\left|\nabla\phi\right|^{2}dt\theta+\int_{Q_{3}}\left|\nabla\phi\right|^{2}dtd\theta\right).$

We postpone the proof of this claim to the appendix (see Lemma A.1). The
inequality (4.7) contradicts (4.5) (by the smooth convergence on $Q_{2}$ and
the Fatou’s lemma) and finishes the proof. $\hfill\Box$

###  4.2. Proof of Theorem 4.1.

Now, we combine the decay of the mean curvature (Corollary 3.4) with Theorem
4.3 to show the decay of the second fundamental form.

###### Proof.

Given $m$ and $q$, fix some $q^{\prime}\in(q,2)$ and let $\beta=m+1$. Thereom
4.3 determines $L_{1}$, $\epsilon$ and $\delta$ depending on $q^{\prime}$ and
$\beta$. Corollary 3.4 determines $L_{0}$ and $\delta_{0}$ depending on
$q^{\prime}$ and $m$. Fix $\delta_{2}<\min(\delta_{0},\epsilon/3)$ and
$L_{2}>\max(L_{1},L_{0})$ satisfying

(4.8) $e^{-(q^{\prime}-q)L}<1/2\qquad\text{for}\quad L>L_{2}.$

For $1<i<k$, if

(4.9) $\int_{Q_{i-1}\cup Q_{i}\cup
Q_{i+1}}|A|^{2}dV_{g}\leq\frac{1}{\delta}\int_{Q_{i-1}\cup Q_{i}\cup
Q_{i+1}}\left|H\right|^{2}dV_{g}.$

Then (4.1) for $i$ holds due to Corollary 3.4.

In what follows, we assume (4.9) is not true for $i$, hence it verifies the
assumption (2) in Theorem 4.3. The assumption (1) holds for $\beta=m+1$ and
$\delta_{2}<1$, and the assumption (3) is obvious because
$\delta_{2}<\epsilon/3$. Hence Theorem 4.3 (for $q^{\prime}$ and
$Q=Q_{i-1}\cup Q_{i}\cup Q_{i+1}$) then implies that

$\int_{Q_{i}}\left|A\right|^{2}dV_{g}\leq
e^{-q^{\prime}L}\left(\int_{Q_{i-1}}\left|A\right|^{2}dV_{g}+\int_{Q_{i+1}}\left|A\right|^{2}dV_{g}\right).$

By (4.8), we have either
$\int_{Q_{i}}|A|^{2}dV_{g}<e^{-qL}\int_{Q_{i-1}}|A|^{2}dV_{g}$ or
$\int_{Q_{i}}|A|^{2}dV_{g}<e^{-qL}\int_{Q_{i+1}}|A|^{2}dV_{g}$. Without loss
of generality, we assume the later one is true. Hence, the proof of (4.1) for
$i$ is reduced to the proof of the same inequality with $i$ replaced by $i+1$.

Now, we may repeat the above argument to consider $i+2$, $i+3$,$\cdots$, until
either (4.9) holds for $i+l$, or $i+l=k$. In both cases, we have (4.1) true
for $i+l$ and hence (4.1) holds for $i$. $\hfill\Box$

##  5\. Positive answers to Q1 and Q2

In this section, we assume $f_{k}$ is a sequence of conformal and Willmore
immersions satisfying A.1)-A.4). Our goal is to give positive answers to Q1
and Q2.

Before that, we prove a claim promised in Remark 1.2.

######  Lemma 5.1.

If $f_{k}:[0,T_{k}]\times S^{1}\to\mathbb{R}^{n}$ can be extended to conformal
and Willmore immersions of closed surfaces $\Sigma^{\prime}_{k}$ with bounded
Willmore energy and genus, then there exists some $\beta>0$ and $T>0$
depending on the sequence such that

