Alexandrov [2, 3] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in Rn≥3 are round hyperspheres

RIGIDITY OF UMBILIC HYPERSURFACES

KWOK-KUN KWONG, HOJOO LEE, AND JUNCHEOL PYO

Abstract. We use the weighted Hsiung-Minkowski integral formulas and Brendle’s inequality to show new rigidity results. First, we prove Alexan- drov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordstr¨om spaces, where the Alexandrov reflection principle is not available. Second, we prove that in the Euclidean space, the only closed immersed self-expanding solitons to the inverse curvature flow of codimension one are round hyperspheres.

1. Motivation and Main Results

A. D. Alexandrov [2, 3] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in R^n≥3 are round hyperspheres. The embeddedness assumption is essential. For instance,R³ admits immersed tori with constant mean curvature, constructed by U. Abresch [1] and H. Wente [36]. R. C.

Reilly [29] and A. Ros [31, 32] presented alternative proofs, employing the Hsiung- Minkowski formula. See also Osserman’s wonderful survey [28].

In 1999, S. Montiel [27] established various general rigidity results in a class of warped product manifolds, including the Schwarzschild manifolds and Gaussian spaces. Some of his results require the additional assumption that the closed hyper- surfaces are star-shaped with respect to the conformal vector field induced from the ambient warped product structure. As a corollary [27, Example 5], he also recovers Huisken’s theorem [13] that the closed, star-shaped, self-shrinking hypersurfaces to the mean curvature flow in R^n≥3 are round hyperspheres. In 2016, S. Brendle [8]

solved the open problem that, inR³, the closed, embedded, self-shrinking topologi- cal spheres to the mean curvature flow are round. The embeddedness assumption is essential. Indeed, in 2015, G. Drugan [11] employed the shooting method to prove the existence of a self-shrinking sphere with self-intersections inR³.

In 2001, H. Bray and F. Morgan [5] proved a general isoperimetric comparison theorem in a class of warped product spaces, including Schwarzschild manifolds. In 2013, S. Brendle [6] showed that Alexandrov Theorem holds in a class of sub-static warped product spaces, including Schwarzschild and Reissner-Nordstr¨om manifolds.

S. Brendle and M. Eichmair [7] extended Brendle’s result to the closed, convex, star- shaped hypersurfaces with constant higher order mean curvature. See also [18] by V. Gimeno, [26] by J. Li and C. Xia, and [35] by X. Wang and Y.-K. Wang.

2010Mathematics Subject Classification. 53C44, 53C42.

Key words and phrases. Embedded hypersurfaces, higher order mean curvatures, integral inequalities.

1

In this paper, we provide new rigidity results (Theorem 1, 2 and 3). First, we asso- ciate the manifoldM^n≥3= Nⁿ⁻¹×[0,r),¯ ¯g=dr²+h(r)²gN

, where (Nⁿ⁻¹, gN) is a compact manifold with constant curvature K. As in [6, 7], we consider four conditions on the warping functionh: [0,¯r)→[0,∞):

(H1) h⁰(0) = 0 andh⁰⁰(0)>0.

(H2) h⁰(r)>0 for all r∈(0,r).¯

(H3) 2^h_h(r)^00(r)−(n−2)^K−h_h(r)^0(r)2 ² is monotone increasing forr∈(0,r).¯ (H4) For allr∈(0,r), we have¯ ^h_h(r)^00(r)+^K−h_h(r)^0(r)2 ² >0.

Examples of ambient spaces satisfying all the conditions include the classical Schwarzschild and Reissner-Nordstr¨om manifolds [6, Section 5].

Theorem 1. Let Σ be a closed hypersurface embedded in M^n≥3 with the k-th normalized mean curvature functionH_k=η(r)>0on Σfor some smooth radially symmetric function η(r). Assume that η(r)is monotone decreasing in r.

(1) k= 1 :Assume (H1), (H2), (H3). ThenΣis umbilic.

(2) k ∈ {2,· · · , n−1} : Assume (H1), (H2), (H3), (H4). If Σ is star-shaped (Section 2), then it is a sliceNⁿ⁻¹× {r0} for some constant r0.

We also prove the following rather general rigidity result for linear combinations of higher order mean cuvatures, with less stringent assumptions on the ambient space.

