n -dimensional area of minimal rotational hypersurfaces in spheres

Linear Algebra Applications Nonlinear Analysis and 125 its (2015) 241–250 466 (2015) 102–116

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

problem hypersurfaces of Jacobi matrix n-dimensionalInverse area of eigenvalue minimal rotational in spheres with mixed data

Oscar M. Perdomo a , Guoxin Wei b,∗ 1 a Ying Wei Department of Mathematics, Central Connecticut State University, New Britain, CT 06050, United States b

School of Mathematical Sciences, South China Normal University, 510631 Guangzhou, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, People’s Republic of China Nanjing 210016, PR China

article

i n faor t i c l e

ianbfs ot r a c t

a b s t r a c t

Article history: Let M R(n) denote theIn setthis of compact minimal hypersurfaces in the (n+1)Article history: paper, the inverserotational eigenvalue problem of reconstructing Received 6 March 2015 Received 16 January dimensional sphere S n+1 . For any compact M ⊂its S n+1 , let us denote 2014 a Jacobi matrix fromhypersurface its eigenvalues, leading principal Accepted 20 May 2015 Accepted 20 September 2014 by |M | its n-dimensional area. Inand this part paperofwethe show that for any integer n ≥ 2, submatrix eigenvalues of its submatrix Communicated by EnzoAvailable Mitidierionline 22 October 2014

the set of numbers Anis=considered. {|M | : M ∈The M R(n)} is a discrete set. We also show that necessary and sufficient conditions for

MSC: 53C42 53A10

Submitted by Y. Weiif M ∈ M R(3), then either |M | = |S 3 |, or |M | = |M (3)|, or |M | > |M (3)|, where 0 derived.  the existence and uniqueness0 of the solution are

MSC: 15A18 15A57

Keywords: Rotational hypersurfaces Keywords: Minimal hypersurfaces Jacobi matrix Area Eigenvalue Inverse problem Submatrix

for 2 ≤ n ≤ 100 and we show ) × S 1 ( √1n ). Finally, M0 (n) = S n−1 ( n−1 Furthermore, a numerical algorithm somenumerically numerical n given. evidence that if M ∈examples M R(n),are then either |M | = |S n |, or |M | = |M0 (n)|, or n |, 2014 Published by Elsevier |M | > 2|M0 (n)|. We strongly believe that for any©M ∈M R(n), either |M | = |SInc. or |M | = |M0 (n)|, or |M | > 2|M0 (n)| for all n. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Let M be a compact minimal hypersurface in S n+1 and S the squared norm of the second fundamental form. Simons [16] proved that if 0 ≤ S ≤ n, then either S ≡ 0 and M is totally geodesic or S ≡ n. In the caseS ≡ n, Chern, do Carmo and Kobayashi [5], Lawson [8] independently proved that M is a Clifford torus

S k ( nk ) × S n−k ( n−k n ). For n ≤ 5, Peng and Terng [13] proved that if there exists δ(n) > 0 such that n ≤ S ≤ n + δ(n), then S ≡ n and M is a Clifford torus. For n ≥ 6, Ding and Xin [6] proved that if there exists δ(n) > 0 such that n ≤ S ≤ n + δ(n), then S ≡ n and M is a Clifford torus. If we assume that S is constant, then we have the well-known Chern’s conjecture: The values of S are discrete if M is a compact minimal hypersurface in S n+1 with constant S-function. E-mail address: [email protected]. On the other hand, by estimating the heat kernel for S n , Cheng, Li and Yau [4] obtained that if M is 1 Tel.: +86 13914485239. a compact minimal hypersurface in S n+1 and the area |M | of M is close to the area |S n | of S n , then M n is totally geodesic Shttp://dx.doi.org/10.1016/j.laa.2014.09.031 . That means, the area value |S n | is discrete among the other area values of compact 0024-3795/© 2014 Published by Elsevier Inc. minimal hypersurfaces in S n+1 . ∗ Corresponding author. E-mail addresses: [email protected] (O.M. Perdomo), [email protected] (G. Wei).

