Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $\tau_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle $\tilde{\tau}_{3,1}$. In the present paper we show in Theorem 1 that this three-parametric family $T_{a,b,c}$ includes in fact all bipolar Lawson tau-surfaces $\tilde{\tau}_{r,m}$. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for $\tilde{\tau}_{3,1}$ and the equilateral torus.


Introduction
Let M be a closed surface and g be a Riemannian metric on M. Let us consider the associated Laplace-Beltrami operator ∆ : C ∞ (M) → C ∞ (M), The spectrum of ∆ is non-negative and consists only of eigenvalues where each eigenvalue has a finite multiplicity and the associated eigenfunctions are smooth. Denote the eigenvalues of ∆ by 0 = λ 0 (M, g) < λ 1 (M, g) ≤ λ 2 (M, g) ≤ λ 3 (M, g) ≤ ..., where eigenvalues are written with multiplicities.
Let us fix the surface M and consider Λ i (M, g) as a functional g → Λ i (M, g) on the space of all Riemannian metrics on M. The eigenvalues possess the following rescaling property, To get scale-invariant functionals on the space of Riemannian metrics one has to normalize the eigenvalue functionals. It is most natural to normalize the functionals by multiplying by the area, The functionals Λ i (M, g) are invariant under the rescaling transformation g → tg.
If we consider the functional Λ i (M, g) over the space of Riemannian metrics g with a fixed surface M, the question about the value of supremum supΛ i (M, g) is interesting. It is a very difficult question with a limited number of known results. It follows from Yang and Yau [21] and Korevaar [14] that this supremum is finite.
where the supremum is taken over the space of Riemannian metrics g on the fixed surface M.
A problem in the study of Λ i (M, g)-maximal metrics is that the functional Λ i (M, g) depends continuously on metric g but is not differentiable. However, for any analytic deformation g t , the left and right derivatives of the functional Λ i (M, g) with respect to t exist (see Bando and Urakawa [1], Berger [2], El Soufi and Ilias [6]).
Definition ( [5], [6], [16]). A Riemannian metric g 0 on a closed surface M is called an extremal metric for the functional Λ i (M, g) if for any analytic deformation g t the following inequality holds, The mentioned metrics above are extremal since they are global maxima. However, extremal metrics are not necessarily maximal. El Soufi and Ilias proved in [5] that the only extremal metric for Λ 1 (T 2 , g) different from the maximal one is the metric on the Clifford torus.
Jakobson, Nadirashvili and Polterovich proved in [9] that the metric on the Klein bottle realized as the bipolar Lawson surfaceτ 3,1 is extremal for Λ 1 (K, g). Using this result El Soufi, Giacomini and Jazar proved in [4] that this metric is the unique extremal metric.
The following extremal metrics on families of tori and Klein bottles were investigated recently: Lapointe investigated metrics on bipolar Lawson surfaces τ r,m S 4 in his 2008 paper [15], these surfaces are described below in Section 2; Penskoi investigated extremal metrics on Lawson surfaces τ m,n S 3 and on Otsuki tori O p q S 3 in his 2012 paper [18] and 2013 paper [19] respectively; Karpukhin investigated metrics on bipolar Otsuki toriÕ p q S 4 and on a family of tori M m,n S 5 in his 2013 papers [12] and [11] respectively; and Karpukhin proved that the metrics on τ m,n ,τ r,m , O p q ,Õ p q , and M m,n are not maximal except metrics on M 1,1 (the equilateral torus) andτ 3,1 in his 2013 paper [10]. Here denotes an immersion.
where K(·) and E(·) are complete elliptic integrals of the first and second kind respectively as defined [7] by: Then the following statements hold: is a minimal compact surface in the 5dimensional sphere (S 5 ).
2) The case b) corresponds to Lawson tau-surfaces denotes the integer part.
The corresponding value of the functional is

4) In the case a) for an integer
Hence, it is sufficient to consider non-negative integers a, b, c satisfying conditions a) such that It was remarked in [20] thatτ 3,1 is isometric to T 1,0,2 but this proof was indirect and based on the uniqueness of the extremal metric for the first eigenvalue of ∆ on the Klein bottle, proved in [4].
The goal of the present paper is to show that in fact all bipolar Lawson surfacesτ r,m are isometric to some T a,b,c . The main result is the following theorem.

