Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 85, pp. 1–5.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.85

REMARK ON ISOLATED REMOVABLE SINGULARITIES OF HARMONIC

MAPS IN TWO DIMENSIONS

CHANGYOU WANG

Abstract. For a ball BR(0) ⊂ R2, we provide sufficient conditions such that a harmonic map

u ∈ C∞(BR(0)\{0}, N), with a self-similar bound on its gradient, belong to C∞(BR(0)). These

conditions also guarantee the triviality of such harmonic maps when R = ∞.

1. Introduction

In this short note, we address a question arising from the recent study [1] on the rigidity for
the steady (simplified) Ericksen-Leslie system in Rn, which seeks to answer the question:

If (u, d) ∈ C∞(Rn \ {0},Rn × Sn−1), n ≥ 2, solves

−∆u+ u · ∇u+∇p = −∇ · (∇d⊙∇d),

∇ · u = 0,

∆d+ |∇d|2d = u · ∇d,

(1.1)

in Rn \ {0}, and satisfies a self-similar bound

|u(x)| ≤ C1(n)

|x|
, |∇d(x)| ≤ C2(n)

|x|
, ∀x ∈ Rn \ {0}, (1.2)

for some constants C1(n), C2(n) > 0, does it follow that (u,∇d) ≡ (0, 0) in Rn?

In [1], we obtained some partial results towards this question. In particular, we proved that when
n ≥ 3, there exists εn > 0 such that if C1(n), C2(n) ≤ εn then ∇d ≡ 0; while u ≡ 0 when
n ≥ 4, or a Landau solution of the steady Navier-Stokes equation when n = 3. When n = 2,
we constructed infinitely many nontrivial solutions of (1.1) and (1.2), that resemble the so-called
Hamel’s solutions of steady Navier-Stokes equation in R2.

A Liouville theorem on harmonic maps plays an important role in [1], that is, for n ≥ 3 if
d ∈ C∞(Rn \ {0}, N) solves the equation of harmonic maps:

∆d+A(d)(∇d,∇d) = 0 in Rn \ {0}, (1.3)

and there exists an ε0(n) > 0 such that

|∇d(x)| ≤ ε0(n)

|x|
, ∀x ∈ Rn \ {0}, (1.4)

then d must be a constant map. Here N ⊂ RL is a compact smooth Riemann manifold without
boundary, and A denotes the second fundamental form of N .

A natural question to ask is whether this Liouville property remains true when n = 2. More
precisely,

Question 1.1. Suppose d ∈ C∞(R2 \ {0}, N) solves (1.3) and satisfies (1.4) for some small
constant ε0(2). Does it follow that d must be constant?

2020 Mathematics Subject Classification. 35J50, 58E20.

Key words and phrases. Harmonic maps; removable isolated singularity.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 18, 2025. Published August 11, 2025.

1



2 C. WANG EJDE-2025/85

To the best of the author’s knowledge, this question has not been addressed in the literature.
In contrast with n ≥ 3, (1.4) alone does not guarantee d has locally finite Dirichlet energy in
dimension two: E(d,B1(0)) =

∫
B1(0)

|∇d|2 < ∞ for the unit ball B1(0) ⊂ R2. Thus, neither the

celebrated theorem by Sacks-Uhlenbeck [3] on the removability of isolated singularity of harmonic
maps in dimension two, nor the regularity theorem by Hélein [2] on weakly harmonic maps can be
applied in two dimensions. Observe that d(x) = x

|x| : R
2 \ {0} → S1 is a harmonic map, satisfying

|∇d(x)| = 1
|x| for x ̸= 0 and E(d,B1(0)) = ∞, while x = 0 is a non-removable singular point. This

example indicates that ε0(2) in Question 1.1 must be chosen sufficiently small.
In this note, we will give a partial answer to Question 1.1. More precisely, let BR(0) ⊂ R2 be

the ball in R2 with center 0 and radius R, we will prove the following.

Theorem 1.2. There exists an ε0 > 0 such that if u : BR(0) \ {0} → N is a smooth harmonic
map, satisfying

|∇u(x)| ≤ ε0
|x|

, ∀x ∈ BR(0) \ {0}, (1.5)

and if, in addition, there exists ri → 0 such that

lim
i→∞

ri

∫
∂Bri

(0)

(
|∂u
∂r

|2 − 1

r2
|∂u
∂θ

|2
)
dσ = 0, (1.6)

then u ∈ C∞(BR(0), N).

As a direct consequence of Theorem (1.2), we establish the following.

Corollary 1.3. There exists an ε0 > 0 such that if u ∈ C∞(R2 \ {0}, N) is a harmonic map,
satisfying

|∇u(x)| ≤ ε0
|x|

, ∀x ∈ R2 \ {0}, (1.7)

and if, in addition, there exists ri → 0 such that

lim
i→∞

ri

∫
∂Bri

(0)

(
|∂u
∂r

|2 − 1

r2
|∂u
∂θ

|2
)
dσ = 0, (1.8)

then u must be a constant map.