http://dx.doi.org/10.1016/j.na.2015.05.017 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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Inspired by the above results, it is natural to ask if the areas of all compact minimal hypersurfaces in a sphere are discrete. In this paper, for rotational hypersurfaces, we prove that it is true. Theorem 1.1. The areas of all compact rotational minimal hypersurfaces in a sphere are discrete. Remark 1.2. There are countably many and infinite compact minimal rotational hypersurfaces in S n+1 , see [10]. In [4], Cheng, Li and Yau also proved that if M is a compact minimal hypersurface in the (n + 1)dimensional sphere S n+1 and M is not totally geodesic, then there exists a constant c(n) > 0, such that |M | > (1+c(n))|S n |. From this fact, we know that the minimum of area of all compact minimal hypersurfaces in a sphere S n+1 is |S n |. That is, the totally geodesic sphere S n ⊂ S n+1 is the hypersurface with least n-area among all compact minimal hypersurfaces in S n+1 . [2] provides another reference for this result. A natural question to ask is if there is a compact minimal hypersurface that achieves a second smallest area. The natural candidates to achieve the second value for the n-area functional, if such a value exists, is one of the  n−k   k  n−k k . When n = 2, it has been shown by Marques and Clifford minimal hypersurfaces S n ×S n

Neves [9] that any non-totally geodesic and closed embedded minimal surface in S 3 has area greater than or equal to 2π 2 , the area of the Clifford torus (cf. [3]). Moreover, this value is only reached by the Clifford torus. In the same paper, as a consequence of their proof of Willmore conjecture, it follows that the area of any non-totally geodesic and compact minimal (not necessarily embedded) minimal torus is greater than 2π 2 . In this paper we study the second value for the n-area functional among compact minimal rotational hypersurfaces in S n+1 and prove 4 3 Theorem 1.3.  If M is a compact minimal  rotational hypersurface in S , then either |M | = |S |, or |M | = |S 2 ( 23 ) × S 1 ( √13 )|, or |M | > |S 2 ( 23 ) × S 1 ( √13 )|.

Moreover, we can also show the following result by using the numerical method. Theorem 1.4. For 2 ≤ n ≤ 100, let M be an  n-dimensional compact minimal rotational hypersurface in  n−1 1 √1 n−1 S n+1 , then either |M | = |S n |, or |M | = |S n−1 ( n−1 ) × S ( )|, or |M | > 2|S ( ) × S 1 ( √1n )|. n n n Motivated by the above observation, we propose the following. Conjecture. The n-area of any compact minimal rotational hypersurface in S n+1 is either |S n |, or |S n−1   n−1 n−1 1 √1 n−1 1 √1 ( ( n ) × S ( n )|, or greater than 2|S n ) × S ( n )|. Remark 1.5. It is evident that the conjecture above cannot be solved numerically for all n, nevertheless we have checked it for 2 ≤ n ≤ 100. About Theorem 1.4, we have all the graphs for 2 ≤ n ≤ 100, but for a matter of space we are only showing them up to n = 20. 2. Preliminaries  1 √1 Without loss of generality, we assume that the hypersurface is neither S n nor S n−1 ( n−1 n ) × S ( n ). From [10,14], we have the following description for every minimal rotational hypersurface in S n+1 which is not trivial in the sense that its principal curvatures are not constant. The set of minimal non trivial complete rotational hypersurfaces can be described as the family 1−n M Rot(n) = {Mc }c>c0 where c0 = n(n−1) n . Notice that the hypersurfaces in M Rot(n) are not necessarily compact.

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Fig. 2.1. The left image shows the stereographic projection of the fundamental portion of the compact surface in S 3 obtained by finding the c that satisfies K(c) = 2π 23 . The right image shows the fundamental curve of this surface.

Let us describe the hypersurface Mc . For any c > c0 , let r(t) be a solution of the following ordinary differential equation (r′ (t))2 = 1 − r(t)2 − c−n r(t)2−2n .

(2.1)

Since c > c0 , we have that the function q(v) = 1 − v 2 − c−n v 2−2n has two positive roots r1 and r2 between 0 and 1. Therefore it is not difficult to check that the solution of the differential equation (2.1), which can be r solved using separation of variables, is a periodic function with period T = 2 r12 √ 1 dv that takes values q(v)

