Extrinsic diameter of immersed flat tori in the 3-sphere II

TL;DR

This paper proves that under specific topological conditions, the extrinsic diameter of immersed flat tori in the 3-sphere is exactly π, based on properties of their asymptotic curves in the Hopf fibration.

Contribution

It establishes a precise diameter bound for immersed flat tori in the 3-sphere under certain topological assumptions, extending previous geometric results.

Findings

Extrinsic diameter of flat tori is π under given conditions

Topological conditions relate to asymptotic curves and Hopf fibration

Provides geometric bounds for immersed flat tori in S^3

Abstract

We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $\pi$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.

Equations

$$\gamma (s+l)=\gamma (s)(s\in \mathbb{R}).$$

$$\underset{t\in \mathbb{R}}{\min }{\kappa }_1(t)>\mu >\underset{t\in \mathbb{R}}{\max }{\kappa }_2(t).$$

$$⟨x,y⟩:=-\frac{1}{2}\mathrm{trace}(xy),x\times y:=\frac{1}{2}[x,y],$$

$$e_1:=\left[\begin{array}{cc}0& \sqrt{-1}\\ \sqrt{-1}& 0\end{array}\right],e_2:=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],e_3:=\left[\begin{array}{cc}\sqrt{-1}& 0\\ 0& -\sqrt{-1}\end{array}\right],$$

$$e_1\times e_2=e_3,e_2\times e_3=e_1,e_3\times e_1=e_2.$$

$$S^2:=\{x\in \mathfrak{su}(2):|x|=1\},$$

$$U{S}^2:=\{(x,y):|x|=|y|=1,⟨x,y⟩=0\},$$

$$p_2(a):=(Ad(a)e_3,Ad(a)e_1).$$

$$\kappa (s):=\frac{\det (\gamma (s),{\gamma }^{\prime }(s),{\gamma }^{\prime \prime }(s))}{|{\gamma }^{\prime }(s){|}^3},$$

$$n(s):=\frac{\gamma (s)\times \gamma {}^{\prime }(s)}{|\gamma {}^{\prime }(s)|}.$$

$$\kappa (s)=\frac{⟨\gamma {}^{\prime \prime }(s),n(s)⟩}{|\gamma {}^{\prime }(s){|}^2}$$

$$n^{\prime }(s)=-\kappa (s){\gamma }^{\prime }(s)$$

$$\hat{\gamma }(s):=\left(\gamma (s),\frac{\gamma {}^{\prime }(s)}{|\gamma {}^{\prime }(s)|}\right).$$

$$p_2\circ c_{\gamma }(s)=\hat{\gamma }(s).$$

$$c_{\gamma}(s)^{-1}c^{\prime}_{\gamma}(s)=\frac{|\gamma{\ {}^{\prime}}(s)|}{2}\biggl{(}e_{2}+\kappa(s)e_{3}\biggr{)}.$$

$$f_{{\gamma }_1,{\gamma }_2}(s_1,s_2):=c_1(0{)}^{-1}{c}_1(s_1)c_2(s_2{)}^{-1}{c}_2(0).$$

$$d{s}^2={d{s}_1}^2+2\cos \omega d{s}_{1}d{s}_2+{d{s}_2}^2,h=2\sin \omega d{s}_{1}d{s}_2,$$

$$\omega (s_1,s_2):={\cot }^{-1}\left({\kappa }_1(s_1)\right)+{\cot }^{-1}\left(-{\kappa }_2(s_2)\right).$$

$$H_{{\gamma }_1,{\gamma }_2}(s_1,s_2)=\frac{1+{\kappa }_1(s_1){\kappa }_2(s_2)}{{\kappa }_1(s_1)-{\kappa }_2(s_2)}.$$

$$\omega (s_1,s_2)=\pi -{\alpha }_1(s_1)+{\alpha }_2(s_2).$$

$${\alpha }_1({\gamma }_1(s))={\bar{\gamma }}_1(s),{\alpha }_2({\gamma }_2(s))={\bar{\gamma }}_2(s).$$

$$Ad({\bar{c}}_i(s))e_3$$

$$Ad({\bar{c}}_i(s))e_1$$

$$f_{{\bar{\gamma }}_1,{\bar{\gamma }}_2}(s_1,s_2)={\bar{c}}_1(0{)}^{-1}{\bar{c}}_1(s_1){\bar{c}}_2(s_2{)}^{-1}{\bar{c}}_2(0)=c_1(0{)}^{-1}{c}_1(s_1)c_2(s_2{)}^{-1}{c}_2(0)=f_{{\gamma }_1,{\gamma }_2}(s_1,s_2).$$

$$p_2\circ c_i=\widehat{{\gamma }_i}(\text{mbox}resp.p_2\circ {\bar{c}}_i=\widehat{{\bar{\gamma }}_i}).$$

$${\bar{\gamma }}_i(s)=Ad({\bar{c}}_i(s))e_3=Ad(g_i)Ad(c_i(s))e_3=Ad(g_i){\gamma }_i(s)(i=1,2).$$

$${\bar{\gamma }}_1(s):=-{\gamma }_2(s),{\bar{\gamma }}_2(s):=-{\gamma }_1(s).$$

$${\bar{\kappa }}_1(s)=-{\kappa }_2(s),{\bar{\kappa }}_2(s)=-{\kappa }_1(s),$$

$${\bar{\kappa }}_1(s)=\frac{⟨{{\bar{\gamma }}_1}^{\prime \prime }(s),{\bar{n}}_1(s)⟩}{|{{\bar{\gamma }}_1}^{\prime }(s){|}^2}=\frac{⟨-{{\gamma }_2}^{\prime \prime }(s),n_2(s)⟩}{|{{\gamma }_2}^{\prime }(s){|}^2}=-{\kappa }_2(s).$$

$$Ad(g)e_1=-e_1,Ad(g)e_2=e_2,Ad(g)e_3=-e_3,$$

$$Ad({\bar{c}}_1(s))e_3$$

Full text

The third author was partly supported by the Grant-in-Aid for Scientific Research (A) 262457005, Japan Society for the Promotion of Science.

1. Introduction

Let $S^{3}$ be the unit sphere in the Euclidean $4$-space ${\mathbb{R}}^{4}$. An immersed surface in $S^{3}$ is called flat if its Gaussian curvature vanishes identically. In [2] and [7], the following problem was posed:

Problem. Is the extrinsic diameter of an immersed flat torus in $S^{3}$ equal to $\pi$?

The problem is affirmative under the assumptions that

• (a)

$f$ is an embedding (cf. [2]), or

• (b)

the mean curvature function of $f$ does not change sign (cf. [7]).

If this problem is solved affirmatively, the rigidity of Clifford tori follows, as pointed out in [2]. (Weiner [10] also noticed this reduction.) The references [1, 11, 12] are also related to the problem. To investigate global properties of immersed flat tori, a representation formula given by the second author [4] is useful: In this paper, a closed regular curve means a smooth map $\gamma:{\mathbb{R}}\to S^{2}$ of period $l(>0)$, that is,

[TABLE]

We may think of $\gamma$ as a map defined on ${\mathbb{R}}/l{\mathbb{Z}}$. Consider two closed regular curves $\gamma_{i}:{\mathbb{R}}/l_{i}{\mathbb{Z}}\to S^{2}$ ($i=1,2$). We denote by $\kappa_{i}(s)$ the geodesic curvature of $\gamma_{i}$ at $s\in{\mathbb{R}}$. We then fix a positive number $\mu$. A pair $(\gamma_{1},\gamma_{2})$ of closed regular curves is said to be $\mu$-admissible if the geodesic curvature of $\gamma_{1}$ is greater than the maximum of that of $\gamma_{2}$, that is,

[TABLE]

The second author showed in [4] that all immersed flat tori can be constructed from $\mu$-admissible pairs (see Section 2 for details). If $f_{\gamma_{1},\gamma_{2}}$ is the immersed flat tori associated to the pair $(\gamma_{1},\gamma_{2})$, then each $\gamma_{i}$ ($i=1,2$) can be interpreted as the projection of a certain asymptotic curve of $f_{\gamma_{1},\gamma_{2}}$ via the Hopf fibration $p:S^{3}\to S^{2}$ (see Remark 2.3 in Section 2). We then solve the problem affirmatively under a certain topological condition for the projections of the asymptotic curves with respect to the Hopf fibration, instead of geometric condition (b) as in the previous paper [7].