2. Proofs of main results

To prove of Theorem 1.2 and Corollary 1.3, we need the following lemma.

Lemma 2.1. If u ∈ C∞(BR(0) \ {0}, N) is a harmonic map, then

ϕ(r) := r

∫
∂Br(0)

(
|∂u
∂r

|2 − 1

r2
|∂u
∂θ

|2
)
dσ (2.1)

is constant for r ∈ (0, R).

Proof. Since u ∈ C∞(BR(0) \ {0}, N) solves the harmonic map equation (1.3), for any 0 < r1 <
r2 < R, we can multiply (1.3) by x · ∇u and integrate the resulting equation over Br2(0) \Br1(0)
to obtain

0 =

∫
Br2

(0)\Br1
(0)

∆u · (x · ∇u)

=

∫
Br2

(0)\Br1
(0)

(ujxiui)j − |∇u|2 − 1

2
xj(|∇u|2)j

=

∫
∂(Br2 (0)\Br1 (0))

(x · ∇u) · (ν · ∇u)− 1

2

∫
∂(Br2 (0)\Br1 (0))

|∇u|2x · ν,

where ν denotes the outward unit normal of ∂(Br2(0) \Br1(0)). This implies that

r2

∫
∂Br2 (0)

(
|∂u
∂r

|2 − 1

2
|∇u|2

)
dσ = r1

∫
∂Br1 (0)

(
|∂u
∂r

|2 − 1

2
|∇u|2

)
dσ.



EJDE-2025/85 ISOLATED REMOVABLE SINGULARITIES 3

Since

|∇u|2 = |∂u
∂r

|2 + 1

r2
|∂u
∂θ

|2,

it follows that

r2

∫
∂Br2

(0)

(
|∂u
∂r

|2 − 1

r2
|∂u
∂θ

|2
)
dσ = r1

∫
∂Br1

(0)

(
|∂u
∂r

|2 − 1

r2
|∂u
∂θ

|2
)
dσ. (2.2)

This implies (2.1). □

Remark 2.2. It is easy to check that if d(x) = x
|x| : R

2 \{0} → S1, then ϕ(r) = −2π for all r > 0.

Proof of Theorem 1.2. From (1.6) and (2.1), we have that∫
∂Br(0)

|∂u
∂r

|2 dσ =
1

r2

∫
∂Br(0)

|∂u
∂θ

|2 dσ (2.3)

for all 0 < r < R.
We will modify the original argument by Sacks-Uhlenbeck [3] to show that x = 0 is a removable

singularity for u.
First, we show that u has finite Dirichlet energy, i.e., u ∈ H1(BR(0)). For this, let 0 < r∗ <

R∗ ≤ R be two given radius. Set K =
[ ln(R∗

r∗ )

ln 2

]
∈ N and define the annulus

Am = B2mr∗(0) \B2m−1r∗(0), 1 ≤ m ≤ K.

We denote the radial harmonic function hm(r) := am+bm ln r : Am → RL, where am and bm ∈ RL

are chosen according to the condition

hm(2mr∗) = −
∫
∂B2mr∗

u dσ, hm(2m−1r∗) = −
∫
∂B2m−1r∗

u dσ,

where

−
∫
∂Br(0)

f dσ =
1

2πr

∫
∂Br(0)

f dσ

denotes the average of f over ∂Br(0).
Note that condition (1.5) implies

oscAm
u ≤ Cε0, ∀1 ≤ m ≤ K.

Now, multiplying (1.3) by u− hm and integrating the resulting equation over Am we obtain∫
Am

|∇(u− hm)|2

=

∫
∂Am

(∂u
∂r

− h′
m(r)

)
· (u− hm) +

∫
Am

A(u)(∇u,∇u) · (u− hm)

=

∫
∂B2mr∗ (0)

∂u

∂r
· (u− hm)−

∫
∂B2m−1r∗ (0)

∂u

∂r
· (u− hm)

+

∫
Am

A(u)(∇u,∇u) · (u− hm)

≤
∫
∂B2mr∗ (0)

∂u

∂r
· (u− hm)−

∫
∂B2m−1r∗ (0)

∂u

∂r
· (u− hm) + Cε0

∫
Am

|∇u|2.

Since hm depends only on r, we can apply (2.3) to obtain that∫
Am

|∇(u− hm)|2 ≥
∫
Am

1

r2
|∂u
∂θ

|2 dσ =
1

2

∫
Am

|∇u|2.

Hence

(
1

2
− Cε0)

∫
Am

|∇u|2 ≤
∫
∂B2mr∗ (0)

∂u

∂r
· (u− hm)−

∫
∂B2m−1r∗ (0)

∂u

∂r
· (u− hm) (2.4)



4 C. WANG EJDE-2025/85

By summing (2.4) over 1 ≤ m ≤ K, we obtain that(1
2
− Cε0

) ∫
B2Kr∗

(0)\Br∗ (0)

|∇u|2 ≤
∫
∂B2Kr∗

(0)

∂u

∂r
· (u− hK)−

∫
∂Br∗ (0)

∂u

∂r
· (u− h1). (2.5)

By Poincarè inequality, (2.3) and (1.5), the terms in the right-hand side of (2.5) can be estimated
as ∣∣ ∫

∂B2Kr∗
(0)

∂u

∂r
· (u− hK)

∣∣
≤ C

(∫
∂B2Kr∗

(0)

|∂u
∂r

|2 dσ
)1/2(∫

∂B2Kr∗
(0)

|u− hK |2 dσ
)1/2

≤ C2Kr∗

(∫
∂B2Kr∗

(0)

|∂u
∂r

|2 dσ
)1/2(∫

∂B2Kr∗
(0)

1

r2
|∂u
∂θ

|2 dσ
)1/2

≤ C2Kr∗

∫
∂B2Kr∗

(0)

|∇u|2 dσ

≤ Cε20,

(2.6)

and, similarly, ∣∣ ∫
∂Br∗ (0)

∂u

∂r
· (u− h1)

∣∣ ≤ Cr∗

∫
∂Br∗ (0)

|∇u|2 dσ ≤ Cε20. (2.7)

Substituting the inequalities (2.6) and (2.7) into (2.5) yields(1
2
− Cε0

) ∫
B2Kr∗

(0)\Br∗ (0)

|∇u|2 ≤ Cε20. (2.8)

Thus, by choosing ε0 < 1
4C and observing R∗

2 ≤ 2Kr∗ ≤ R∗, we obtain that∫
BR∗/2(0)\Br∗ (0)

|∇u|2 ≤ Cε20. (2.9)

Since (2.9) holds for any two 0 < r∗ < R∗ ≤ R, we conclude that∫
BR/2(0)

|∇u|2 ≤ Cε20 < ∞. (2.10)

Next, with the help of (2.10), we can repeat the above arguments to obtain t he Hölder conti-
nuity of u near x = 0. In fact, after labeling r = 2Kr∗ so that r∗ = 2−Kr, (2.5), (2.6) and (2.7)
imply that for any 0 < r < R,∫

Br(0)\B2−Kr(0)

|∇u|2 ≤ Cr

∫
∂Br(0)

|∇u|2 dσ + C2−Kr

∫
∂B2−Kr(0)

|∇u|2 dσ. (2.11)

On the other hand, from (2.10) it follows that

lim
K→∞

2−Kr

∫
∂B2−Kr(0)

|∇u|2 dσ = 0.

Hence, after sending K → ∞ in (2.11), we obtain that for any 0 < r < R,∫
Br(0)

|∇u|2 ≤ Cr

∫
∂Br(0)

|∇u|2 dσ. (2.12)

This implies the existence of an α ∈ (0, 1) such that∫
Br(0)

|∇u|2 ≤ (
r

R
)2α

∫
BR/2(0)

|∇u|2, 0 < r ≤ R

2
.

This, combined with u ∈ C∞(B1 \ {0}), yields u ∈ Cα(BR
2
(0)). By the higher order regularity of

harmonic maps, u ∈ C∞(BR/2(0)) (see, for example, [3]). □



EJDE-2025/85 ISOLATED REMOVABLE SINGULARITIES 5

Proof of Corollary 1.3. It follows from Theorem 1.2 and (2.12) that u ∈ C∞(R2), and∫
BR(0)

|∇u|2 ≤ CR

∫
∂BR(0)

|∇u|2 dσ ≤ Cε20, ∀R > 0. (2.13)

By choosing sufficiently small ε0 in (2.13) and applying the ε0-gradient estimate for harmonic
maps, we obtain that

∥∇u∥L∞(BR(0)) ≤
Cε0
R

, ∀R > 0.

Sending R → ∞, this yields that u must be constant. □

Acknowledgments. The work was partially supported by the NSF DMS grants 2101224 and
2453789, and by the Simons Travel Grant TSM-00007723.

References

[1] J. Bang, C. Y. Wang; On rigidity of the steady Ericksen-Leslie system. arXiv:2502.05326, Proceedings of America

Mathematical Society, to appear.
[2] F. Hélein; Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne,

C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591-596.

[3] J. Sacks, K. Uhlenbeck; The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no.
1, 1-24.

Changyou Wang

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Email address: wang2482@purdue.edu


	1. Introduction
	2. Proofs of main results
	Acknowledgments

	References