Theorem 2. Suppose (Mⁿ,g)¯ (n ≥ 3) satisfies (H2) and (H4). Let Σ be a closed star-shaped k-convex (H_k > 0) hypersurface immersed in Mⁿ, {ai(r)}^l−1_i=1 and {bj(r)}^k_j=l (2 ≤l < k ≤n−1) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least oneai(r)and onebj(r)are positive). Suppose

l−1

X

i=1

a_i(r)H_i =

k

X

j=l

b_j(r)H_j. ThenΣis totally umbilic.

Theorem 2 contains the case where ^H_H^k

l = η(r) for some monotone decreasing function η and k > l. We notice that the same result also applies to the space forms Rⁿ, Hⁿ andSⁿ+ (open hemisphere) without the star-shapedness assumption (Theorem 6). Our result extends [21, Theorem B] by S.-E. Koh, [22, Corollary 3.11]

by the first named author, and [37, Theorem 11] by J. Wu and C. Xia.

We next prove, in Section 4, a rigidity theorem for self-expanding soliton to the inverse curvature flow. Let us first recall the well-knowninverse curvature flow of hypersurfaces

d

dtF= σ_k−1 σk

ν, (1.1)

where ν denotes the outward pointing unit normal vector field and σk the k-th symmetric function of the principal curvature functions. We point out that the inverse curvature flow has been used to prove various geometric inequalities and rigidities: Huisken-Ilmanen [14], Ge-Wang-Wu [12], Li-Wei-Xiong [25], Kwong-Miao [20], Brendle-Hung-Wang [9], Guo-Li-Wu [19], and Lambert-Scheuer [23].

In the Euclidean space, the long time existence of smooth solutions to (1.1) was proved by Gerhardt in [17] and by Urbas in [33], when the initial closed hypersurface is star-shaped andk-convex (σk>0). Furthermore, they showed that the rescaled hypersurfaces converge to a round hypersphere ast→ ∞.

Theorem 3. Let Σ be a closed hypersurface immersed in R^n≥3. If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

In the proof of our main results, we shall use several integral formulas and in- equalities. Theorem 1 requires the embeddedness assumption as in the classical Alexandrov Theorem and is proved for the space forms in [22]. Theorem 2 and 3 require no embeddedness assumption and Theorem 3 is proved in [10] for the inverse mean curvature flow.

2. Preliminaries

Let (Nⁿ⁻¹, g_N) be an (n−1)-dimensional compact manifold with constant curva- tureK. Our ambient space is the warped product manifoldM^n≥3=Nⁿ⁻¹×[0,r)¯ equipped with the metric ¯g=dr²+h(r)²g_N. The precise conditions on the warping functionhwill be stated separately for each result.

In this paper, all hypersurfaces we consider are assumed to be connected, closed, and orientable. On a given hypersurface Σ in M, we define the normalized k-th mean curvature function

Hk:=Hk(Λ) = 1

n−1 k

σk(Λ), (2.1)

where Λ = (λ1,· · ·, λ_n−1) are the principal curvature functions on Σ and the ho- mogenous polynomialσk of degreekis thek-th elementary symmetric function

σk(Λ) = X

i₁<···<ik

λi1· · ·λi_k. We adopt the usual conventionσ0=H0= 1.

Definition 1(Potential function and conformal vector field). In our ambient warped product manifold M, we define the potential function f(r) = h⁰(r) > 0.

We define the vector field X =h(r)_∂r^∂ =∇ψ, where ψ⁰(r) = h(r) and ∇ is the connection onM. We note that it is conformal: LXg= 2f g [6, Lemma 2.2].

Definition 2 (Star-shapeness). For a hypersurface Σ oriented by the outward pointing unit normal vector field ν, we say that it is star-shaped when hX, νi ≥0.

A useful tool in studying higher order mean curvatures is thek-th Newton trans- formationTk :TΣ→TΣ (cf. [29, 30]). If we write

T_k(e_j) =

n−1

X

i=1

(T_k)ⁱ_je_i, then (Tk)ⁱ_j are given by

(T_k)_jⁱ= 1 k!