Theorem 2.
(1) If rm ≡ 0 (mod 2) then the bipolar Lawson torusτ r,m is isometric to the surface T a,b,c where a = r − m, b = 0, c = r + m.

Construction of bipolar Lawson surfaces
Let us now recall the construction of bipolar Lawson surfaceτ r,m following Lapointe's paper [15]. The explicit formula forĨ = I ∧ I * : R 2 → S 5 ⊂ R 6 is theñ −m sin v cos v r sin v cos v −r cos 2 v sin mu cos ru − m sin 2 v sin ru cos mu r cos 2 v cos mu sin ru + m sin 2 v cos ru sin mu −r cos 2 v sin mu sin ru + m sin 2 v cos ru cos mu r cos 2 v cos mu cos ru − m sin 2 v sin ru sin mu It is known thatτ r,m actually lies in S 4 , seen as an equator of S 5 .
In [15] Lapointe proved that for the bipolar surfaceτ r,m of a Lawson torus or Klein bottle τ r,m , (1) If rm ≡ 0 (mod 2),τ r,m is a torus with an extremal metric for Λ 4r−2 .
The value of functional Λ i (τ r,m ) can be calculated as follows [15].

Case rm ≡ 0 (mod 2)
Let us prove that the bipolar Lawson surfaceτ r,m when rm ≡ 0 (mod 2) is isometric to the surface T a,b,c where a = r − m, b = 0, and c = r + m.
The induced metric g on T a,b,c is given by the formula [20] Set b = 0 and apply the change of variable sin y = sn(z, k), where k = a √ a 2 −c 2 and sn(z, k) is a Jacobi elliptic function [7]. This implies Let us recall that the bipolar Lawson surfaceτ r,m has the metric [15] Begin by setting r = a+c 2 , m = c−a 2 to rewrite the metric ofτ r,m as Similarly, apply the change of variable sin v = sn(w,k), wherek = 2 √ ac a+c arriving at the metric g = ((a + c) 2 − 4ac sn 2 (w,k)) 2 + (c 2 − a 2 ) 2 (a + c) 2 − 4ac sn 2 (w,k) The task is now to find the change of variable between metrics (1) and (2). Let us use the following transformation, Then we obtain To continue, let k ′ = √ 1 − k 2 . We use the following identities ([7], 13.22-23), .
Let us now apply these identities to (4) and simplify, When rm ≡ 0 (mod 2), a = r − m and c = r + m are both odd since (r, m) = 1. We have that T a,b,c is a torus andF a,b,c : R 2 /L → T a,b,c is a one-to-one map (Theorem 1). Apply change of variable sin y = sn(z, k), and we alternatively haveF a,b,c : R 2 /L → T a,b,c , whereL = {(2nπ, 4mK(k))|n, m ∈ Z}.
There is now a one-to-one correspondence between the rectangle [0, 2π) × [K(k), 5K(k)) and T a,b,c . Our linear transformation (3) maps this rectangular domain as follows: Let us remark that when rm ≡ 0 (mod 2) and after change of variable sin v = sn(w,k), the bipolar Lawson torusτ r,m has a one-to-one correspondence with [0, 2π) × [0, 2K(k)).
Thus we obtained the desired isometry.
Using the values Λ 2(a+c)−2 (T a,b,c ) and Λ 4r−2 (τ r,m ) from [20] and [15], which are twice the areas of these surfaces, we can check that the areas of corresponding surfaces are the same. Note that S(a, b, c) = S(b, a, c) since T b,a,c ∼ = T a,b,c .

Case rm ≡ 1 (mod 4)
Let us prove that the bipolar Lawson surfaceτ r,m when rm ≡ 1 (mod 4) is isometric to the three-parametric surface T a,b,c where a = r−m 2 , b = 0, and c = r+m 2 .
As before, after change of variable sin y = sn(z, k), the induced metric g on T a,b,c is given by the formula g = 1 2 (c 2 − a 2 + 2a 2 sn 2 (z, k)) dx 2 + dz 2 c 2 − a 2 .
The change of variable between metrics (5) and (6) is Let us remark that after change of variable sin v = sn(w,k), the bipolar Lawson Klein bottleτ r,m has a one-to-one correspondence with [0, π 2 )×[0, 2K(k)).
Thus we obtained the desired isometry.
Finally, we can again use the values Λ a+c−2 (T a,b,c ) and Λ r−2 (τ r,m ) to check that the areas of the corresponding surfaces are the same. This completes the proof.