(5.1) $\sup_{k}\|\nabla u_{k}\|_{L^{\infty}([T,T_{k}-T])}<\beta.$

###### Proof.

The Gauss-Bonnet theorem implies that
$\left\|A_{\Sigma^{\prime}_{k}}\right\|_{L^{2}}$ is uniformly bounded. By
Theorem 7.1 in [11] (or Theorem 6.4 in [17]), for any sequence $t_{k}$
satisfying $t_{k}\to+\infty$ and $T_{k}-t_{k}\rightarrow+\infty$, we can find
$\lambda_{k}$ and $c_{k}$, such that
$\hat{f}_{k}=\lambda_{k}(f_{k}(t_{k}+t,\theta)-c_{k})$ converges in $W^{2,2}$
to a conformal map. According to Helein’s convergence theorem, at this point,
we have $|u_{k}|$ is bounded on any $(-T,T)\times S^{1}$, and further, due to
the $\epsilon$-regularity (cf. [12, Theorem 2.10],[27, Theorem I.5]),
$\hat{f}_{k}$ converges smoothly. Hence, if $\hat{u}_{k}$ is defined by
$e^{2\hat{u}_{k}}(dt^{2}+d\theta^{2})=\hat{f}_{k}^{*}(g_{\mathbb{R}^{n}})$, we
have

$\sup_{k}\sup_{\left\\{0\right\\}\times
S^{1}}\left|\nabla\hat{u}_{k}\right|<\infty.$

If (5.1) were not true, then after taking a subsequence, we can find $t_{k}$,
such that $t_{k}\rightarrow+\infty$, $T_{k}-t_{k}\rightarrow+\infty$, and
$\sup_{\\{t_{k}\\}\times S^{1}}|\nabla u_{k}|\rightarrow+\infty$. This is a
contradiction because

$\left|\nabla\hat{u}_{k}\right|(t,\theta)=\left|\nabla
u_{k}\right|(t+t_{k},\theta).$

$\hfill\Box$

Now, let’s discuss Q1 and Q2. We observe that the positive answers to Q1 an Q2
follow from the following claim:

Claim. For any $\varepsilon>0$, there exist $T>0$ and $M>0$ such that for any
$k>M$, we have

$\int_{[t,t+1]\times S^{1}}\left|A_{k}\right|^{2}dV_{k}\leq
C\varepsilon\left(e^{-q(t-T)}+e^{-q(T_{k}-T-t)}\right)\qquad\text{for}\quad
t\in[T,T_{k}-T].$

The proof of the claim uses Theorem 1.3. The assumption (a) follows from A.4)
and Theorem 2.1. By A.2), we have $T^{\prime}>0$ and $M^{\prime}>0$ such that
for $k>M^{\prime}$,

$\sup_{t\in[T^{\prime},T_{k}-T^{\prime}]}\int_{[t,t+1]\times
S^{1}}\left|A_{k}\right|^{2}dV_{k}<\min(\delta_{1},\varepsilon),$

which is the assumption (c). The proof of the claim is then reduced to the
verification of (b), which is the next lemma. Assuming the lemma, we can take
$T=\max(T^{\prime},T^{\prime\prime})$ and
$M=\max(M^{\prime},M^{\prime\prime})$ and apply Theorem 1.3 to the restriction
of $f_{k}$ on $[T,T_{k}-T]\times S^{1}$ to see that the claim holds.

######  Lemma 5.2.

Assume that $f_{k}$ satisfies A.1)-A.3). For any $\varepsilon>0$, there exist
$T^{\prime\prime}>0$ and $M^{\prime\prime}>0$ such that for any
$k>M^{\prime\prime}$, there is $m_{k}\in\mathbb{Z}\setminus\left\\{0\right\\}$
with $\left|m_{k}\right|\leq\beta$ such that if
$f_{k}^{*}(g_{\mathbb{R}^{n}})=e^{-2m_{k}t+2v_{k}}(dt^{2}+d\theta^{2})$, then
we have

(5.2) $\sup_{[T^{\prime\prime},T_{k}-T^{\prime\prime}]}\left|\nabla
v_{k}\right|<\varepsilon.$

###### Proof.

We consider first an apparently weaker statement: for any $\varepsilon>0$,
there exist $T^{\prime\prime}>0$ and $M^{\prime\prime}>0$ such that for any
$k>M^{\prime\prime}$, for any
$t_{0}\in[T^{\prime\prime},T_{k}-T^{\prime\prime}]$, we have
$m=m(k,t_{0})\in\mathbb{Z}\setminus\left\\{0\right\\}$ and
$f_{k}^{*}(g_{\mathbb{R}^{n}})=e^{2u_{k}}(dt^{2}+d\theta^{2})=e^{-2mt+2v_{k}}(dt^{2}+d\theta^{2})$
satisfying $\left|m\right|\leq\beta$ and