To consider the problem, we may assume that $\gamma_{1}$ and $\gamma_{2}$ admit only transversal double points (called crossings), without loss of generality (cf. Proposition 2.17). We denote by $\#(\gamma_{i})$ ($i=1,2$) the number of crossings of $\gamma_{i}$. If either $\#(\gamma_{1})$ or $\#(\gamma_{2})$ is an even number, then the extrinsic diameter of the immersed tori associated with $(\gamma_{1},\gamma_{2})$ is equal to $\pi$ (see Fact 2.14). So we may assume that $\#(\gamma_{1})$ and $\#(\gamma_{2})$ are odd integers. Thus the first non-trivial case happens when $\#(\gamma_{1})=\#(\gamma_{2})=1$. The topological type of such $\gamma_{i}$ ($i=1,2$) are uniquely determined (that is, the image of $\gamma_{i}$ ($i=1,2$) is homeomorphic to that of the eight-figure curve). Let $p:S^{2}\setminus\{(0,0,1)\}\to{\mathbb{R}}^{2}$ be the stereographic projection. Then we can draw such a pair as plane curves via this projection, as in Figure 1.

Unfortunately, even in this particular case, the techniques in the previous paper [7] do not seem to be sufficient. One of the important results of this paper is the following assertion:

Proposition A. Suppose that $(\gamma_{1},\gamma_{2})$ is a $\mu$-admissible pair so that $\#(\gamma_{1})=\#(\gamma_{2})=1$ $($as in Figure 1$)$. Then the immersed flat torus associated to the pair $(\gamma_{1},\gamma_{2})$ has extrinsic diameter $\pi$.

This assertion is a special case of our main theorem (cf. Theorem B in Section 4) on extrinsic diameters of flat tori, which is proved by showing the existence of an admissible bi-tangent $(P,Q)$ of the second kind (cf. Definition 2.15) between $\gamma_{1}$ and $\gamma_{2}$ as in Figure 2 (right) under a certain topological assumption. We remark that our theorem (i.e. Theorem B in Section 4) covers not only the case $\#(\gamma_{1})=\#(\gamma_{2})=1$ (as in Proposition A), but also infinitely many topological types of $\mu$-admissible pairs. For example, if $\#(\gamma_{1})$ and $\#(\gamma_{2})$ are both less than or equal to $3$, then there are, in total, $10$ topological types as possibilities of each $\gamma_{i}$ (cf. Figure10 of Appendix B). However, there are only two exceptional topological types $3_{3},3_{6}$ with 3-crossings. If $\gamma_{1}$ and $\gamma_{2}$ are not of these two types (ignoring their orientation indicated in Figure 10), then the immersed flat torus associated to the pair $(\gamma_{1},\gamma_{2})$ has extrinsic diameter $\pi$ (cf. Corollary 4.5). Such exceptional topological types for $\#(\gamma_{1}),\#(\gamma_{2})\leq 5$ are also discussed in Section 4.

The paper is organized as follows: We recall several fundamental properties of $\mu$-admissible pairs in Section 2. In Section 3, we prove a key fact (cf. Proposition 3.5) for use in the proof of our main theorem, and we prove it in Section 4. At the end of this paper, two appendices are prepared. In the first appendix, we show that the topological type of $\mu$-admissible pairs representing a given flat torus is not uniquely determined, in general. In the second appendix, a table (Figure10) of closed spherical curves within $3$-crossings and a table (Figure11) of closed spherical curves with $5$-crossings having a special property are given.

2. $\mu$-admissible pairs of closed curves

2.1. Kitagawa’s representation formula

Let $\operatorname{SU}(2)$ be the group of all $2\times 2$ unitary matrices with determinant $1$. Its Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ consists of all $2\times 2$ skew Hermitian matrices of trace zero. The adjoint representation of $\operatorname{SU}(2)$ is given by $Ad(a)x:=axa^{-1},$ where $a\in\operatorname{SU}(2)$ and $x\in\mathfrak{s}\mathfrak{u}(2)$. For $x,y\in\mathfrak{s}\mathfrak{u}(2)$, we set

[TABLE]

where [ , ] denotes the Lie bracket on $\mathfrak{s}\mathfrak{u}(2)$. Then $\langle{\ ,\ }\rangle$ is the canonical $Ad$-invariant (positive definite) inner product on $\mathfrak{s}\mathfrak{u}(2)$. By setting

[TABLE]

$\{e_{1},e_{2},e_{3}\}$ gives an orthonormal basis of $\mathfrak{s}\mathfrak{u}(2)$, and it satisfies the following:

[TABLE]

Using this, we can identify $\mathfrak{s}\mathfrak{u}(2)$ with the Euclidean space ${\mathbb{R}}^{3}$. We endow $\operatorname{SU}(2)$ with a Riemannian metric $\langle{\ ,\ }\rangle$ such that $\langle{E_{i},E_{j}}\rangle=\delta_{ij}$, where $E_{i}$ denotes the left invariant vector field on $\operatorname{SU}(2)$ corresponding to $e_{i}$. Then $\operatorname{SU}(2)$ is isometric to the unit 3-sphere $S^{3}$. Henceforth, we identify $S^{3}$ with $\operatorname{SU}(2)$.

Let $S^{2}$ be the unit 2-sphere in $\mathfrak{s}\mathfrak{u}(2)$ defined by

[TABLE]

and let $US^{2}$ be the unit tangent bundle of $S^{2}$. Note that $US^{2}$ is identified with a subset of $\mathfrak{s}\mathfrak{u}(2)\times\mathfrak{s}\mathfrak{u}(2)$ in the usual way, i.e.,

[TABLE]

where the canonical projection $p_{1}:US^{2}\to S^{2}$ is given by $p_{1}(x,y):=x$. Define a map $p_{2}:S^{3}\to US^{2}$ by

[TABLE]

Then the map $p_{2}$ is a double covering satisfying $p_{2}(a)=p_{2}(-a)$.

We consider a regular curve $\gamma:{\mathbb{R}}\to S^{2}$. The geodesic curvature $\kappa(s)$ of $\gamma$ is defined by

[TABLE]

where we regard $\gamma(s)\in{\mathbb{R}}^{3}$ and $\gamma^{\prime}(s):=d\gamma/ds$, $\gamma^{\prime\prime}(s):=d^{2}\gamma/ds^{2}$. The unit normal vector field of $\gamma(s)$ is given by

[TABLE]

Then

[TABLE]

and

[TABLE]

hold. We then consider the curve $\hat{\gamma}(s)$ in $US^{2}$ defined by

[TABLE]

Since $p_{2}:S^{3}\to US^{2}$ is a covering, there exists a regular curve $c_{\gamma}(s)$ in $S^{3}$ such that

[TABLE]

The curve $c_{\gamma}(s)$ is called a lift of the curve $\hat{\gamma}(s)$ with respect to the covering $p_{2}$. (Such a lift is determined up to a sign, that is, $-c_{\gamma}$ is also a lift of $\gamma$.) It follows that

[TABLE]

In particular, $|{\gamma}^{\prime}|^{2}(1+\kappa^{2})$ is equal to $4$ if and only if $s$ is the arc-length parameter of $c_{\gamma}$ with respect to the canonical Riemannian metric of $S^{3}$. Regarding this, we give the following definition of ‘admissible pairs’.

Definition 2.1* ([4], [7]).*

A pair $(\gamma_{1},\gamma_{2})$ of regular curves in $S^{2}$ defined on ${\mathbb{R}}$ is called an admissible pair if it satisfies the following conditions :

• (1)

$|{\gamma_{i}}^{\prime}|^{2}(1+\kappa_{i}^{2})=4\quad\text{for \ }i=1,2$,

• (2)

$\kappa_{1}(s_{1})>\kappa_{2}(s_{2})\quad\text{for all \ }s_{1},s_{2}\in{\mathbb{R}}$,

where $\kappa_{i}$ denotes the geodesic curvature of $\gamma_{i}$.

Let $(\gamma_{1},\gamma_{2})$ be an admissible pair, and let $c_{i}:{\mathbb{R}}\to S^{3}$ be the lift of the curve $\widehat{\gamma_{i}}$ with respect to the covering $p_{2}:S^{3}\to US^{2}$. Using the group structure on $S^{3}$, we define a map $f_{\gamma_{1},\gamma_{2}}:{\mathbb{R}}^{2}\to S^{3}$ by

[TABLE]

Since $p_{2}:S^{3}\to US^{2}$ is a double covering, there are two possibilities for the lift $c_{i}$ of $\hat{\gamma}_{i}$. However, the map $f_{\gamma_{1},\gamma_{2}}$ is independent of the choices of the lifts $c_{1}$ and $c_{2}$. The first fundamental form $ds^{2}$ and the second fundamental form $h$ of $f_{\gamma_{1},\gamma_{2}}$ are given by

[TABLE]

respectively, where

[TABLE]

Since $f_{\gamma_{1},\gamma_{2}}$ is flat, the Gaussian curvature of $f_{\gamma_{1},\gamma_{2}}$ vanishes identically. On the other hand, the mean curvature function $H_{{\gamma_{1},\gamma_{2}}}$ satisfies

[TABLE]

Using the function $\alpha_{i}(s)$ given by $\tan\alpha_{i}(s):=\kappa_{i}(s)\,\,(|\alpha_{i}(s)|<{\pi}/2),$ we obtain

[TABLE]

The following assertion holds:

Fact 2.2** ([4, Theorem 4.3]).**

For a complete flat immersion $f:{\mathbb{R}}^{2}\to S^{3}$ whose mean curvature function is bounded, there exists an admissible pair $(\gamma_{1},\gamma_{2})$ such that $f_{\gamma_{1},\gamma_{2}}$ is congruent to $f$ in $S^{3}$.