between r1 and r2 . Moreover, since the differential equation (2.1) does not depend on t explicitly, then, for any k we have that r(t − k) is a solution, provided r(t) is a solution. Therefore we can assume that r(0) = r1 and r( T2 ) = r2 . If we define  t − n 1−n (τ ) c 2r θ(t) = dτ, 2 (τ ) 1 − r 0 the hypersurface φ : S n−1 × [0, T ] → S n+1 given by     φ(y, t) = r(t) y, 1 − r2 (t) cos(θ(t)), 1 − r2 (t) sin(θ(t)) is called  the fundamental portion  of Mc and we will denote it by F Pc . We will also call the curve  α(t) = 1 − r2 (t) cos(θ(t)), 1 − r2 (t) sin(θ(t)) , the profile curve of Mc . It turns out that the whole hypersurface Mc is the union of rotations of the fundamental portion F Pc . We also conclude that the hypersurface Mc is compact if and only if the number  T2 − n 1−n c 2r (τ ) r K(c) = θ(T ) = 2 dτ = 2π , (2.2) 2 1 − r (τ ) s 0 for some pair of relatively prime integers r and s. In this case Mc is made out of exactly s copies of the fundamental portion F Pc (see Fig. 2.1). When K(c) 2π is not a rational number we have that not only the hypersurface is not compact but it is not properly immersed. A direct computation (also in [14]) using the ordinary differential equation (2.1) shows that the vector ∂φ ∂t has length 1 therefore we have the following remark (see also [7]). Remark 2.1. The n-dimensional area F A(c) of F Pc is given by  T2   F A(c) = 2 rn−1 (t)dydt = 2σn−1 0

S n−1

where σn−1 denotes the (n − 1)-area of S n−1 .

0

T 2

rn−1 (t)dt,

(2.3)

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Fig. 3.1. The three images show the graph of K as a function of a when n = 2, 3 and 4 respectively. When n = 2, a0 is 14 , when √ 4 27 n = 3, a0 is 27 and when n = 4, a0 is 256 . In each case the horizontal lines at 2π and π are shown. These graphs were obtained by numerically solving the integral in Remark 3.2 using Mathematica 8.

3. Discrete area result In this section, we will prove that the set of possible n-areas of compact minimal rotational hypersurfaces of S n+1 is discrete. Let us start with the following lemma due to Otsuki [11,12]. A proof for the particular case n = 2 can also be found in [1] and in [15]. Lemma 3.1. The function K(c) given in (2.2) is strictly decreasing and differentiable on (c0 , ∞) and √ lim K(c) = π. lim K(c) = 2π, c→c0

c→∞

In this paper, instead of considering the variable c we will consider the variable a = c1n . With this change, √ we have that the function K increases from π to 2 π. The following remark gives us an expression that allows us to numerically graph K as a function of a in a relatively easy way. Remark 3.2. If a = c1n , then a moves from a0 = (n−1) to 0 when c moves from c0 to ∞. Moreover, If nn R1 < R2 are the only two roots in the interval (0, 1) of the polynomial p(R) = Rn−1 − Rn − a, then, for any a ∈ (0, a0 ) we have that    R2 √ a dR 1 ˜ √ √ = . (3.1) K(a) =K cn (1 − R) R Rn−1 − Rn − a R1 n−1

The remark follows by making the substitution r = r(t) in the integral that defines the function K in the expression given in Eq. (2.2), and then, doing the substitution R = r2 . We will do these two substitutions in the proof of Lemma 3.3. About K, we have Fig. 3.1. From now on we will abuse the notation and use ˜ K(a) to denote the function K(a). The same notation will be made with other functions that depend on c. We now study the n-area of the fundamental portion of Mc . Lemma 3.3. The function F A(c) given in (2.3) is differentiable on (c0 , ∞) and   √ √ (n − 1) n−1 2 π Γ n2 lim F A(c) = σn−1 2 π lim F A(c) = σn−1  n+1  , n c→c0 c→∞ n2 Γ 2  ∞ x−1 −t where Γ (x) = 0 t e dt. In particular the limit when c goes to infinity of the function F A(c) is 4π 2 2 when n = 2, is 2π when n = 3, is 8π3 when n = 4, and so on (see Fig. 3.2).

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245

Fig. 3.2.√The three images show the graph of F A as a function of a when n = 2, 3 and 4 respectively. In each case the horizontal line √ √ σn−1 π Γ ( n ) 2 and σn−1 2 π a0 are shown. These graphs were obtained by numerically solving an integral using Mathematica 8. at Γ ( n+1 ) 2

Proof. By doing first the substitution r = r(t) and then R = r2 we obtain that,  r2 n−1  R2  T2 3 r Rn− 2 dR F A(c)  √ rn−1 (t)dt = 2 dr = , =2 σn−1 Rn−1 − Rn − c−n q(r) 0 r1 R1

(3.2)

where R1 = r12 and R2 = r22 can also be viewed as the only two roots between 0 and 1 of the polynomial p(R) = Rn−1 − Rn − c−n .