$\sup_{\left\\{t_{0}\right\\}\times S^{1}}\left|\nabla
v_{k}\right|<\varepsilon.$

Assuming this “weaker” statement, if we fix $k>M^{\prime\prime}$, by the
continuity of $\nabla u_{k}$ and

$\sup_{\left\\{t_{0}\right\\}\times
S^{1}}\left|\partial_{t}(u_{k})+m(k,t_{0}))\right|<\varepsilon,\qquad\text{for
all}\quad t_{0}\in[T^{\prime\prime},T_{k}-T^{\prime\prime}],$

we know $m(k,t_{0})$ is independent of $t_{0}$. Hence, the lemma follows from
this “weaker” statement, which we prove below by contradiction.

Assume that there is $\varepsilon_{0}>0$, a subsequence $k_{l}$, such that if
$u_{l}$ is defined by

$f^{*}_{k_{l}}(g_{\mathbb{R}^{n}})=e^{2u_{l}}(dt^{2}+d\theta^{2}),$

we have $t_{l}\in[l,T_{k}-l]$ satisfying

(5.3) $\sup_{\left\\{t_{l}\right\\}\times
S^{1}}\left|\partial_{t}(u_{l}+mt)\right|\geq\varepsilon_{0}\qquad\text{for
any}\quad
m\in\mathbb{Z}\setminus\left\\{0\right\\},\,\,\,\,\left|m\right|\leq\beta.$

For $\lambda_{l}=e^{-u_{l}(t_{l},0)}$, we set
$\hat{f}_{l}(t,\theta)=\lambda_{l}(f_{k_{l}}(t+t_{l},\theta)-(2\pi)^{-1}\int_{0}^{2\pi}f_{k_{l}}(t_{l},\theta)d\theta)$.
Let $e^{2\hat{u}_{l}}$ be the conformal factor of
$\hat{f}^{*}_{l}(g_{\mathbb{R}^{n}})$. By our choice of $\lambda_{l}$ and
A.3), $\hat{u}_{l}$ is uniformly bounded on $[-l_{0},l_{0}]\times S^{1}$ when
$l\geq l_{0}$. By A.2), if we take $l_{0}$ sufficiently large, then for
$l>l_{0}$, we have

$\sup_{t\in[-l_{0}/2,l_{0}/2]}\int_{[t,t+1]\times
S^{1}}\left|A_{l}\right|^{2}dtd\theta<\epsilon$

for any $\epsilon$. Therefore, the $\epsilon$-regularity provides any $C^{k}$
bound for $\hat{f}_{l}$ on $[-l_{0}/3,l_{0}/3]\times S^{1}$. Hence, by taking
subsequence (still denoted by $\hat{f}_{l}$), we may assume that $\hat{f}_{l}$
converges locally smoothly to a limit $f_{\infty}$ defined on
$\mathbb{R}\times S^{1}$.

$f_{\infty}$ is a conformal Willmore immersion from $\mathbb{R}\times S^{1}$
to $\mathbb{R}^{n}$ with vanishing second fundamental form. If
$(f_{\infty})^{*}(g_{\mathbb{R}^{n}})=e^{2u}(dx^{2}+dy^{2})$, then $u$ is a
harmonic function defined on $\mathbb{R}\times S^{1}$ satisfying $\left|\nabla
u\right|\leq\beta$. By the well known expansion for harmonic function

(5.4)
$u(t,\theta)=-mt+b+\sum_{k=1}^{\infty}\left((a_{k}e^{-kt}+b_{k}e^{kt})\cos
k\theta+(a^{\prime}_{k}e^{-kt}+b^{\prime}_{k}e^{kt})\sin k\theta\right),$

we must have $u(t,\theta)=-mt+b$ for $m\in\mathbb{R}$ with
$\left|m\right|\leq\beta$. By modifying $\lambda_{l}$, we may assume that
$b=0$ and hence $u(t,\theta)\equiv-mt$.