Remark 2.3*.*

By (2.7), each $c_{i}$ ($i=1,2$) is an asymptotic curve of $f$. Since $p:=p_{1}\circ p_{2}:S^{3}\to S^{2}$ gives the Hopf fibration, each $\gamma_{i}$ can be considered as the projection of an asymptotic curve of $f$ by $p$.

We next give a necessary and sufficient condition for two given admissible pairs to give the same immersed flat torus: Let $({\gamma_{1},\gamma_{2}})$ and $({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ be admissible pairs. We denote by $({\gamma_{1},\gamma_{2}})\equiv({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ the existence $\alpha_{1},\alpha_{2}\in\operatorname{SO}(3)$ satisfying

[TABLE]

The following assertion holds:

Proposition 2.4**.**

$({\gamma_{1},\gamma_{2}})\equiv({\bar{\gamma}_{1},\bar{\gamma}_{2}})$* if and only if $f_{{\gamma_{1},\gamma_{2}}}=f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}$.*

Proof.

Suppose that $({\gamma_{1},\gamma_{2}})\equiv({\bar{\gamma}_{1},\bar{\gamma}_{2}})$. Since the map $Ad:S^{3}(=\operatorname{SU}(2))\to\operatorname{SO}(3)$ is surjective, there exist $g_{1},g_{2}\in S^{3}$ such that $Ad(g_{i})\gamma_{i}(s)=\bar{\gamma}_{i}(s)$ ($i=1,2$). Let $c_{i}(s)$ be a lift of $\widehat{\gamma_{i}}(s)$ with respect to the covering $p_{2}:S^{3}\to US^{2}$, and let $\bar{c}_{i}(s)$ be a curve in $S^{3}$ defined by $\bar{c}_{i}(s):=g_{i}c_{i}(s)$. Then we have

[TABLE]

This shows that $p_{2}\circ\bar{c}_{i}=\widehat{\bar{\gamma}_{i}}$. So we obtain

[TABLE]

Conversely, we suppose that $f_{{\gamma_{1},\gamma_{2}}}=f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}$. Let $c_{i}$ (resp. $\bar{c}_{i}$) be a curve in $S^{3}$ such that

[TABLE]

Since $f_{{\gamma_{1},\gamma_{2}}}(s,0)=f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s,0)$ and $f_{{\gamma_{1},\gamma_{2}}}(0,s)=f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(0,s)$, we obtain $\bar{c}_{i}(s)=g_{i}c_{i}(s)$ ($i=1,2$), where $g_{i}=\bar{c}_{i}(0)c_{i}(0)^{-1}$. This shows that

[TABLE]

Hence $({\gamma_{1},\gamma_{2}})\equiv({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ holds. ∎

We prove the following:

Proposition 2.5**.**

Let $(\gamma_{1},\ \gamma_{2})$ be an admissible pair, and let

[TABLE]

Then the pair $(\bar{\gamma}_{1},\ \bar{\gamma}_{2})$ is an admissible pair such that

[TABLE]

and there exists $g\in S^{3}$ such that $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})=g^{-1}\left(f_{{\gamma_{1},\gamma_{2}}}(s_{2},s_{1})\right)^{-1}g$. In particular, $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})$ is congruent to $f_{{\gamma_{1},\gamma_{2}}}(s_{2},s_{1})$ in $S^{3}$.

Proof.

Since ${\bar{\gamma}_{1}}^{\prime}(s)=-{\gamma_{2}}^{\prime}(s)$, we obtain $\bar{n}_{1}(s)=n_{2}(s)$. Hence

[TABLE]

By the same way, we obtain $\bar{\kappa}_{2}(s)=-\kappa_{1}(s)$. Hence, $\bar{\kappa}_{1}(s_{1})>\bar{\kappa}_{2}(s_{2})$, and so $({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ is an admissible pair. Let $c_{i}$ be a curve in $S^{3}$ such that $p_{2}\circ c_{i}=\widehat{\gamma_{i}}$ ($i=1,2$). We choose $g\in S^{3}(=\operatorname{SU}(2))$ such that

[TABLE]

and set $\bar{c}_{1}(s):=c_{2}(s)g,\ \bar{c}_{2}(s):=c_{1}(s)g$. Then we have

[TABLE]

This shows that $p_{2}\circ\bar{c}_{1}(s)=\widehat{\bar{\gamma}_{1}}(s)$. By the same way, we obtain $p_{2}\circ\bar{c}_{2}(s)=\widehat{\bar{\gamma}_{2}}(s)$. Hence

[TABLE]

This completes the proof. ∎

2.2. Periodic admissible pairs

and $\mu$-admissible pairs

Let $\gamma:{\mathbb{R}}/l{\mathbb{Z}}\to S^{2}$ be a closed regular curve (cf. (1.1)). In this paper, a closed interval $[a,b](\subset{\mathbb{R}})$ means a proper sub-interval of $S^{1}:={\mathbb{R}}/l{\mathbb{Z}}$ satisfying $|a-b|<l$.

Definition 2.6*.*

An admissible pair $(\gamma_{1},\gamma_{2})$ is called periodic if $\gamma_{1}$ and $\gamma_{2}$ are both periodic, that is, there exist two positive numbers $l_{i}$ ($i=1,2$) such that $\gamma_{i}$ has period $l_{i}$.

Remark 2.7*.*

We dropped the condition (1) of Definition 2.1 to define periodic admissible pairs in the previous paper [7]. So the above definition is stronger than that given in [7].

Proposition 2.8** ([4]).**

The flat immersion $f_{{\gamma_{1},\gamma_{2}}}:{\mathbb{R}}^{2}\to S^{3}$ induced by $({\gamma_{1},\gamma_{2}})$ gives a flat torus if and only if $(\gamma_{1},\gamma_{2})$ is a periodic admissible pair.

Proof.

Suppose that $f_{\gamma_{1},\gamma_{2}}$ gives a torus. Kitagawa [4, Theorem A] showed that asymptotic curves on an immersed flat torus are all closed curves. Since $c_{1},c_{2}$ induced by $\gamma_{1},\gamma_{2}$ are asymptotic curves of $f_{\gamma_{1},\gamma_{2}}$ by (2.7), we have that $c_{1},c_{2}$ are closed, and so $\gamma_{1},\gamma_{2}$ are periodic by (2.4). Conversely, if $\gamma_{1},\gamma_{2}$ are periodic, (2.3) and (2.4) yield that $c_{i}(s+l_{i})=\pm c_{i}(s)$ for $i=1,2$. In particular, we have $c_{i}(s+2l_{i})=c_{i}(s)$ ($i=1,2$), and then $f_{\gamma_{1},\gamma_{2}}$ is doubly periodic by (2.6), proving the assertion. ∎

If $\gamma_{i}$ ($i=1,2$) are periodic, then $\min_{s\in{\mathbb{R}}}\kappa_{i}(s)$ and $\max_{s\in{\mathbb{R}}}\kappa_{i}(s)$ exist. So (2) of Definition 2.1 yields the following:

Proposition 2.9**.**

If $(\gamma_{1},\gamma_{2})$ is a periodic admissible pair, then there exists $\mu\in{\mathbb{R}}$ such that

[TABLE]

The following assertion holds:

Proposition 2.10**.**

In the setting as in (2.11), we may assume $\mu>0$.

Proof.

Let $(\gamma_{1},\gamma_{2})$ be a periodic admissible pair. If $\mu=0$, then we may adjust $\mu$ to be positive satisfying $\min_{s\in{\mathbb{R}}}\kappa_{1}(s)>\mu>0.$ If $\mu<0$, then the $\mu$-admissible pair $(\bar{\gamma}_{1},\bar{\gamma}_{2})$ given by (2.9) has the property (cf. Proposition 2.5)

[TABLE]

where $\bar{\mu}:=-\mu>0$. By replacing $(\gamma_{1},\gamma_{2})$ by $(\bar{\gamma}_{1},\bar{\gamma}_{2})$, we get the conclusion. ∎

If $(\gamma_{1},\gamma_{2})$ is a periodic admissible pair, then its parameter $s$ must satisfy (1) of Definition 2.1. To remove this, we prepare the following terminology:

Definition 2.11*.*

A pair of closed regular curves $(\gamma_{1},\gamma_{2})$ is said to be $\mu$-admissible if it satisfies (2.11) for positive $\mu$.