(3.3)

As it was done earlier, let us denote n−1 (n − 1)n−1 and R0 = . (3.4) nn n Notice that a moves from a0 to 0 when c moves from c0 to ∞. A direct computation shows that the polynomial p given in (3.3) when a = a0 (or equivalently, when c = c0 ) has a zero at R = R0 with multiplicity 2, therefore the Corollary 4.2 in [14] does apply. A direct computation shows that when a = a0 a = c−n

and a0 =

p(R0 ) = p′ (R0 ) = 0

and p′′ (R0 ) = −

(n − 1)n−2 . nn−3

Therefore using Corollary 4.2 in [14] we get that  n−1  2n−3 n−1 2 √ √ F A(c) √ (n − 1) 2 n lim = 2a0 π. = 2π  n  12 = 2 π c→c0 σn−1 2 n−2 n (n−1) nn−3

Therefore the first part of the lemma follows. The second part is a direct computation. We have   √  1 n−2 π Γ n2 F A(c) F A(c) R 2   .  √ = lim = lim = c→∞ σn−1 a→0 σn−1 Γ n+1 1−R 0 2 Lemma 3.4. For any n let us define f : [0, a0 ] → R given by f (a) = 2π

F A(a) . K(a)

(3.5)

If M ∈ M R(n), that is, if M is a compact minimal rotational hypersurface in S n+1 with non-constant principal curvatures, then, the n-area of M , denoted by |M | is equal to f (a) r for some integer r greater than 1, and some a ∈ (0, a0 ). Moreover we have (cf. Fig. 3.3),       n−1 1  n−1  lim f (a) = S × S1 √  a→a0  n n 

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Fig. 3.3. The three images show the graph of f as a function of a when n = 2, 3 and 4 respectively. In each case the horizontal line √ Γ( n ) 2 π Γ ( n+1 and |M0 (n)| are shown. These graphs were obtained by numerically solving integrals using Mathematica 8. )

at 2 σn−1

2

and lim f (a) = 2 σn−1

a→0

√

  Γ n2 π  n+1  . Γ 2

Proof. As explained in Section 2, we have that if M ∈ M R(n) then M = Mc for some c ∈ (c0 , ∞) and K(c) = 2π rs for some pair of relatively prime integers r and s. By Lemma 3.1 we have that r cannot be 1. Moreover, since r and s are relatively prime numbers, then M is the union of exactly s rigid motions of the fundamental portion F P (c). Therefore the n-dimensional area of Mc is given by |Mc | = s F A(c) =

2πr F A(c) = f (c) r. K(c)

The other part of the lemma follows from Lemmas 3.1 and 3.3.



Theorem 3.5. For any fixed n, the set A = {|M | : M ∈ M R(n)} is discrete. Proof. By Lemma 3.4 we have that the function f : [0, a0 ] → R is continuous. Since f is positive, then, there exists a positive number m such that f (a) ≥ m for all a ∈ [0, a0 ]. We will prove the theorem by showing that for any positive r0 , the number of hypersurfaces in M R(n) with n-area less than r0 m is finite. Given r0 > 1, for any positive integer r < r0 , let us define r  r Br = : K(a) = 2π for some integer k and a ∈ (0, a0 ) k  kr √  = : 2 r < k < 2r for some integer k and a ∈ (0, a0 ) . k r=[r ]

In the last equality we have used Lemma 3.1. Let B = ∪r=2 0 Br . Clearly each set Br and B has finitely many elements. We will prove that there are as many hypersurfaces in M R(n) with area less than mr0 as elements in B. Let us assume that Mc ∈ M R(n) has area less than r0 m. Since Mc is compact then K(c) = 2π kr with k and r a pair of relatively prime numbers. As pointed out in the proof of Lemma 3.4, |Mc | = f (c)r, since we are assuming that |Mc | < mr0 and f (c) ≥ m then we conclude that r < r0 and therefore kr ∈ Br . Therefore, there are as many hypersurfaces in M R(n) with area less than mr0 as elements in B. Notice that once we know that there are finitely many hypersurface with n-area less than mr0 we can prove that the set of n-areas A is discrete by contradiction. If there is a sequence of areas bi ∈ A that converges to a point b, then we can pick r0 such that b < mr0 , getting a contradiction with the fact that there are finitely many elements in A less than r0 m. 