Since $A_{f_{\infty}}\equiv 0$, the image of $f_{\infty}$ lies in a plane. By
a rotation of $\mathbb{R}^{n}$, we may assume that
$f_{\infty}(t,\theta)=(w(t,\theta),v(t,\theta),0,\cdots,0)$ and $w+iv$ is a
holomorphic function. By $u(t,\theta)\equiv-mt$, we obtain

$\left|\partial_{t}w\right|^{2}+\left|\partial_{\theta}w\right|^{2}=\left|\partial_{t}v\right|^{2}+\left|\partial_{\theta}v\right|^{2}=e^{-2mt},\qquad\text{for
all}\quad(t,\theta)\in\mathbb{R}\times S^{1}.$

This equation, together with a similar expansion of $w$ as (5.4), we conclude
that $m\in\mathbb{Z}\setminus\left\\{0\right\\}$. This is a contradiction to
(5.3) because $\nabla u_{l}(t+t_{l},\theta)=\nabla\hat{u}_{l}(t,\theta)$
converges to $\nabla u(t,\theta)$ on near $\left\\{0\right\\}\times S^{1}$
smoothly but

$\sup_{\left\\{0\right\\}\times S^{1}}\left|\partial_{t}u+m\right|=0.$

$\hfill\Box$

##  6\. Applications

In this section, we derive two applications of the decay estimates that were
proved in the previous sections.

###  6.1. Removability of singularities

The problem of removability of isolated singularities for Willmore surfaces
was first studied by Kuwert and Schätzle for the codimension one case. In
their works [13, 14], they demonstrated that under the assumption of finite
$L^{2}$ integral of the second fundamental form and finite area, such surfaces
are $C^{1,\alpha}$ smooth. Moreover, the surface can be extended smoothly to
be an embedded surface when the singularity residue is zero and the area
density is less than 2. Later, Rivière proved that such results are valid for
higher codimensions [27]. Unlike harmonic mappings, $C^{1,\alpha}$ (for any
$\alpha\in(0,1)$) is the best we can hope for isolated singularities of
Willmore surfaces without additional conditions (e.g., the inversion of a
catenoid). In [2], Bernard and Rivière explored whether Willmore surfaces with
isolated singularities could be represented as the image sets of smooth
mappings.

Based on the discussions in [13, 14], to prove the removability of
singularities, we only need to demonstrate the decay of the second fundamental
form.

######  Corollary 6.1.

Let $f:D\setminus\left\\{0\right\\}\rightarrow\mathbb{R}^{n}$ be a conformal
and Willmore immersion. If

$\left\|A_{f}\right\|_{L^{2}(D\setminus\left\\{0\right\\})}<\infty\qquad\text{and}\qquad\mbox{Area}(f)<\infty,$

then for any $q\in(0,2)$, we have

(6.1) $\int_{(t,t+1)\times S^{1}}|A_{f}|^{2}dV_{g}\leq
C(q,f)e^{-qt}\qquad\text{for}\quad t>0.$

This corollary will imply that $f$ can be extended to a $C^{1,\alpha}$ map
from $D$ into $\mathbb{R}^{n}$. Hence, we have reproved a result due to Kuwert
and Schätzle for the case of $n=3$ [13, 14] and Rivière for the general case
[27].

Ideally, a possible approach of this corollary is to apply Theorem 1.3 to the
restriction of $f$ on $[0,k]\times S^{1}$ and take the limit $k\to\infty$.
Unfortunately, it is not possible to verify the assumption $\tau_{1}(f,c)=0$
for all $c\in\mathbb{R}^{n}$. Indeed, for the inverted catenoid, this is not
true, but it satisfies the assumptions of Corollary 6.1. The only place that
we need $\tau_{1}(f,c)=0$ is in the proof of Theorem 3.1 (and hence Corollary
3.4 for the decay of the mean curvature).

Here the solution is to prove the decay of the mean curvature directly without
appealing to Theorem 3.1. This is possible because we work with a special
solution $f$ defined on $D\setminus\left\\{0\right\\}$. If we run the argument
of Theorem 3.1 for a sequence of scaling of $f$, thanks to the known result of
Kuwert and Li [11], we do not need any translation in the proof, which spares
us the need of having $\tau_{1}(f,c)=0$ (See Remark 3.3.)