By definition, a periodic admissible pair satisfying (2.11) with $\mu>0$ is $\mu$-admissible. Conversely the following assertion holds:

Proposition 2.12**.**

Let $(\gamma_{1},\gamma_{2})$ be a $\mu$-admissible pair. Then after suitable changes of parameters of $\gamma_{i}$ ($i=1,2$), $(\gamma_{1},\gamma_{2})$ gives a periodic admissible pair.

Proof.

Suppose that the parameter $t$ of $\gamma_{i}(t)$ ($i=1,2$) is general, perhaps not satisfying (1) of Definition 2.1. Since $\gamma_{i}$ is periodic, there exists $d_{i}(>0)$ such that

[TABLE]

We set

[TABLE]

Since $s_{i}:{\mathbb{R}}\to{\mathbb{R}}$ is a bijection satisfying $ds_{i}/dt>0$, we can choose $s_{i}$ as a new parameter of $\gamma_{i}$. Then $|\gamma^{\prime}_{i}(s_{i})|\sqrt{1+\kappa_{i}(s_{i})^{2}}=2$ and $\gamma_{i}(s_{i}+l_{i})=\gamma(s_{i})$ hold, where $l_{i}:=s_{i}(d_{i})$ ($i=1,2$). ∎

2.3. The regular homotopical invariant $I$

of closed spherical curves

For a closed regular curve $\gamma(t)$ in $S^{2}$ of period $l$, we have shown the existence of a new parameter $s$ so that

[TABLE]

as seen above. Then we set

[TABLE]

Definition 2.13*.*

A closed regular curve $\gamma$ is said to be generic if all of its self-intersections are transversal double points. Moreover, a $\mu$-admissible pair $(\gamma_{1},\gamma_{2})$ is said to be generic if each $\gamma_{i}$ ($i=1,2$) is generic.

If $\gamma$ is generic, the number of crossings (i.e. self-intersections) is finite, and we denote it by $\#(\gamma)$. Then it is well-known that

[TABLE]

Suppose that $(\gamma_{1},\gamma_{2})$ is a periodic admissible pair, and each $\gamma_{i}$ has the period $l_{i}(>0)$, and $I(\gamma_{1})=1$. Then

[TABLE]

Similarly, $I(\gamma_{2})=1$ also implies that $f_{{\gamma_{1},\gamma_{2}}}(s_{1},s_{2}+l_{2})=-f_{{\gamma_{1},\gamma_{2}}}(s_{1},s_{2}).$ Consequently, we get the following:

Fact 2.14**.**

The extrinsic diameter of $f_{\gamma_{1},\gamma_{2}}$ is equal to $\pi$ if $I(\gamma_{1})=1$ or $I(\gamma_{2})=1$.

Thus, to prove Proposition A, we may assume that $I(\gamma_{1})=I(\gamma_{2})=0$.

Let $\gamma_{i}:{\mathbb{R}}/l_{i}{\mathbb{Z}}\to S^{2}$ ($i=1,2$) be two closed regular curves of period $l_{i}(>0)$ in $S^{2}$ with unit normal vector fields $n_{i}$ along $\gamma_{i}$. Let $a_{i},b_{i}$ $(0<a_{i}<b_{i}<l_{i})$ be real numbers satisfying

[TABLE]

If it holds that

[TABLE]

or

[TABLE]

then the pair of points $(P,Q)$ on $S^{2}$ is called a bi-tangent between $\gamma_{1}$ and $\gamma_{2}$. Moreover, if the former case (i.e. (2.15)) happens, then the bi-tangent is said to be admissible.

Definition 2.15*.*

We set $\check{\gamma}_{i}:=(\gamma_{i},n_{i}):{\mathbb{R}}\to US^{2}\,\,(i=1,2).$ An admissible bi-tangent $(P,Q)$ is called of the first kind if the restriction $\check{\gamma}_{i}|_{[a_{1},b_{1}]}$ on the interval $[a_{1},b_{1}]$ is regular homotopic to $\check{\gamma}_{2}|_{[a_{2},b_{2}]}$ keeping the properties (2.14) and (2.15). If $(P,Q)$ is not of the first kind, it is called of the second kind (see Figure 3 right).

The following assertion was proved in [7, Proposition 2.3]:

Fact 2.16**.**

Let $(\gamma_{1},\gamma_{2})$ be a $\mu$-admissible pair. Then the following two assertions are equivalent.

1. (1)

The flat immersed torus $f_{\gamma_{1},\gamma_{2}}$ associated to $(\gamma_{1},\gamma_{2})$ has extrinsic diameter $\pi$. 2. (2)

There exists an orientation preserving isometry $\varphi$ on $S^{2}$ such that a subarc of $\gamma_{1}$ and a subarc of $\varphi\circ\gamma_{2}$ have an admissible bi-tangent of the second kind.

We have the following reduction for the problem given in the introduction.

Proposition 2.17**.**

The problem in the introduction is solved affirmatively if there exists an admissible bi-tangent of the second kind for each generic $\mu$-admissible pair.

Proof.

This assertion follows from the fact that a non-generic $\mu$-admissible pair can be obtained by the limit of a sequence of generic pairs. So we obtain the conclusion regarding the fact that $f_{\gamma_{1},\gamma_{2}}$ depends on $\gamma_{1},\gamma_{2}$ continuously. ∎

2.4. A tool to control the behavior of $\gamma_{1}$

Let $C$ be an oriented circle in $S^{2}$ of positive geodesic curvature. Then $C$ separates $S^{2}$ into two open regions. The smaller one is called the interior domain and denote it by $\Delta_{C}$. Let $O$ be the center of $C$ in the interior domain of $C$ and $P$ a point on $C$. Then the great circle passing through $P$ and $O$ meets $C$ at the antipodal point $P^{\prime}$ with respect to the circle $C$. Let $[P,P^{\prime}]$ (resp. $[P^{\prime},P]$) be the subarc of $C$ from $P$ to $P^{\prime}$ (resp. $P^{\prime}$ to $P$). Then $C$ is a union of two arcs $[P,P^{\prime}]$ and $[P^{\prime},P]$. We call $[P,P^{\prime}]$ (resp. $[P^{\prime},P]$) the future part (resp. the past part) of the oriented circle $C$ with respect to P. The following assertion was proved in [7], which is a useful tool for investigating the behavior of the curve $\gamma$ whose geodesic curvature is greater than $\mu(>0)$.

Fact 2.18** ([7, Lemma 4.5]).**

Let $C$ be an oriented circle of positive constant geodesic curvature $\mu(>0)$ centered at a point $O$ in $S^{2}$. Let $\gamma:[a,b]\to S^{2}$ be a simple regular arc (i.e. an arc without self-intersections) whose geodesic curvature is greater than $\mu$ for all $t\in[a,b]$. Suppose that $\gamma((a,b))$ lies in $\Delta_{C}$, and the two endpoints $\gamma(a),\,\,\gamma(b)$ both lie on the circle $C$. Then $\gamma(b)$ lies in the past part of $C$ with respect to $\gamma(a)$ $($see Fig 4$)$.

3. Bi-tangent $\mu$-circles inscribed in a shell

We give here the concept of ‘shell’, which is used throughout this paper:

Definition 3.1*.*

A regular arc $\gamma:[a,b]\to S^{2}$ is called a shell if

• •

$\gamma([a,b))$ has no self-intersection,

• •

$\gamma(a)=\gamma(b)$, and

• •

$\gamma^{\prime}(a)$ and $\gamma^{\prime}(b)$ are linearly independent.

The point $\gamma(a)=\gamma(b)$ is called the node of the shell. Moreover, the shell is called positive (resp. negative) if the determinant function satisfies

[TABLE]

Since the image of the shell $\gamma$ is a simple closed curve, it separates $S^{2}$ into two open regions. The domain whose interior angle at the node is less than $\pi$ is called the interior domain of the shell and is denoted it by $\Delta_{\gamma}$. Then the shell $\gamma:[a,b]\to S^{2}$ is positive (resp. negative), if $\Delta_{\gamma}$ lies in the left-hand side (resp. the right-hand side) of $\gamma$.

The classical four vertex theorem assets that there are four vertex (i.e. four critical points of the curvature function) on a given simple closed curve in ${\mathbb{R}}^{2}$. By the stereographic projection, circles in $S^{2}$ are corresponding to circles or lines in ${\mathbb{R}}^{2}$. In particular, the critical points of the geodesic curvature function (resp. the osculating circles) of a given spherical regular curve corresponds to the vertices (resp. the osculating circles) of the corresponding plane curve. An osculating circle at a point on a curve is a circle which has second order contact with the curve at the point. As a refinement of the four vertex theorem, Kneser [5] proved that there are two distinct inscribed osculating circles and two distinct circumscribed osculating circles of a simple closed spherical curve. As an analog of this, the following fact is known (cf. [7, Proposition B.1]):

Fact 3.2** (Strong version of Jackson’s lemma).**

Suppose that a closed regular spherical curve $\gamma$ has a shell on a closed interval $[a,b](\subset S^{1})$. Then there exists $c\in(a,b)$ such that the osculating circle $C$ of $\gamma$ at $\gamma(c)$ lies in the closure of the interior domain of the shell $\gamma([a,b])$.