X

1≤i1,···,i_k≤n−1 1≤j₁,···,j_k≤n−1

δⁱⁱ_jj^1...ik

1...jkA^j_i¹

1· · ·A^j_i^k

k

where (A^j_i) is the second fundamental form of Σ. If{ei}ⁿ⁻¹_i=1 consist of eigenvectors ofAwith

A(ej) =λjej, then we have

T_k(e_j) = Λ_je_j, where

Λ_j= X

1≤i1<···<ik≤n−1, j /∈{i1,···,ik}

λ_i1· · ·λ_ik=σ_k(λ₁,· · ·, λ_j−1, λ_j+1,· · · , λ_n−1). (2.2)

One also definesT0= Id, the identity map. We have the following basic facts:

Lemma 1. Let Σbe a closed hypersurface in a warped product manifold M satis- fying the condition (H2).

(1) OnΣ, there is an elliptic point, where all principal curvatures are positive.

(2) Assume thatΣis p-convex (H_p>0). Then the following assertions hold (a) For allk∈ {1,· · · , p−1}, we have T_k>0 andH_k >0. For anyj ∈

{1,· · · , n−1}, we haveHk;j :=Hk(λ1,· · · , λ_j−1, λj+1,· · ·, λ_n−1)>0.

(b) The inequality ^H_H^p−1

p ≥_H¹

1 >0holds.

(c) For1≤i < j≤kand for any l={1,· · · , n−1}, jHiH_j−1;l> iHjH_i−1;l.

Proof. The first assertion is proved in [24, Lemma 4]. As in the proof of [4, Propo- sition 3.2],T_k>0 when k∈ {1,· · ·, p−1}, which implies

Hk(λ1,· · · , λj−1, λj+1,· · ·, λn−1)>0 by (2.2). Also, Hk = ¹

(n−1−k)(ⁿ⁻¹_k )trΣ(Tk) > 0. The classical Newton-Maclaurin inequalityH1H_p−1≥Hp then gives ^H_H^p−1

p ≥ _H¹

1 >0.

We now show (2c). Let λ =λl, m = n−1, and σi;l = ^m−1_i

Hi;l. Note that σi=λσ_i−1;l+σi;l, which implies

H_i= i

mλH_i−1;l+m−i m H_i;l.

Using this identity, (2a), and the Newton-Maclaurin inequality, we have jH_iH_j−1;l−iH_jH_i−1;l

=j i

mλH_i−1;l+m−i m Hi;l

H_j−1;l−i j

mλH_j−1;l+m−j m Hj;l

H_i−1;l

=j(m−i)

m Hi;lHj−1;l−i(m−j)

m Hj;lHi−1;l

=(j−i)H_j−1;lH_i−1;l+i(m−j)

m (Hi;lH_j−1;l−Hj;lH_i−1;l)

>0.

For the reader’s convenience, let us also record the following Heintze-Karcher- type inequality due to Brendle [6, Theorem 3.5 and 3.11], which is crucial in our proof of Theorem 1.

Proposition 1 (Brendle’s Inequality). Suppose the warped product manifold (M,¯g) satisfies (H1), (H2), and (H3). LetΣbe a closed embedded hypersurface in (M,¯g) with positive mean curvature. Then

Z

Σ

f H₁ ≥

Z

Σ

hX, νi.

The equality holds if and only ifΣis umbilic. If, futhermore, (H4) is satisfied, then Σis a sliceN× {r0}.

3. Proof of Theorem 1 and 2 The following formulas will play an essential role in our proof.

Proposition 2. Let φ be a smooth function on a closed hypersurfaceΣ in a Rie- mannian manifold Mⁿ.

(1) (Weighted Hsiung-Minkowski formulas) For k ∈ {1,· · ·, n−1}, we have

Z

Σ

φ(f H_k−1−H_khX, νi) + 1 k ⁿ⁻¹_k

Z

Σ

φ(div_ΣT_k−1) (ξ)

=− 1

k ⁿ⁻¹_k Z

Σ

hTk−1(ξ),∇Σφi.

(3.1)

Here, ξ=X^T is the tangential projection of the conformal vector field X ontoTΣ. (Note thatdiv(T₀) = 0.)