###### Proof.

Using cylinder coordinate, we regard $f$ as defined on $[0,\infty)\times
S^{1}$. Fix any $q^{\prime}\in(q,2)$ and fix any $L>L_{0}$, where $L_{0}$ is
given in Theorem 3.1. Recall that $[0,\infty)\times S^{1}$ is the union of
$Q_{i}$ for $i=1,2,\cdots$.

Claim. There is some $i_{0}>0$ such that for any $i>i_{0}$, we have

(6.2) $\int_{Q_{i}}\left|H\right|^{2}dV_{g}\leq
e^{-q^{\prime}L}\left(\int_{Q_{i-1}}\left|H\right|^{2}dV_{g}+\int_{Q_{i+1}}\left|H\right|^{2}dV_{g}\right).$

Assuming the claim, it follows from the discussion at the beginning of Section
3 and the finiteness of the integral $\int_{[0,\infty)\times
S^{1}}\left|H\right|^{2}dV_{g}$ that we have the decay estimate

$\int_{Q_{i}}\left|H\right|^{2}dV_{g}\leq Ce^{-iqL}\qquad\text{for}\quad
i>i_{0}.$

The proofs in Section 4 can be repeated line by line for
$f_{k}=f|_{[i_{0},i_{0}+k]\times S^{1}}$ to get the dacay estimate (6.1) for
$f$.

The rest of the proof is devoted to the proof of the claim. If otherwise, we
find a subsequence $i_{k}$ such that (6.2) is not true for $i_{k}$.

By Theorem 3.1 of [11], we know $f\in W^{2,2}(D)$ and there exists
$m\in\mathbb{Z}_{+}$ and $u\in C^{0}(D)$ such that

$f^{*}(g_{\mathbb{R}^{n}})=e^{2u}\left|z\right|^{2(m-1)}(dx^{2}+dy^{2}).$

Without loss of generality, we assume $f(0)=0$ and $u(0)=0$. It is also proved
there that for any sequence $r_{k}\to 0$, if we set
$f_{k}(z)=\frac{f(r_{k}z)}{r_{k}^{m}}$, then $|\nabla f_{k}|$ is bounded, and
hence $f_{k}$ sub-converges to $\frac{z^{m}}{m}$ in $C^{0}_{loc}(D)\cap
C^{\infty}_{loc}(D\setminus\\{0\\})$ after a rotation that is independent of
$k$.

In terms of $(t,\theta)$ coordiante, the above result means that for
$t_{k}=i_{k}L$,

$\hat{f}_{k}(t,\theta):=e^{mt_{k}}f(t_{k}+t,\theta)$

converges (after a rotation independent of $k$) to

$f_{\infty}(t,\theta)=\frac{e^{-mt}}{m}(\cos m\theta,\sin m\theta,0,\cdots,0)$

locally smoothly on $\mathbb{R}\times S^{1}$. There is a similar part in the
proof of Lemma 5.2.

Since $\tau_{2}$ is independent of the scaling of $f$, we obtain

$\tau_{2}(f,S)=\tau_{2}(\hat{f}_{k},S)\to\tau_{2}(f_{\infty},S)=0\qquad\text{as}\quad
k\to\infty.$

In particular, $\tau_{2}(\hat{f}_{k},S)=0$ for any
$S\in\mathfrak{so}(\mathbb{R}^{n})$.

Set
$\phi_{k}=\frac{H_{\hat{f}_{k}}}{\left\|H_{\hat{f}_{k}}e^{-mt}\right\|_{L^{2}(Q_{1})}}$
and regard them as defined on $Q=Q_{0}\cup Q_{1}\cup Q_{2}$. By our choice of
$i_{k}$, we have

(6.3)
$\int_{Q_{1}}\left|\phi_{k}\right|^{2}e^{-2mt}dt\theta>e^{-qL}\left(\int_{Q_{0}}\left|\phi_{k}\right|^{2}e^{-2mt}dtd\theta+\int_{Q_{2}}\left|\phi_{k}\right|^{2}e^{-2mt}dtd\theta\right).$