Jackson [3] proved that the geodesic curvature function attains at least one local maximum on a given positive shell. The above fact is a refinement of this assertion.

Throughout this section, we fix a $\mu$-admissible pair $(\gamma_{1},\gamma_{2})$ such that $\gamma_{2}$ has a positive shell, that is, the restriction of $\gamma_{2}$ to a closed interval $[a_{2},b_{2}](\subset{\mathbb{R}}/l_{2}{\mathbb{Z}})$ gives a positive shell. We now fix this to be so, and set

[TABLE]

which is the image of the positive shell.

Definition 3.3*.*

A circle on $S^{2}$ is called a $\mu$-circle if its geodesic curvature is $\mu(>0)$.

Lemma 3.4**.**

There exists a $\mu$-circle which is inscribed in $\Sigma_{2}$ (i.e. the $\mu$-circle lies in the closure of the interior domain of the positive shell) and meets $\Sigma_{2}$ at more than one point.

We call such a $\mu$-circle a bi-tangent circle of $\Sigma_{2}$.

Proof.

By Fact 3.2, we can find an inscribed osculating circle $\Gamma$ which meets $\Sigma_{2}$ at

[TABLE]

We let $\Gamma_{1}$ be the $\mu$-circle which is tangent to $\Sigma_{2}$ at $P_{0}$. Since $\mu$ is greater than the maximum of the geodesic curvature of $\gamma_{2}$, $\Gamma_{1}$ lies in the left-hand side of $\Gamma$. In particular, $\Gamma_{1}$ is a $\mu$-circle inscribed in $\Sigma_{2}$. For each $t\in(a_{2},t_{0}]$, we let $C_{t}$ be the $\mu$-circle which is tangent to $\Sigma_{2}$ at $\gamma_{2}(t)$. Then $\Gamma_{1}=C_{t_{0}}$ holds. We set

[TABLE]

Since $\mu$ is greater than the maximum of the geodesic curvature of $\gamma_{2}$, the circle $C_{t_{1}}$ meets $\Sigma_{2}$ somewhere other than $\gamma(t_{1})$. (Otherwise, we can roll $C_{t_{1}}$ towards the node of the shell, and then we can find $C_{t}$ ($t<t_{1}$) which is inscribed in $\Sigma_{2}$.) So we get the assertion. ∎

For the sake of simplicity, we set $C:=C_{t_{1}}$ and assume that the orientation of $C$ is compatible with the orientation of the positive shell $\Sigma_{2}$. By definition, we have

[TABLE]

We then set

[TABLE]

and

[TABLE]

By definition, it holds that $a_{2}<t_{1}<t_{2}<b_{2}.$ We call the angle of the subarc of $C$ from $P$ to $Q$ the bi-tangent angle of $C$. We denote it by $\theta$ $($see Figure 6$)$. The following assertion plays an important role in the next section:

Proposition 3.5**.**

The bi-tangent angle $\theta$ of the $\mu$-circle $C$ is greater than or equal to $\pi$.

Proof.

We prove the assertion by way of contradiction, so we suppose the bi-tangent angle of $C(:=C_{t_{1}})$ is less than $\pi$. Then we define a $C^{1}$-shell joining

• •

the subarc of $\Sigma_{2}$ from the node $N_{2}$ of the shell $\Sigma_{2}$ to $P$,

• •

the subarc of $C$ from $P$ to $Q$, and

• •

the subarc of $\Sigma_{2}$ from $Q$ to $N_{2}$.

We denote this $C^{1}$-differentiable shell by $\Sigma^{\prime}_{2}$. We let $\mu^{\prime}$ be a number satisfying $\mu>\mu^{\prime}$. Let $C_{1}$ (resp. $C_{2}$) be the $\mu^{\prime}$-circle which is tangent to $\Sigma_{2}$ at $Q$ (resp. at $P$). We denote by $\Delta_{\Sigma^{\prime}_{2}}$ the interior domain bounded by $\Sigma^{\prime}_{2}$. Let $X,Y$ be the intersections of the two circles $C_{1}$ and $C_{2}$. If we take $\mu^{\prime}$ sufficiently close to $\mu$, then one of $\{X,Y\}$ lies in the complement of the closure of the interior of $\Sigma^{\prime}_{2}$ and the other lies in the interior of $\Sigma^{\prime}_{2}$. Without loss of generality, we may assume $X$ lies in $\Delta_{\Sigma^{\prime}_{2}}$. We then define a $C^{1}$-differentiable shell joining

• •

the subarc of $C_{2}$ from $X$ to $P$,

• •

the subarc of $C$ from $P$ to $Q$, and

• •

the subarc of $C_{1}$ from $Q$ to $X$.

We denote by $\Omega$ the interior domain of this newly defined $C^{1}$-differentiable shell. If we choose $\mu^{\prime}$ sufficiently close to $\mu$, then $\Omega\subset\Delta_{\Sigma^{\prime}_{2}}$ holds. We let $Q^{\prime}$ (resp. $P^{\prime}$) be the midpoint of the arc bounded by $X$ and $Y$ on $C_{1}$ (resp. $C_{2}$). Since the bi-tangent angle of $C$ is less than $\pi$, we have $Q^{\prime},P^{\prime}\in\partial\Omega$, where $\partial\Omega$ is the boundary of the domain $\Omega$. Moreover, there exists a unique circle $\Gamma$ which is tangent to $\partial\Omega$ at two points $Q^{\prime},P^{\prime}$ satisfying (cf. Figure 7)

• •

$\Gamma$ lies in $\overline{\Omega}$, and

• •

the geodesic curvature of $\Gamma$ is less than $\mu$ and greater than $\mu^{\prime}$.

We continuously expand $\Gamma$ keeping the property that it is tangent to $C_{1}$ at $Q^{\prime}$, and get a new circle $\Gamma^{\prime}$ such that

• •

the geodesic curvature of $\Gamma^{\prime}$ is less than that of $\Gamma$,

• •

$\Gamma^{\prime}$ lies in $\overline{\Delta_{\Sigma^{\prime}_{2}}\cap\Delta_{C_{1}}}$,

• •

$\Gamma^{\prime}$ is tangent to $\gamma_{2}((a_{2},t_{1}))$ at a certain point $\tilde{P}$.

We then consider the $\mu$-circle $\tilde{C}$ which is inscribed in $\Gamma^{\prime}$ and is tangent to $\Gamma^{\prime}$ at $\tilde{P}$. Since $\Gamma^{\prime}$ is inscribed in $\Sigma^{\prime}_{2}$, so is $\tilde{C}$. Then $\tilde{C}$ must meet $\Sigma^{\prime}_{2}$ only at $\tilde{P}$. Since the image of $\Sigma^{\prime}_{2}$ lies in the closure of the interior domain of the positive shell $\Sigma_{2}$, $\tilde{C}$ is a $\mu$-circle inscribed in $\Sigma_{2}$ which is tangent to $\Sigma_{2}$ only at $\tilde{P}$. This contradicts the definition of $t_{1}$. ∎

4. The main theorem, its proof and applications

We fix a $\mu$-admissible pair $(\gamma_{1},\gamma_{2})$. The second and third authors showed in [7] that the extrinsic diameter of $f_{\gamma_{1},\gamma_{2}}$ is $\pi$ under the assumption that its mean curvature function does not change sign. This assumption corresponds to the condition that

[TABLE]

which is stronger than the condition (1.2). In this case, we can find a positive shell on $\gamma_{1}$ (cf. [7, Prop. 3.7]) and can show the existence of an admissible bi-tangent of the second kind between $\gamma_{2}$ and the positive shell on $\gamma_{1}$. However, under our weaker assumption (1.2), the argument given in [7] is not sufficient to show the following assertion:

Theorem B. Let $(\gamma_{1},\gamma_{2})$ be a generic $\mu$-admissible pair. Suppose that

1. (a)

either $\#(\gamma_{1})$ or $\#(\gamma_{2})$ is even, or 2. (b)

*each of $\#(\gamma_{1}),\#(\gamma_{2})$ is odd, and $\gamma_{1}$ *$($resp. $\gamma_{2})$ has a negative $($resp. positive$)$ shell.

*Then the extrinsic diameter of $f_{\gamma_{1},\gamma_{2}}$ is equal to $\pi$. *

We prove this using a new idea different from the one used in the proof of the main theorem in [7]. Proposition A in the introduction follows from this assertion, since a closed curve having only one crossing has a positive shell and a negative shell at the same time.