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247

4. Lower bound for the n-area of minimal rotational hypersurfaces In this section, we study the second value for the n-area functional among hypersurfaces in compact minimal rotational hypersurfaces in S n+1 . Fig. 3.3, indicates that for n = 2, 3 and 4, the area of any element in M R(n) is not only greater than the area of M0 (n) (which is the value of the function f at a0 ) but also greater than twice the area of M0 (n). Recall that by Lemma 3.4, the area of any hypersurface in M R(n) is at least twice f (a) for some a ∈ (0, a0 ). The graphs in Fig. 3.3 show that the minimum of f is |M0 (n)| for n = 2, 3, 4. Motivated by this observation, we pose the following conjecture. Conjecture. The n-area of any compact minimal rotational hypersurface in S n+1 with non-constant principal curvatures is greater than 2|M0 (n)|. Notice that for a given n, the conjecture will follow if we can show that the function f is decreasing because in this case the minimum of f will be |M0 (n)|. We can compute the limit when n goes to infinity of the ratio between the initial value of the function f, f (0), and the final value f (a0 ) = |M0 (n)|. We have    Γ n2 2e f (0)   = lim √ √ ≈ 1.31549. = lim n→∞ n→∞ f (a0 ) π a0 πΓ n+1 2

Remark 4.1. Fig. 4.1 shows the graphs of the function conjecture holds true for these values of n.

1 2π f

(4.1)

for n = 2, . . . , 20. This image shows that the

The next theorem shows that for n = 3, the 3-area of any non-totally geodesic and minimal rotational compact hypersurface is greater than the 3-area of the 3 dimensional minimal Clifford M0 (3). Theorem 4.2. If M is a compact minimal rotational hypersurface in S 4 with non-constant principal curvatures, then       1  2 |M | > S 2 × S 1 √ . 3 3

Proof. Even though n = 3 in this theorem, we will use n instead of 3 to emphasize that some computations work for any n. Let Mc be a compact minimal rotational hypersurface in S 4 with non-constant principal curvatures. By Lemma 3.4 we have that |Mc | = f (a) r with r ≥ 2. Recall that, also by Lemma 3.4, we have that  √ √ 4 16 3 2 f (a0 ) = 2 σ2 a0 π = 2 × 4π × π= π 27 9  is the 3-area of S 2 ( 23 ) × S 1 ( √13 ). Therefore, we only need to show that 2f (a) > f (a0 )

for any a ∈ (0, a0 ).

(4.2)

Using the definition of the function f (a) and the expression found for F A(a) in the proof of Lemma 3.3 we have that the last equation is equivalent to the following equation, (here n = 3) √  R2  R2 √ √ Rn−1 dR a dR √ √ √ √ 4πσn−1 > σn−1 a0 2πK(a) = σn−1 a0 2π n−1 n n−1 − Rn − a R R −R −a R1 R1 (1 − R) R R

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and this last inequality is equivalent to 

√

2 R1



 R2 √ √ 3 a0 a dR Rn− 2 dR √ √ , > n−1 n R −R −a R (1 − R) Rn−1 − Rn − a R1 √ √ 2Rn−1 (1 − R) − a0 a dR √ √ > 0. (1 − R) R Rn−1 − Rn − a

R2

R2

R1

(4.3) (4.4)

In the previous three inequalities we have used the fact that after doing two substitutions for the integral that defines K(c), (in the same way that we did for the expression F A(c)) we obtain that for any positive a smaller than a0 , √  R2 a dR √ √ K(a) = . n−1 − Rn − a R1 (1 − R) R R The inequality (4.3) holds true for any a > a40 because if p(R) = Rn−1 −Rn −a, then, we know that p(R) > 0 on (R1 , R2 ) and √ √ √ √ √ √ √ 2Rn−1 (1 − R) − a0 a = 2(p(R) + a) − a a0 = 2p(R) + a(2 a − a0 ) > 0. Therefore, we have proven the inequality (4.2) for all a ≥ a40 . Notice that the right hand side of inequality √ √ (4.3) is a0 K(a) and we know that K(a) < 2π, therefore, in order to prove the inequality (4.2) for all a < a40 , we will prove the equivalent inequality (4.3) by showing that 

√

R1

Notice that R1 < R0 = assume that

n−1 n

3

R2

2

√ Rn− 2 dR > 2a0 π. Rn−1 − Rn − a

(4.5)