Now we can obtain a contradiction using the same argument in the proof of
Theorem 3.1. To avoid repetition, we list only a few major points. $\phi_{k}$
satisfies some linear elliptic equation whose coefficients are bounded, hence
the above $L^{2}$ bound on $Q$ implies local smooth convergence on $Q$ that
gives us a harmonic function $\phi$. Being the limit of normal vectors, we may
write

$\phi=(0,0,\phi_{3},\cdots,\phi_{n}).$

As a consequence of $\tau_{2}(\hat{f}_{k},S)=0$ for any
$S\in\mathfrak{so}(\mathbb{R}^{n})$, we find that in the expansion of $\phi$,
the coefficients before $e^{mt}\cos m\theta$ and $e^{mt}\sin m\theta$ are
zero. Hence, by Lemma 3.2, (3.6) holds for $\phi$. This contradicts the limit
of (6.3). $\hfill\Box$

###  6.2. A gap theorem

In the discussion of the bubble tree problem, the gap theorem is very crucial,
because it ensures that only a finite number of bubbles appear. To study the
compactness of a sequence of conformal Willmore immersions with bounded
Willmore energy and topology, we need to have some version of gap theorem for
conformal and Willmore surfaces defined on
$\mathbb{C}\setminus\\{0\\}\rightarrow\mathbb{R}^{n}$. As an application of
our decay estimate, we provide such a theorem. Other discussions on the gap
theorem can be found in [12], [4]. Moreover, in [15], the authors classified
all Willmore spheres in $\mathbb{R}^{3}$ with no more than $3$ singularities
(counting multiplicity), which naturally implies a gap theorem. A
generalization of the result from Lamm-Nguyen can be found in [21].

######  Corollary 6.2.

For any $\beta>0$, there exists a positive number $\tau$ depending on $\beta$.
Let $f$ be a conformal and Willmore immersion from $\mathbb{R}\times S^{1}$
into $\mathbb{R}^{n}$. Let
$f^{*}(g_{\mathbb{R}^{n}})=e^{2u}(dt^{2}+d\theta^{2})$ and assume

(6.4) $\sup_{\mathbb{R}\times S^{1}}\left|\nabla u\right|<\beta.$

If $\|A_{f}\|_{L^{2}(\mathbb{R}\times S^{1})}<\tau$, then $A_{f}\equiv 0$ .

######  Remark 6.3.

We do not know if this gap $\tau$ can be proved to be independent of $\beta$.
It is an interesting question. Nevertheless, it is not a problem when we use
it to discuss the compactness of Willmore immersions with bounded Willmore
energy and bounded genus, because we can derive such upper bound for $\beta$
once the energy upper bound is fixed.

###### Proof.

Fix $q\in(0,2)$ and let $\delta_{1}$ be given in Theorem 1.3.

If there is no such $\tau$, then we have a sequence of $f_{k}$ satisfying all
the assumptions of Corollary 6.2 and

$\lim_{k\to\infty}\left\|A_{f_{k}}\right\|_{L^{2}(\mathbb{R}\times S^{1})}\to
0,$

but each $A_{f_{k}}$ is not zero somewhere. This sequence satisfies A.1)-A.3)
when restricted to any segments $[A_{k},B_{k}]\times S^{1}$ with
$B_{k}-A_{k}\to\infty$. Lemma 5.2 then implies the existence of
$M^{\prime\prime}$ such that the assumption (b) of Theorem 1.3 holds for
$f_{k}|_{[-l,l]\times S^{1}}$ for any $l\in\mathbb{N}$ as long as
$k>M^{\prime\prime}$. Obviously, if $M^{\prime\prime}$ is chosen larger, the
assumption (c) also holds. Now, fix some $k>M^{\prime\prime}$, if we can apply
Theorem 1.3 to $f_{k}|_{[-l,l]\times S^{1}}$ for any $l\in\mathbb{N}$, we
could derive $A_{f_{k}}\equiv 0$ and obtain a contradiction.

Threrefore, it remains to prove the assumption (a) of Theorem 1.3:

(6.5) $\tau_{1}(f_{k},c)=\tau_{2}(f_{k},S)=0\qquad\text{for all}\quad
c\in\mathbb{R}^{n},\,\,\,\,S\in\mathfrak{so}(\mathbb{R}^{n}).$

With $k$ fixed, we omit the subscript for simplicity and write $f$ for
$f_{k}$. Let $u$ and $K$ be the corresponding conformal factor and the
Gaussian curvature. If $k$ has been chosen large enough, we may assume

$\left|\int_{\mathbb{R}\times
S^{1}}Ke^{2u}dtd\theta\right|\leq\frac{1}{2}\int_{\mathbb{R}\times
S^{1}}\left|A\right|^{2}e^{2u}dtd\theta<1/4.$

Hence, for any $t^{\prime}<t^{\prime\prime}$, it follows from the equation
$-\Delta u=Ke^{2u}$ that

(6.6) $\left|\int_{\\{t^{\prime}\\}\times S^{1}}\frac{\partial u}{\partial
t}d\theta-\int_{\\{t^{\prime\prime}\\}\times S^{1}}\frac{\partial u}{\partial
t}d\theta\right|=\left|\int_{(t_{1},t_{2})\times
S^{1}}Ke^{2u}dtd\theta\right|<\frac{1}{4}.$

Given $t_{j}\rightarrow+\infty$, by (6.4) and the $\epsilon$-regularity, we
can find $\lambda_{j}$ and $c_{j}$ such that (after passing to subsequence)
$\hat{f}_{j}(t,\theta):=\lambda_{j}(f(t-t_{j},\theta)-c_{j})$ converges
smoothly locally to a conformal Willmore immersion $f_{\infty}$ defined on
$\mathbb{R}\times S^{1}$ and the second fundamental form of $f_{\infty}$ is
zero. By the same argument as in Lemma 5.2, we find that if $u_{\infty}$ and
$\hat{u}_{j}$ are the conformal factors of $f_{\infty}$ and $\hat{f}_{j}$
respectively, then

$\partial_{t}u_{\infty}=m\qquad\text{for some}\quad
m\in\mathbb{Z}\setminus\left\\{0\right\\}.$

Hence,

$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial u}{\partial
t}(t_{j},\theta)d\theta=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial\hat{u}_{j}}{\partial
t}(0,\theta)d\theta\rightarrow m.$

By (6.6), we have

$2\pi m-\frac{1}{4}\leq\frac{d}{dt}\int_{0}^{2\pi}u(t,\theta)d\theta\leq 2\pi
m+\frac{1}{4},\qquad\text{for all}\quad t\in\mathbb{R}.$

By a scaling of $f$ if necessary, we may assume
$\int_{0}^{2\pi}u(0,\theta)d\theta=0$, which implies that

$(2\pi m-\frac{1}{4})t\leq\int_{0}^{2\pi}u(t,\theta)d\theta\leq(2\pi
m+\frac{1}{4})t,\qquad\text{for all}\quad t\in\mathbb{R}.$

Together with our assumption that $|\nabla u|<\beta$, we get

$-\pi\beta+(m-\frac{1}{8\pi})t\leq u\leq\pi\beta+(m+\frac{1}{8\pi})t.$

Depending on the sign of $m$, either $Area(f|_{(0,+\infty)\times S^{1}})$ or
$Area(f|_{(-\infty,0)\times S^{1}})$ is infinity. Assume without loss of
generality that $Area(f|_{(0,+\infty)\times S^{1}})=\infty$ and by the lower
bound of $u$, the surface is complete at this end. Combining Theorem 4.2.1 in
[24] and the $\epsilon$-regularity of [12, 27], we know that for some
$\tilde{t}_{l}\to\infty$ and $\tilde{\lambda}_{l}$ uniformly comparable with
$e^{-m\tilde{t}_{l}}$, the sequence

$\tilde{f}_{l}(t,\theta):=\tilde{\lambda}_{l}(f(t+\tilde{t}_{l},\theta)-f(\tilde{t}_{l},0))$

converges smoothly locally on $\mathbb{R}\times S^{1}$ to a rotation of

$f_{\infty}(t,\theta)=\frac{e^{mt}}{m}(\cos m\theta,\sin m\theta,0,\cdots,0).$

Hence, for any $c\in\mathbb{R}^{n}$,

$\tau_{1}(\tilde{f}_{l},c)=\tilde{\lambda}_{l}^{-1}\tau_{1}(f,c)\to\tau_{1}(f_{\infty},c)=0\qquad\text{as}\quad
l\to\infty$

which implies that $\tau_{1}(f,c)=0$. Similarly,

$\tau_{2}(f,S)=\tau_{2}(\tilde{f}_{l},S)\to\tau_{2}(f_{\infty},S)=0\qquad\text{as}\quad
l\to\infty$

for any $S\in\mathfrak{so}(\mathbb{R}^{n})$, which implies that
$\tau_{2}(f,S)=0$. Hence, we have finished the proof of (6.5) and therefore
the whole proof of Corollary 6.2.

$\hfill\Box$

##  Appendix A A 3-circle lemma for harmonic function

In this section, we prove a 3-circle lemma for the Dirichlet energy of
harmonic functions. Such a result is well-known. There are many versions of
such results in the literature, but we haven’t found a proof for the following
version that is needed in this paper. As a service to the readers, we provide
the statement and a proof here.

######  Lemma A.1.

For any $q\in(0,2)$, there exists $L_{0}(q)$ such that if $L>L_{0}$, the
following holds. Define $Q_{i}:=[(i-1)L,iL]\times S^{1}$. Let $v$ be a
harmonic function from $Q=Q_{1}\cup Q_{2}\cup Q_{3}$. If

(A.1)
$\int_{Q_{2}}\left(|\partial_{t}v|^{2}-\left|\partial_{\theta}v\right|^{2}\right)dtd\theta=0,$

then

$\int_{Q_{2}}\left|\nabla v\right|^{2}dtd\theta\leq
e^{-qL}\left(\int_{Q_{1}}\left|\nabla
v\right|^{2}dtd\theta+\int_{Q_{3}}\left|\nabla v\right|^{2}dtd\theta\right).$

###### Proof.

Multiplying both sides of $(\partial_{t}^{2}+\partial_{\theta}^{2})v=0$ by
$\partial_{t}v$ and integrating by parts, we get

$\partial_{t}\int_{\left\\{t\right\\}\times
S^{1}}\left(|\partial_{t}v|^{2}-\left|\partial_{\theta}v\right|^{2}\right)d\theta=0\qquad\text{for
all}\quad t\in[0,3L].$

Hence, (A.1) implies that

(A.2) $\int_{\left\\{t\right\\}\times
S^{1}}\left(|\partial_{t}v|^{2}-\left|\partial_{\theta}v\right|^{2}\right)d\theta=0\qquad\text{for
all}\quad t\in[0,3L].$

By (A.2) and integrating over $t$, it suffices to prove

$\int_{\left\\{t\right\\}\times
S^{1}}\left|\partial_{\theta}v\right|^{2}d\theta\leq
e^{-qL}\left(\int_{\left\\{t-L\right\\}\times
S}\left|\partial_{\theta}v\right|^{2}d\theta+\int_{\left\\{t+L\right\\}\times
S^{1}}\left|\partial_{\theta}v\right|^{2}d\theta\right).$

By translation invariance, this is reduced to the special case $t=0$,

(A.3) $\int_{\left\\{0\right\\}\times
S^{1}}\left|\partial_{\theta}v\right|^{2}d\theta\leq
e^{-qL}\left(\int_{\left\\{-L\right\\}\times
S^{1}}\left|\partial_{\theta}v\right|^{2}d\theta+\int_{\left\\{L\right\\}\times
S^{1}}\left|\partial_{\theta}v\right|^{2}d\theta\right).$

To see this, we notice that $\partial_{\theta}v$ is a special harmonic
function in the sense that in its expansion, there is no constant term or
linear term, that is,

$\partial_{\theta}v=\sum_{k=1}^{\infty}(a_{k}e^{kt}+b_{k}e^{-kt})\cos
k\theta+\sum_{k=1}^{\infty}(a_{k}^{\prime}e^{kt}+b_{k}^{\prime}e^{-kt})\sin
k\theta,$

By Parseval’s theorem, proving (A.3) amounts to verify that for any
$a,b\in\mathbb{R}$, we have

$(a+b)^{2}\leq e^{-qL}\left((ae^{L}+be^{-L})^{2}+(ae^{-L}+be^{L})^{2}\right)$

as long as $L>L_{0}$. The existence of such $L_{0}$ is elementary.

$\hfill\Box$

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