Remark 4.1*.*

The main theorem of [7] is independent of Theorem B. In fact, let $\gamma_{1}$ be a closed regular spherical curve of positive geodesic curvature greater than $1$ whose topological type is $3_{6}$ as in Appendix B. Then $\gamma_{1}$ has three independent positive shells but has no negative shells. We let $\gamma_{2}$ be the closed curve obtained by reversing the orientation of $\gamma_{1}$. Then, by definition

[TABLE]

Since $\gamma_{1}$ has no negative shells, $\gamma_{2}$ has no positive shells. So the $\mu$-admissible pair ($\mu=1$) $(\gamma_{1},\gamma_{2})$ does not satisfy the assumption of Theorem B, but satisfies (4.1), and so the extrinsic diameter of $f_{\gamma_{1},\gamma_{2}}$ is $\pi$ by [7, Proposition 4.7].

As pointed out in Fact 2.14, the case (a) is obvious. So, to prove Theorem B, we may assume (b). Then we can find a negative shell $\gamma_{1}|_{[a_{1},b_{1}]}:[a_{1},b_{1}]\to S^{2}$, where $[a_{1},b_{1}]$ is a subarc of ${\mathbb{R}}/l_{1}{\mathbb{Z}}$. We set

[TABLE]

To prove Theorem B, we prepare three lemmas:

Lemma 4.2**.**

There exists a $\mu$-circle $C$ such that $\Sigma_{1}$ is inscribed in $C$ and meets $\Sigma_{1}$ at exactly two points.

We call such a $\mu$-circle a bi-tangent $\mu$-circle of $\Sigma_{1}$.

Proof.

If we reverse the orientation of $\Sigma_{1}$, then it becomes a positive shell. So by the same argument as in Lemma 3.4, we can find a bi-tangent $\mu$-circle $C$ of $\Sigma_{1}$. It is sufficient to show that $C$ meet $\Sigma_{1}$ at exactly two points. If not, there are three points $P_{1},P_{2},P_{3}$ on $C$ in this cyclic order and three values $t_{1},t_{2},t_{3}\in[a_{1},b_{1}]$ $(t_{1}<t_{2}<t_{3})$ such that

[TABLE]

By Fact 2.18, $P_{3}$ (resp. $P_{1}$) lies in the past part (resp. the future part) of $P_{2}$, and so the cyclic order of the three points on $C$ must be $P_{3},P_{2},P_{1}$, a contradiction. ∎

Let $C$ be the $\mu$-circle given in Lemma 4.2. We let $P,Q\in\Sigma_{1}$ be the two bi-tangent points of $C$ on $\Sigma_{1}$ such that

[TABLE]

For each $t\in[d_{1},b_{1}]$, we let $\hat{C}_{t}$ be the $\mu$-circle which is tangent to $\Sigma_{1}$ at $\gamma_{1}(t)$ such that $\gamma_{1}(t^{\prime})$ ($t^{\prime}\neq t$) lies in the interior domain of $\hat{C}_{t}$ whenever $t^{\prime}$ is sufficiently close to $t$ (see Figure 8, left). Since $\gamma_{1}(t)$ lies in the interior domain of $C$, $\hat{C}_{t}$ meets $C$ at exactly two points on $S^{2}$.

Lemma 4.3**.**

Suppse that there exists a point $X=\gamma_{1}(x)$ $(x\in[d_{1},b_{1}]\setminus\{t\})$ which lies on $\hat{C}_{t}$. If $t<x\leq b_{1}$ $($resp. $d_{1}\leq x<t)$, then the subarc $\mathcal{A}$ of $\hat{C}_{t}$ from $\gamma_{1}(t)$ to $X$ (resp. from $X$ to $\gamma_{1}(t))$ meets the circle $C$ twice $($Figure 8, right in the case of $x>t)$.

Proof.

If $x>t$ (resp. $x<t$), then we set

[TABLE]

Then $\gamma_{1}(x_{0})\in\hat{C}_{t}$ holds. We let $\mathcal{A}_{0}$ be the subarc of $\mathcal{A}$ between $\gamma_{1}(t)$ and $\gamma_{1}(x_{0})$. Let $l$ be the length of the $\mu$-circle $\hat{C}_{t}$. Suppose that $\mathcal{A}$ lies in $\overline{\Delta_{C}}$. Since $C$ and $\hat{C}_{t}$ have common radius and $\mathcal{A}\subset\overline{\Delta_{C}\cap\Delta_{\hat{C}_{t}}}$, the length of $\mathcal{A}$ must be less than $l/2$. Since the open subarc of $\gamma_{1}$ between $t$ and $x_{0}$ lies in the interior of $\hat{C}_{t}$, Fact 2.18 yields that $\gamma_{1}(x_{0})$ must lie in the past (resp. the future) part of $\hat{C}_{t}$ with respect to $\gamma_{1}(t)$ if $x_{0}>t$ (resp. $x_{0}<t$). Thus, the length of $\mathcal{A}_{0}$ must be greater than $l/2$. This creates a contradiction, proving the conclusion. ∎

Lemma 4.4**.**

For each $t\in[d_{1},b_{1}]$, $\hat{C}_{t}$ never meets $\gamma_{1}((t,b_{1}])$.

Proof.

Suppose that $\hat{C}_{t}$ meets $\gamma_{1}((t,b_{1}])$ at $X=\gamma_{1}(x)$ ($t<x\leq b_{1}$). Let $\mathcal{A}$ be the subarc of $\hat{C}_{t}$ from $\gamma_{1}(t)$ to $X$. By Lemma 4.3, $\mathcal{A}$ meets $C$ twice. So we denote by $Y\in C$ the point where $\mathcal{A}$ starting from $\gamma_{1}(t)$ meets $C$ the first time. We then denote by $\mathcal{A}^{\prime}$ the subarc of $\mathcal{A}$ from $\gamma_{1}(t)$ to $Y$. Let $\mathcal{B}$ be the subarc of $\hat{C}_{t}$ from $X$ to $\gamma_{1}(t)$. Then

[TABLE]

Let $\mathcal{E}$ be the subarc of $C$ from $P$ to $Y$. We let $\Omega(\subset\overline{\Delta_{C}})$ be a domain bounded by the simple closed curve $\mathcal{A}^{\prime}\cup\gamma_{1}([d_{1},t])\cup\mathcal{E}$ (see Figure 8 right, where $\hat{C}_{t}$ is drawn so that it does not look like a circle). Then any point on $\mathcal{B}$ sufficiently close to $\gamma_{1}(t)$ lies in $\Omega$. Since $X\not\in\overline{\Omega}$, $\mathcal{B}$ must meet the boundary $\partial\Omega$. However

• •

$\mathcal{B}\setminus\{\gamma_{1}(t)\}$ cannot meet $\mathcal{A}^{\prime}$, since $\hat{C}_{t}$ has no self-intersections.

• •

$\mathcal{B}$ cannot meet $C$, since $\mathcal{A}$ meets $C$ twice and the radius of $\hat{C}_{t}$ and $C$ is common.

So there must be a point $Z=\gamma_{1}(z)$ ($z\in(d_{1},t)$) such that $Z\neq\gamma_{1}(t)$ and $Z\in\mathcal{B}$. By Lemma 4.3, $\mathcal{B}$ meets $C$ twice. Then $\hat{C}_{t}(=\mathcal{A}\cup\mathcal{B})$ meets $C$ four times, a contradiction. ∎

Proof of Theorem B.

Let $\Sigma_{2}=\gamma_{2}([a_{2},b_{2}])$ (cf. (3.1)) be the image of the positive shell given in Section 3, and let $C_{t_{1}}$ be the bi-tangent $\mu$-circle inscribed in $\Sigma_{2}$. To find the desired bi-tangent between $\gamma_{1}$ and $\gamma_{2}$, the initial position of $\gamma_{1}$ can be arbitrarily chosen. So we may set the $\mu$-circle $C$ associated with $\Sigma_{1}$ to coincide with $C_{t_{1}}$, and $\Sigma_{1}$ (cf. (4.2)) is tangent to $\Sigma_{2}$ at the point $P$. We set (cf. Figure 9, left)

[TABLE]

Since

• •

the bi-tangent angle of $C$ of $\Sigma_{2}$ is greater than or equal to $\pi$ (cf. Proposition 3.5) and

• •

$P$ is the past part of $Q$ on $C$ (cf. Fact 2.18),

the three points $P,Q,Q^{\prime}$ lie on the circle $C(=C_{t_{1}})$ in this cyclic order. We denote by $[PQ],[QQ^{\prime}]$ and $[Q^{\prime}P]$ the subarcs of $C$ from $P$ to $Q$, $Q$ to $Q^{\prime}$ and $Q^{\prime}$ to $P$, respectively. Consider the $C^{1}$-differentiable simple closed curve given by $\mathcal{S}:=[Q^{\prime}P]\cup\gamma_{2}([c_{2},d_{2}]).$ Let $\Omega$ be the open domain bounded by $\mathcal{S}$ which lies in the closure of the interior domain of $\Sigma_{2}$.