< R2 < 1 are root of the equation Rn−1 − Rn − a = 0. Therefore we may

a = R1n−1 − R1n

where 0 < R1 < R0 , if we want 0 < a < a0

1 1 a will move from 0 to a40 = 27 when R1 moves from 0 to the first positive root of the equation R2 −R3 − 27 = 0. ⋆ This root can be computed explicitly and will be denoted by R1 , it is near 0.217568. For any R1 we can explicitly compute R2 due to the fact that

R2 − R3 − a = R2 − R3 − (R12 − R13 ) = (R − R1 )(R − R2 + R1 − RR1 − R12 ) and the only positive root of the polynomial R − R2 + R1 − RR1 − R12 is    1 R2 = 1 − R1 + 1 + 2R1 − 3R12 . 2 Let us define the function 

3

R2

√

w(a) = 2 R1

It is evident that w(a) < 2

 R2 R1



Rn− 2 dR =2 Rn−1 − Rn

√R

2n−3 2

dR . Rn−1 −Rn −a

R2

2 R1



R2

R1

3

R 2 dR √ R2 − R3

for any a ∈ (0, a0 ).

The function w(a) can be explicitly obtained since

√ √   R dR 2 √ = 2 arcsin R − 2 R(1 − R)|R R1 1−R

and therefore, if R2 is defined by Eq. (4.6), then         w(a) = 2 arcsin R2 − arcsin R1 − R2 (1 − R2 ) + R1 (1 − R1 ) .

(4.6)

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249

Fig. 4.1. The image shows the graphs of the function 2πσ1n−1 f . Since the minimum of each graph is reached at the end point a0 , the image indicates that the n-area of all element in M R(n) is greater than 2|M0 (n)| for n = 2 . . . , 20.

A direct computation shows that w(a) is decreasing as a function of R1 . Therefore for any R1 ∈ (0, R1⋆ ) (or equivalently, for any a ∈ (0, a40 )) we have that  R2 3 √ Rn− 2 dR √ 2 > w(a) = w(R1 ) > w(R1⋆ ) ≈ 2.19961 > 2a0 π ≈ 1.71007. Rn−1 − Rn − a R1 Therefore inequality (4.5) holds true and the theorem follows.



We next give all the graphs for 2 ≤ n ≤ 20. Acknowledgments The authors would like to express their thanks to the referees for their valuable comments and suggestions. The first author would like to thank the Department of Mathematics of South Normal China University for its hospitality during his visit. The second author was supported by NSFC (Grant No. 11371150) and the project of Pearl River New Star of Guangzhou (Grant No. 2012J2200028). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

B. Andrews, H. Li, Embedded constant mean curvature tori in the three-sphere, J. Differential Geom. 99 (2015) 169–189. S. Brendle, A sharp bound for the area of minimal surfaces in the unit ball, Geom. Funct. Anal. 22 (2012) 621–626. S. Brendle, Embedded minimal tori in S 3 and the Lawson conjecture, Acta Math. 211 (2013) 177–190. S.Y. Cheng, P. Li, S.T. Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984) 1033–1065. S.S. Chern, M. do Carmo, S. Kobayashi, Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length. 1970 Functional Analysis and Related Fields, Springer, New York, pp. 59–75. Q. Ding, Y.L. Xin, On Chern’s problem for rigidity of minimal hypersurfaces in the spheres, Adv. Math. 227 (2011) 131–145. M. do Carmo, M. Dajczer, Hypersurfaces in space of constant curvature, Trans. Amer. Math. Soc. 277 (1983) 685–709. H.B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. 89 (1969) 187–197. F.C. Marques, A. Neves, Min-Max theory and the Willmore conjecture, Ann. of Math. 179 (2014) 683–782. T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970) 145–173. T. Otsuki, On integral inequalities related with a certain non linear differential equation, Proc. Japan Acad. 48 (1972) 9–12.

250

[12] [13] [14] [15] [16]

O.M. Perdomo, G. Wei / Nonlinear Analysis 125 (2015) 241–250

T. Otsuki, On a differential equation related with differential geometry, Mem. Fac. Sci. Kyushu Univ. 47 (1993) 245–281. C.K. Peng, C.L. Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983) 105–113. O. Perdomo, Embedded constant mean curvature hypersurfaces on spheres, Asian J. Math. 14 (2010) 73–108. O. Perdomo, Rotational surfaces in S 3 with constant mean curvature, Preprint. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968) 62–105.