(2) Suppose (Mⁿ,¯g) is the warped product manifold in Section 2. Then, for k∈ {2,· · ·, n−1},

(div_ΣT_k−1)(ξ) =− n−3

k−2 ⁿ⁻¹

X

j=1

H_k−2;jξ^jRic(e_j, ν). (3.2) Here,{ej}ⁿ⁻¹_j=1 and{λj}ⁿ⁻¹_j=1 are the principal directions and principal curva- tures ofΣ, respectively, andHk−2;j =Hk−2(λ1,· · ·, λj−1, λj+1,· · · , λn−1).

If Σis star-shaped and (H4) is satisfied, then, for eachj ∈ {1,· · · , n−1},

−ξ^jRic(ej, ν)≥0. (3.3)

Proof. Letξ = X^T =X − hX, νiν, and recall that X is conformal: LXg = 2f g.

By [22, Proposition 3.1], we have

divΣ(φTk−1(ξ)) = (n−k)f σk−1φ−kσkφhX, νi+φ(divΣTk−1)(ξ) +hTk−1(ξ),∇Σφi.

Integrating this equation, we get (3.1).

We now show (2). Take a local orthonormal frame ν, e₁, · · ·, e_n−1, so that e₁,

· · ·,e_n−1are the principal directions of Σ. By the proof of [7, Proposition 8] (note thatT^(k) in [7] is the (k−1)-th Newton transformation), we have

(divΣT_k−1)ξ=−n−k n−2

n−1

X

j=1

σ_k−2;jξ^jRic(ej, ν),

whereσ_k−2;j=σ_k−2(λ₁,· · · , λ_j−1, λ_j+1,· · · , λ_n−1), which is equivalent to (3.2).

It remains to show (3.3). As in [6, Equation (2)], we compute Ric =−

h⁰⁰(r)

h(r) −(n−2)K−h⁰(r)² h(r)²

g−(n−2)

h⁰⁰(r)

h(r) +K−h⁰(r)² h(r)²

dr².

By the assumption (H4) and star-shaped conditionh_∂r^∂ , νi>0, we have

−ξ^jRic(ej, ν) = (n−2)

h⁰⁰(r)

h(r) +K−h⁰(r)² h(r)²

(ξ^j)² h(r)h∂

∂r, νi ≥0.

Theorem 4(=Theorem 1). Suppose(M^n≥3,¯g)is the warped product manifold in Section 2. LetΣbe a closed hypersurface embedded inMⁿ with thep-th normalized mean curvature H_p=η(r)>0 for some smooth radially symmetric function η(r).

Assume thatη(r)is monotone decreasing.

(1) p= 1 :Assume (H1), (H2), (H3). ThenΣis umbilic.

(2) p∈ {2,· · ·, n−1} : Assume (H1), (H2), (H3), (H4). IfΣ is star-shaped, then it is a sliceNⁿ⁻¹× {r0} for some constant r0.

Proof. Taking the radial weightφ(r) =_η(r)¹ >0 on Σ, we haveφ⁰(r)≥0 and fHp−1

Hp

− hX, νi=φ(f H_p−1−HphX, νi). (3.4) Assume firstp∈ {2,· · ·, n−1}. By Proposition 2 (2) and Lemma 1, (divΣT_p−1)ξ≥ 0. So by Proposition 2 (1), we have

Z

Σ

φ(f H_p−1−H_phX, νi)≤ − 1 (n−p) ⁿ⁻¹_p−1

Z

Σ

hT_p−1(X^T),∇_Σφi

=− 1

(n−p) ⁿ⁻¹_p−1 Z

Σ

h(r)φ⁰(r)hTp−1(∇Σr),∇Σri

≤0

(3.5)

as Tp−1 >0 by Lemma 1. We note that this inequality also holds for p = 1 by Proposition 2 (without assuming (H4) and the star-shapedness). Combining (3.4) and (3.5), we deduce

Z

Σ

fH_p−1

H_p − hX, νi

= Z

Σ

φ(f H_p−1−H_phX, νi)≤0.

From Lemma 1, we have ^H_H^p−1

p ≥_H¹

1 >0. Then, we obtain the inequality Z

Σ

f

H₁− hX, νi

≤0.

However, Brendle’s inequality (Proposition 1) is the reverse inequality Z

Σ

f H1

− hX, νi

≥0.

These two inequalities imply the equality in Brendle’s inequality. We conclude that Σ is umbilic and that, in the case when the condition (H4) holds, it is a slice.

Remark 1. Recently, Brendle’s inequality is extended in several ways, for instance, see [26, 34, 35]. We observe that the proof of (1) in Theorem 1 works on more general warped product manifold Mⁿ = Nⁿ⁻¹×[0,¯r), which admits the property that Brendle’s inequality holds. For instance, the fiber Nⁿ⁻¹ can be a compact Einstein manifold, as in Brendle’s paper [6].

We now give another rigidity result which contains as a special case where the ratio of two distinct higher order mean curvatures is a radial function.

Theorem 5 (= Theorem 2). Suppose (Mⁿ,g)¯ (n ≥ 3) is the warped product manifold in Section 2 satisfying (H2) and (H4). Let Σbe a closed star-shaped k- convex (H_k > 0) hypersurface immersed in Mⁿ, {ai(r)}^l−1_i=1 and {bj(r)}^k_j=l (2 ≤ l < k≤n−1) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one a_i(r) and oneb_j(r) are positive). Suppose

l−1

X

i=1

ai(r)Hi =

k

X

j=l

bj(r)Hj. ThenΣis totally umbilic.

Proof. Letξ=X^T andAp=−_{(n−1)(n−2)}¹ ξ^pRic(ep, ν). Since we are assuming (H2) and (H4), we can apply Lemma 1 (2) and Proposition 2 to have, for eachiandj,

Z

Σ

ai(r) (f Hi−1−HihX, νi) + (i−1) Z

Σ

ai(r)

n−1

X

p=1

ApHi−2;p

=− 1

i ⁿ⁻¹_i Z

Σ

hT_i−1(ξ),∇Σaii=− 1 i ⁿ⁻¹_i

Z

Σ

h(r)a⁰_i(r)hT_i−1(∇Σr),∇Σri ≥0 (3.6) and

Z

Σ

bj(r) (f Hj−1−HjhX, νi) + (j−1) Z

Σ

bj(r)

n−1

X

p=1

ApHj−2;p

=− 1

j ⁿ⁻¹_j Z

Σ

hT_j−1(ξ),∇Σbji=− 1 j ⁿ⁻¹_j

Z

Σ

h(r)b⁰_j(r)hT_j−1(∇Σr),∇Σri ≤0.

(3.7) Summing (3.6) overiand (3.7) overj, and then taking the difference gives

0 = Z





k

X

j=l

bj(r)Hj−

l−1

X

i=1

ai(r)Hi



hX, νi

≥ Z

Σ

f





k

X

j=l

b_j(r)H_j−1−

l−1

X

i=1

a_i(r)H_i−1





+ Z

Σ n−1

X

p=1

A_p





k

X

j=l

(j−1)b_j(r)H_j−2;p−

l−1

X

i=1

(i−1)a_i(r)H_i−2;p



.

(3.8)

Note thatHj >0 forj≤kby Lemma 1. Let 1≤i < j≤k, then multiplying the Newton’s inequalityHiHj−1≥Hi−1Hj byai(r)bj(r) and summing overi, j gives

l−1

X

i=1

a_i(r)H_i

k

X

j=1

b_j(r)H_j−1≥

l−1

X

i=1

a_i(r)H_i−1

k

X

j=1

b_j(r)H_j. SincePl−1

i=1ai(r)Hi =Pk

j=lbj(r)Hj>0, we deduce

k

X

j=l

bj(r)Hj−1≥

l−1

X

i=1

ai(r)Hi−1. (3.9)

Similar to (3.9), we can obtain from Lemma 1 (2c) the inequality

k

X

j=l

(j−1)bj(r)H_j−2;p−

l−1

X

i=1

(i−1)ai(r)H_i−2;p>0. (3.10) On the other hand, Ap ≥ 0 by Proposition 2. Combining this with (3.10) and (3.9), we conclude that all the integrands in (3.8) are zero. This implies (3.9) is an equality and hence Σ is totally umbilic by the Newton-Maclaurin inequality.

Due to the analogous, but simpler, weighted Hsiung-Minkowski integral formu- las in the space forms (cf. [22]), without assuming the star-shapedness condition, following the proof of Theorem 5, we can deduce

Theorem 6. Let Σ be a closed k-convex hypersurface immersed in Mⁿ = Rⁿ, Hⁿ or Sⁿ+ (open hemisphere). Let r be the distance in Mⁿ from a fixed point p₀ ∈ M (chosen to be the center if M = Sⁿ+). Let {a_i(r)}^l−1_i=1 and {b_j(r)}^k_j=l (2 ≤ l < k ≤ n−1) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one ai(r) and onebj(r)are positive). Suppose

l−1

X

i=1

ai(r)Hi =

k

X

j=l

bj(r)Hj. Then it is a geodesic hypersphere.

4. Proof of Theorem 3

We consider theinverse curvature flow in Euclidean spaceR^n≥3: d

dtF= σ_k−1 σk

ν. (4.1)

Here, ν denotes the outward pointing unit normal vector field and σ_k the k-th symmetric function of the principal curvatures. Whenk= 1, the evolution (4.1) is equivalent to the inverse mean curvature flow.

Definition 3. We say that a hypersurfaceΣwith σk>0 is a self-expander to the inverse curvature flow when we have

σ_k−1 σk

=ChX, νi onΣ (4.2)

for some constant C >0.

Theorem 7(=Theorem 3). LetΣbe a closed hypersurface immersed inR^n≥3. If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

Proof. Letp=hX, νidenote the support function. We shall repeatedly use Hsiung–

Minkowski integral formula [15, 16] for the normalized higher order mean curvature functionsH_i= ¹

(ⁿ⁻¹i )σ_i. Then, the soliton equation (4.2) is equivalent to H_k−1

Hk

=µp (4.3)

for some constant µ >0. By thek-convexity assumptionσk >0, the item (2) in Lemma (1) implies that σi >0 and Hi >0 for i ∈ {0,· · ·, k}. We claim µ= 1.

Indeed, the Hsiung–Minkowski integral formula shows that 0 =

Z

Σ

(H_k−1−H_kp) =

1−1 µ

Z

Σ

H_k−1.

SinceHk−1>0 on Σ, we haveµ= 1. The soliton equation (4.3) reduces to Hk−1=Hkp.

Notice thatp>0. We distinguish two cases: k= 1 andk∈ {2,· · ·, n−1}. When k= 1, by Newton-Maclaurin inequality, we have

H2p= H2

H₁ ≤H1

H₀ =H1. (4.4)

Integrating this inequality gives Z

Σ

H2p≤ Z

Σ

H1. (4.5)

However, by the Hsiung-Minkowski formula, we have the equality Z

Σ

H₁= Z

Σ

H₂p.

Whenk∈ {2,· · ·, n−1}, sinceHi>0 fori≤k, by Newton-Maclaurin inequality, H0= 1 = Hk

H_k−1p≤ H_k−1

H_k−2p≤ · · · ≤ H1

H0

p=H1p. (4.6)

Integrating this gives

Z

Σ

H0≤ Z

Σ

H1p. (4.7)

However, by the Hsiung-Minkowski inequality, we have the equality Z

Σ

H₁p= Z

Σ

H₀.

In both cases, the Hsiung-Minkowski formula assures that the equality in (4.5) or (4.7) actually holds, and so the equality in (4.4) or (4.6) holds. Therefore, the closed hypersurface Σ must be umbilic and so is round.

Acknowledgements. Part of our work was done while the three authors were visiting the National Center for Theoretical Sciences in Taipei, Taiwan in November 2015. We would like to thank the NCTS for their financial support and warm hospitality during our visit. The research of K.-K. Kwong is partially supported

by Ministry of Science and Technology in Taiwan under grant MOST103-2115-M- 006-016-MY3. J. Pyo is partially supported by the National Research Foundation of Korea (NRF-2015R1C1A1A02036514).

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Department of Mathematics, National Cheng Kung University, Tainan City 701, Tai- wan

E-mail address:kwong@mail.ncku.edu.tw

Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, Korea

E-mail address:momentmaplee@gmail.com

Department of Mathematics, Pusan National University, Busan 46241, Korea E-mail address:jcpyo@pusan.ac.kr