For each $t\in[d_{1},b_{1}]$, there exists a unique orientation-preserving isometry $\varphi_{t}:S^{2}\to S^{2}$ such that

1. (1)

$\varphi_{t}(\gamma_{1}(t))=P$, 2. (2)

$\varphi_{t}(\Sigma_{1})$ is tangent to $\Sigma_{2}$ at $P$, and 3. (3)

$\varphi_{t}(\hat{C}_{t})=C$.

We then set

[TABLE]

Since $d_{1}\in A$, the set $A$ is non-empty. By (1.2), we have $\tau>d_{1}$. By the center of Figure 9, $\tau<b_{1}$ holds. Suppose that $\varphi_{\tau}(\Sigma_{1})$ meets $[Q^{\prime}P]$. We know that $\varphi_{\tau}\circ\gamma_{1}((\tau,b_{1}])$ lies in $\Delta_{C}$ (cf. Lemma 4.4). In particular, it cannot meet $[Q^{\prime}P]$. Since the angle of subarc of $C$ from $Q^{\prime}$ to $P$ is less than or equal to $\pi$ (cf. Proposition 3.5), $\varphi_{\tau}\circ\gamma_{1}([a_{1},\tau])$ also cannot meet $[Q^{\prime}P]$, by Fact 2.18. Since $\varphi_{\tau}\circ\gamma_{1}((\tau,b_{1}])$ lies in $\Delta_{C}$, there exist $c^{\prime}_{1}\in[a_{1},\tau)$ and $c^{\prime}_{2}\in(c_{2},d_{2}))$ such that $\varphi_{\tau}(\Sigma_{1})$ meets $\gamma_{2}((c_{2},d_{2}))$ at the point

[TABLE]

Then $(P,P^{\prime})$ gives an admissible bi-tangent of the second kind (cf. Figure 9, right), since this bi-tangent is exactly the case of Figure 3, right. ∎

As a corollary, we can prove the following, which contains Proposition A as a special case:

Corollary 4.5**.**

Let $(\gamma_{1},\gamma_{2})$ be a $\mu$-admissible pair such that $\#(\gamma_{i})\leq 3$ $(i=1,2)$. Suppose that

• •

$\gamma_{1}$* is not of type $3_{j}$ $(j=3,6)$ $($ignoring their orientation$)$ as in Figure 10 of Appendix B, and*

• •

$\gamma_{2}$* is not of type $3_{j}$ $(j=3,6)$ as in Figure 10 including their orientation.*

Then $f_{\gamma_{1},\gamma_{2}}$ has extrinsic diameter $\pi$.

Proof.

If $\#(\gamma_{1})$ or $\#(\gamma_{2})$ is even, the conclusion is obvious, and the case $\#(\gamma_{1})=\#(\gamma_{2})=1$ was the case of Proposition A. So we need to consider only the case $\#(\gamma_{1})=3$ or $\#(\gamma_{2})=3$. If $\gamma_{i}$ ($i=1,2$) is not of type $3_{j}$ ($j=3,6$), then $\gamma_{i}$ has a positive shell and a negative shell at the same time. So we can find a negative shell on $\gamma_{1}$ and a positive shell on $\gamma_{2}$.

We suppose $\gamma_{1}$ is of type $3_{3}$ or $3_{6}$. Since $\gamma_{1}$ has positive geodesic curvature, $\gamma_{1}$ has a positive shell (cf. [7, Prop. 3.7]), and in this case, $\gamma_{1}$ has no negative shells and does not satisfy the assumption of Theorem B. We next suppose that $\gamma_{2}$ is of type $3_{3}$ or $3_{6}$ with the orientation indicated in Figure 10. Then $\gamma_{2}$ does not admit positive shells and does not satisfy the assumption of Theorem B. (However, if $\gamma_{2}$ is of type $3_{3}$ or $3_{6}$ but the orientation is opposite, then $\gamma_{2}$ has a positive shell.) So we get the conclusion. ∎

There are, in total, $10$ types of spherical curves at most 3-crossings, and only the two cases $3_{3},3_{6}$ are exceptional. Similarly, we can prove the following:

Corollary 4.6**.**

Let $(\gamma_{1},\gamma_{2})$ be a $\mu$-admissible pair such that $\#(\gamma_{i})\leq 5$ $(i=1,2)$. Suppose that

• •

$\gamma_{1}$* is not of type $3_{j}$ ($j=3,6$) as in Figure 10, nor a 5-crossing curve as in one of the 17-curves as in Figure 11 $($ignoring their orientation$)$ of Appendix B, and*

• •

$\gamma_{2}$* is not of types $3_{j}$ ($j=3,6$), nor one of the 5-crossing curves given in the list of Figure 11 of Appendix B including their orientation.*

Then $f_{\gamma_{1},\gamma_{2}}$ has extrinsic diameter $\pi$.

There are, in total, $105$ types of spherical curves with at most 5-crossings, and the exceptional case (i.e. curves that do not admit a positive shell and a negative shell at the same time) are $3_{3},3_{6}$ and the $17$ curves listed in Figure 11 of Appendix B. So the cases for which Theorem B can be applied share

[TABLE]

Since the number of curves with even crossings amongst $105$-curves is $22=1+2+19$. So, if we do not apply Theorem B, we can judge only

[TABLE]

So, at least for spherical curves within 5-crossings, the percentage has extremely increased by Theorem B.

Appendix A Properties of admissible pairs

In this appendix, we show that the assumption of Theorem B does depend on the choice of $\mu$-admissible pair which represents a common flat torus. For this purpose, we prepare a lemma on parallel families of curves consisting of $\mu$-admissible pairs.

Let $\gamma:{\mathbb{R}}\to S^{2}$ be a regular curve. For each $\theta\in{\mathbb{R}}$, the family of curves $\gamma^{\theta}(s)$ ($\theta\in[0,2\pi)$) in $S^{2}$ defined by

[TABLE]

is called the parallel curve of the curve $\gamma(s)$. It follows from (2.2) that

[TABLE]

where $\kappa(s)$ denotes the geodesic curvature of $\gamma(s)$.

Lemma A.1**.**

Let $(\gamma_{1},\gamma_{2})$ be an admissible pair. If the real number $\theta$ satisfies

[TABLE]

for all $s\in{\mathbb{R}}$, then the curve $\gamma_{i}^{\theta}(s)$ is regular and its geodesic curvature $\kappa_{i}^{\theta}(s)$ satisfies the following:

• (1)

${\displaystyle\kappa_{i}^{\theta}(s)=\frac{\sin\theta+\kappa_{i}(s)\cos\theta}{|\cos\theta-\kappa_{i}(s)\sin\theta|}}$,

• (2)

$|(\gamma_{i}^{\theta})^{\prime}(s)|^{2}(1+\kappa_{i}^{\theta}(s)^{2})=4$.

Proof.

Since $|\cos\theta-\kappa(s)\sin\theta|>0$, it follows from (A.2) that the curve $\gamma^{\theta}(s)$ is regular and satisfies the following relation:

[TABLE]

Hence, the unit normal vector field of $\gamma^{\theta}(s)$ is given by

[TABLE]

So, using (2.2), we obtain

[TABLE]

On the other hand, the Frenet formula implies $(n^{\theta})^{\prime}=-\kappa^{\theta}(\gamma^{\theta})^{\prime}.$ Hence the assertion (1) follows from (A.2) and (A.3). Furthermore, the assertion (2) follows from (A.2) and (1). ∎

Proposition A.2**.**

Let $({\gamma_{1},\gamma_{2}})$ be an admissible pair, and let $\theta$ be a real number such that the geodesic curvature of $\gamma_{i}$ satisfies

[TABLE]

We set $\bar{\gamma}_{i}(s):=\gamma_{i}^{\theta}(s)$. Then the pair $({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ is an admissible pair, and $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})=gf_{{\gamma_{1},\gamma_{2}}}(s_{1},s_{2})g^{-1}$ holds, where $g=\exp\left(\theta e_{1}/2\right)\in S^{3}$.

Proof.

It follows from Lemma A.1 that

[TABLE]

and

[TABLE]

So the pair $({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ is an admissible pair. Furthermore, it follows from the definition of $g\in S^{3}$ that

[TABLE]

Let $c_{i}(s)$ be the lift of $\widehat{\gamma_{i}}(s)$ with respect to the covering $p_{2}:S^{3}\to US^{2}$, and let $\bar{c}_{i}(s)=c_{i}(s)g^{-1}$. Since the unit normal vector field of $\gamma_{i}(s)$ is given by $n_{i}(s)=Ad(c_{i}(s))e_{2}$, it follows from (A.4) that

[TABLE]

Since $Ad(g^{-1})e_{1}=e_{1}$ and $\cos\theta-\kappa_{i}(s)\sin\theta>0$, (A.2) implies

[TABLE]

Therefore $p_{2}\circ\bar{c}_{i}=\widehat{\bar{\gamma}_{i}}$, and so we obtain

[TABLE]

Hence $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})=gf_{{\gamma_{1},\gamma_{2}}}(s_{1},s_{2})g^{-1}$. ∎

Remark A.3*.*

Let $({\gamma_{1},\gamma_{2}})$ be an admissible pair such that the geodesic curvatures $\kappa_{1}$ and $\kappa_{2}$ are bounded. Define $\alpha\in(0,\pi)$ and $\beta\in(-\pi,0)$ by

[TABLE]

If $\beta<\theta<\alpha$, then $\cos\theta-\kappa_{i}(s)\sin\theta>0$.

Proposition A.4**.**

Let $({\gamma_{1},\gamma_{2}})$ and $({\bar{\gamma}_{1},\bar{\gamma}_{2}})$ be admissible pairs. Suppose that

[TABLE]

for some $g\in S^{3}$. Then there exists a real number $\theta$ such that the geodesic curvature of $\gamma_{i}$ satisfies $\cos\theta-\kappa_{i}(s)\sin\theta>0\ (i=1,2),$ and $({\bar{\gamma}_{1},\bar{\gamma}_{2}})\equiv(\gamma_{1}^{\theta},\gamma_{2}^{\theta})$ holds.

Proof.

Since $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}=gf_{{\gamma_{1},\gamma_{2}}}g^{-1}$, we obtain $\partial_{i}f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(0,0)=Ad(g)\partial_{i}f_{{\gamma_{1},\gamma_{2}}}(0,0)$, and so

[TABLE]

where $\partial_{i}:=\partial/\partial s_{i}$ ($i=1,2$).

On the other hand, it follows from (2.5) and (2.6) that

[TABLE]

Hence

[TABLE]

Similarly, we obtain

[TABLE]

where $\bar{\kappa}_{i}$ denotes the geodesic curvature of $\bar{\gamma}_{i}$. So it follows from (A.5) that $Ad(g)e_{1}=e_{1}.$ Then there exists a real number $\theta$ such that

[TABLE]

Let $c_{i}$ (resp. $\bar{c}_{i}$) be a curve in $S^{3}$ such that $p_{2}\circ c_{i}=\widehat{\gamma_{i}}$ (resp. $p_{2}\circ\bar{c}_{i}=\widehat{\bar{\gamma}_{i}}$). Since $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s,0)$ coincides with $gf_{\gamma_{1},\gamma_{2}}(s,0)g^{-1}$, we obtain $\bar{c}_{1}(0)^{-1}\bar{c}_{1}(s)=gc_{1}(0)^{-1}c_{1}(s)g^{-1}$. Hence

[TABLE]

Similarly, it follows from $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(0,s)=gf_{\gamma_{1},\gamma_{2}}(0,s)g^{-1}$ that

[TABLE]

Hence it follows from (A.6) that

[TABLE]

This shows that $({\bar{\gamma}_{1},\bar{\gamma}_{2}})\equiv(\gamma_{1}^{\theta},\gamma_{2}^{\theta})$. Furthermore, it follows from (A.2) that

[TABLE]

On the other hand, since

[TABLE]

we have $\cos\theta-\kappa_{i}(s)\sin\theta=|\bar{\gamma}_{i}{{}^{\prime}}(s)|/|\gamma_{i}{{}^{\prime}}(s)|>0$. ∎

Proposition A.5**.**

Let $({\gamma_{1},\gamma_{2}})$ be an admissible pair. Suppose that there exists a real number $\theta\ (0<\theta<\pi)$ satisfying

[TABLE]

We set $\bar{\gamma}_{1}(s):=\gamma_{1}^{\theta}(s)$ and $\bar{\gamma}_{2}(s):=\gamma_{2}^{\theta}(-s)$. Then

• (1)

$({\bar{\gamma}_{1},\bar{\gamma}_{2}})$* is an admissible pair such that $\bar{\kappa}_{1}(s_{1})>\cot\theta>\bar{\kappa}_{2}(s_{2})$,*

• (2)

there exists $a\in S^{3}$ such that $f_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})=af_{{\gamma_{1},\gamma_{2}}}(s_{1},-s_{2})a^{-1}$,

• (3)

the mean curvature functions satisfy $H_{{\bar{\gamma}_{1},\bar{\gamma}_{2}}}(s_{1},s_{2})=-H_{{\gamma_{1},\gamma_{2}}}(s_{1},-s_{2})$.

Proof.

By the assumption (A.7), we obtain

[TABLE]

So it follows from (A.2) that the curves $\gamma_{i}^{\theta}$ are regular curves, and

[TABLE]

hold. Since the geodesic curvatures of $\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ are given by

[TABLE]

it follows from Lemma A.1 that

[TABLE]

This implies the assertion (1). Let $c_{i}$ be a curve in $S^{3}$ such that $p_{2}\circ c_{i}=\widehat{\gamma_{i}}$. We set

[TABLE]

where $g:=\exp(\theta e_{1}/2)$ and $h$ is a point of $S^{3}$ such that

[TABLE]

Since $Ad(g^{-1})e_{3}=(\sin\theta)e_{2}+(\cos\theta)e_{3}$ and $Ad(c_{i}(s))e_{2}=n_{i}(s)$, we obtain

[TABLE]

Since $Ad(g^{-1})e_{1}=e_{1}$, it follows from (A.8) that

[TABLE]

Similarly, $Ad(\bar{c}_{2}(s))e_{1}={\bar{\gamma}_{2}{{}^{\prime}}(s)}/{|\bar{\gamma}_{2}{{}^{\prime}}(s)|}$ holds. This shows that $p_{2}\circ\bar{c}_{i}=\widehat{\bar{\gamma}_{i}}$. Hence

[TABLE]

holds, where $a=h^{-1}g$. This implies the assertion (2). The assertion (3) follows from (2). ∎

The following assertion implies that the assumption of Theorem B depends on the choice of $\mu$-admissible pairs representing the same immersed torus, in general:

Corollary A.6**.**

There exist two $\mu$-admissible pairs $(\gamma_{1},\gamma_{2})$ and $(\bar{\gamma}_{1},\bar{\gamma}_{2})$ satisfying the following properties;

1. (1)

$f_{\gamma_{1},\gamma_{2}\equiv f_{\bar{\gamma}_{1},\bar{\gamma}_{2}}}$, 2. (2)

$\gamma_{1}$* has no negative shells and $\gamma_{2}$ has no positive shells, but* 3. (3)

$\bar{\gamma}_{2}$* has negative shells and $\bar{\gamma}_{1}$ has positive shells.*

Proof.

For $a>0$, we set

[TABLE]

which is a closed regular curve in ${\mathbb{R}}^{2}$ with three crossings with positive curvature. Consider a stereographic projection $p:S^{2}\to{\mathbb{R}}^{2}$ from the north pole, and set

[TABLE]

that gives a closed spherical curve of type $3_{6}$ in Figure 10 of Appendix B. Obviously $\gamma^{a}$ has no negative shells. If $a$ is sufficiently small, $\tilde{\sigma}_{a}$ is positively curved. Moreover, one can easily check that $n_{a}(t)=\gamma^{\pi/2}_{a}$ is a closed curve of type $3_{4}$ with positive geodesic curvature, and has three positive shells and three negative shells at the same time. We set

[TABLE]

then $(\gamma_{1},\gamma_{2})$ is a $\mu$-admissible pair for sufficiently small $\mu>0$ (for example $a=1/7$), since $\gamma_{1}$ (resp. $\gamma_{2}$) has positive (resp. negative) geodesic curvature if $a$ is sufficiently small, for example $a\leq 1/7$. By definition, $\gamma_{1}$ has no negative shells, and $\gamma_{2}$ has no positive shells. We set $\mu:=\cot\theta$ ($\theta\in[0,\pi/2)$). Then $\bar{\gamma}_{1}(t)$ and $\bar{\gamma}_{2}(t)$ are close to $n_{a}(t)$ and $n_{a}(-t)$ if $\mu$ is chosen to be sufficiently close to zero. Since $n_{a}$ is of type $3_{4}$, $\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ have positive shells and negative shells at the same time. ∎

Appendix B

Tables of closed regular spherical curves

In [6], a table of generic closed spherical curves with at most five crossings is given. Figure 10 is the table within 3-crossings:

There are in total $19$ (resp. $76$) curves with 4-crossings (resp. 5-crossings). Figure 11 is the list of curves with 5-crossings amongst them whose shells are of the same kind: