On mean-convex Alexandrov embedded surfaces in the 3-sphere

Abstract

We investigate non-compact mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.

Appendix: Proof of Lemma 3.2

Proof

We shall construct a Killing field $\vartheta$ with the desired properties, which rotates $\gamma$ around two antipodes of $\gamma$. The corresponding rotated geodesics $\gamma _\vartheta (s,\cdot )$ belong to a unique great 2-sphere $\mathbb S^2\subset \mathbb S^3$. The corresponding paths $s\mapsto p(s)$ and $s\mapsto q(s)$ move along the intersection of this 2-sphere $\mathbb S^2$ with M. Hence we can calculate all derivatives on this sphere.

We parameterize this 2-sphere by the real parameter s of the family $s\mapsto \gamma _{\vartheta }(s,\cdot )$ of rotated geodesics, and the real arc length parameter t of these geodesics. We choose the equator as the points corresponding to $t=0$ with distance $\frac{\pi }{2}$ to the rotation axis. Let $t_p$ and $t_q$ denote the values of this parameter t at the points p(s) and q(s). Hence ${{\mathrm{dist}}}_N(p,q)=|t_p-t_q|$. The vector fields $\vartheta$ and the geodesic vector field $\dot{\gamma }$ along the geodesics $\gamma _\vartheta (s,\cdot )$ form an orthogonal basis of the tangent spaces of this 2-sphere away from the zeros of $\vartheta$. The vector fields $\vartheta$ and $\dot{\gamma }$ have at (s, t) the scalar products

$$\begin{aligned} g(\vartheta ,\vartheta )&=\cos ^2(t),\quad g(\vartheta ,\dot{\gamma })=0,\quad g(\dot{\gamma },\dot{\gamma })=1. \end{aligned}$$

Since $\dot{\gamma }$ is a geodesic vector field the derivative $\nabla _{\dot{\gamma }}\dot{\gamma }$ vanishes. Moreover, the geodesic curvature in $\mathbb S^2$ of the integral curve of $\vartheta$ starting at (s, t) is equal to $\tan (t)$. Therefore at (s, t) we have

$$\begin{aligned} \nabla _{\vartheta }\vartheta&=\cos ^2(t)\tan (t)\dot{\gamma } =\cos (t)\sin (t)\dot{\gamma },&\nabla _{\vartheta }\dot{\gamma }&=-\tan (t)\vartheta ,\\ \nabla _{\dot{\gamma }}\vartheta&=-\tan (t)\vartheta ,&\nabla _{\dot{\gamma }}\dot{\gamma }&=0. \end{aligned}$$

We parameterize a neighbourhood of the geodesic from p to q in such a way that the corresponding vector field $\dot{\gamma }$ points inward to N at p and outward of N at q, respectively. The derivatives of $s\mapsto p(s)$ and $s\mapsto q(s)$ are equal to

$$\begin{aligned} p'&=\vartheta (p)-\dot{\gamma }(p) \frac{g(\mathfrak {N}(p),\vartheta (p))}{g(\mathfrak {N}(p),\dot{\gamma }(p))}\quad \text{ and } \quad q'=\vartheta (q)-\dot{\gamma }(q) \frac{g(\mathfrak {N}(q),\vartheta (q))}{g(\mathfrak {N}(q),\dot{\gamma }(q))}. \end{aligned}$$

The lengths $|p'|$ and $|q'|$ depend on the angles $\sphericalangle (\mathfrak {N}(p),\vartheta (p))$ and $\sphericalangle (\mathfrak {N}(q),\vartheta (q))$. Since $\vartheta$ is orthogonal to $\dot{\gamma }$ and $\sphericalangle (\mathfrak {N}(p),\dot{\gamma }(p))=\chi _p$ and $\sphericalangle (\mathfrak {N}(q),\dot{\gamma }(q))=\chi _q$ these angles obey $\sphericalangle (\mathfrak {N}(p),\vartheta (p))\in [\frac{\pi }{2}-\chi _p,\frac{\pi }{2}+\chi _p]$ and $\sphericalangle (\mathfrak {N}(q),\vartheta (q))\in [\frac{\pi }{2}-\chi _q,\frac{\pi }{2}+\chi _q]$. Then we have

$$\begin{aligned} |p'|&\le \frac{|\cos (t_p)|}{\cos (\chi _p)}\qquad |q'|\le \frac{|\cos (t_q)|}{\cos (\chi _q)}\\ d'&= \frac{g(\mathfrak {N}(p),\,\vartheta (p))}{g(\mathfrak {N}(p),\,\dot{\gamma }(p))}- \frac{g(\mathfrak {N}(q),\,\vartheta (q))}{g(\mathfrak {N}(q),\,\dot{\gamma }(q))}= \frac{g(\mathfrak {N}(p),\,\vartheta (p))}{\cos (\chi _p)}- \frac{g(\mathfrak {N}(q),\,\vartheta (q))}{\cos (\chi _q)}.\nonumber \end{aligned}$$

(6.1)

Along the paths p and q with $X=p'$ and $X=q'$, respectively, we have at (s, t)

$$\begin{aligned} \nabla _X\frac{g(\mathfrak {N},\,\vartheta )}{g(\mathfrak {N},\,\dot{\gamma })}&= \frac{g(\nabla _X\mathfrak {N},\,\vartheta )+ g(\mathfrak {N},\,\nabla _X\vartheta )}{g(\mathfrak {N},\,\dot{\gamma })}- \frac{g(\mathfrak {N},\vartheta )( g(\nabla _X\mathfrak {N},\,\dot{\gamma })+ g(\mathfrak {N},\,\nabla _X\dot{\gamma }))}{(g(\mathfrak {N},\,\dot{\gamma }))^2}\\&=\frac{g(\nabla _X\mathfrak {N},X)+g(\mathfrak {N},\nabla _X\vartheta )}{g(\mathfrak {N},\dot{\gamma })}- \frac{g(\mathfrak {N},\vartheta ) g(\mathfrak {N},\nabla _X\dot{\gamma })}{g(\mathfrak {N},\dot{\gamma })^2}\\&=-\frac{\mathfrak {h}(X,\,X)}{g(\mathfrak {N},\dot{\gamma })} +\cos (t)\sin (t) +2\tan (t)\left( \frac{g(\mathfrak {N},\,\vartheta )}{g(\mathfrak {N},\,\dot{\gamma })}\right) ^2. \end{aligned}$$

Hence the second derivative is equal to

$$\begin{aligned} d''&=-\frac{\mathfrak {h}(p',\,p')}{\cos (\chi _p)}-\frac{\mathfrak {h}(q',\,q')}{\cos (\chi _q)} +\frac{\sin (2t_p)-\sin (2t_q)}{2}\\&\quad +2\tan (t_p)\left( \frac{g(\mathfrak {N}(p),\,\vartheta (p))}{g(\mathfrak {N}(p),\,\dot{\gamma }(p))}\right) ^2 -2\tan (t_q)\left( \frac{g(\mathfrak {N}(q),\,\vartheta (q))}{g(\mathfrak {N}(q),\,\dot{\gamma }(q))}\right) ^2. \end{aligned}$$

If along the rotation of the geodesic for $s\in [0,s_0]$ the following inequalities are satisfied

$$\begin{aligned} -\tfrac{\pi }{2}\le t_p&\le 0,&0\le t_q&\le \tfrac{\pi }{2},&\tfrac{c}{2}&\le d=t_q-t_p,\quad \text{ and }&\left( \sin ^2(\chi _p)+\sin ^2(\chi _q)\right) ^{\frac{1}{2}}&\le \tfrac{1}{2}, \end{aligned}$$

(6.2)

then $\min \{\cos (\chi _p),\cos (\chi _q)\}\ge \tfrac{\sqrt{3}}{2}$ implies the third inequality of (3.4):

$$\begin{aligned} |p'|+|q'|&\le \frac{\cos (t_p)}{\cos (\chi _p)} +\frac{\cos (t_p)}{\cos (\chi _p)} \le \tfrac{2}{\sqrt{3}}(\cos (t_p)+\cos (t_q))\\&=\tfrac{4}{\sqrt{3}}\cos \left( \tfrac{d}{2}\right) \cos \left( \tfrac{t_p+t_q}{2}\right) \le 3\cos (\tfrac{d}{2}). \end{aligned}$$

Furthermore, the last two terms of $d''$ are bounded by

$$\begin{aligned} \left| \tan (t_p)\left( \frac{g(\mathfrak {N}(p),\,\vartheta (p))}{g(\mathfrak {N}(p),\,\dot{\gamma }(p))}\right) ^2\right|&\le \sin (|t_p|)\cos (t_p)\tan ^2(\chi _p) \le \frac{\sin (2|t_p|)}{2\cdot 3}\\ \left| \tan (t_q)\left( \frac{g(\mathfrak {N}(q),\,\vartheta (q))}{g(\mathfrak {N}(q),\,\dot{\gamma }(q))}\right) ^2\right|&\le \sin (|t_q|)\cos (t_q)\tan ^2(\chi _q) \le \frac{\sin (2|t_q|)}{2\cdot 3}. \end{aligned}$$

Due to $\sin (2t_q)-\sin (2t_p) = 2\sin (t_q-t_p)\cos (t_p+t_q)$ and we arrive at

$$\begin{aligned} d''(s)&\le -\frac{\mathfrak {h}(p',\,p')}{\cos (\chi _p)}-\frac{\mathfrak {h}(q',\,q')}{\cos (\chi _q)} -\sin (d)\cos (t_p+t_q)\left( 1-\tfrac{2}{3}\right) . \end{aligned}$$

(6.3)

Now we claim that the second inequality of (3.4) is implied by (6.2) and the existence of $\delta$ which satisfy

$$\begin{aligned} \delta&\le \frac{1}{9}\sin \left( \tfrac{c}{2}\right) \cos (t_p+t_q),\quad -\frac{\mathfrak {h}(p',\,p')}{|p'|^2} \le \frac{\delta }{2}\quad \text{ and } \quad -\frac{\mathfrak {h}(q',\,q')}{|q'|^2} \le \frac{\delta }{2}. \end{aligned}$$

(6.4)

The assumption (6.2) implies $t_p\le -d+\frac{\pi }{2}$ and $d-\frac{\pi }{2}\le t_q$.

For $d\in [\frac{\pi }{2},\pi )$ we use $\cos (t_p)\le \sin (d)$ and $\cos (t_q)\le \sin (d)$ and for $d\in [\frac{c}{2},\frac{\pi }{2})$ we use $\cos (t_p)\le 1$ and $\cos (t_q)\le 1$ to obtain

$$\begin{aligned} \tfrac{1}{2\cdot 9}\sin \left( \tfrac{c}{2}\right) \max \left\{ \tfrac{|p'|^2}{\cos (\chi _p)},\tfrac{|q'|^2}{\cos (\chi _q)}\right\} \le \tfrac{3\sqrt{3}}{8\cdot 9}\sin \left( \tfrac{c}{2}\right) \max \left\{ \tfrac{|p'|^2}{\cos (\chi _p)},\tfrac{|q'|^2}{\cos (\chi _q)}\right\} \le \tfrac{1}{9}\sin (d). \end{aligned}$$

Together with (6.4) we can estimate the first two terms in (6.3):

$$\begin{aligned} -\tfrac{\mathfrak {h}(p',\,p')}{\cos (\chi _p)}\le \tfrac{\delta }{2}\tfrac{|p'|^2}{\cos (\chi _p)}&\le \tfrac{1}{9}\sin (d)\cos (t_p+t_q)&-\tfrac{\mathfrak {h}(q',\,q')}{\cos (\chi _q)}\le \tfrac{\delta }{2}\tfrac{|q'|^2}{\cos (\chi _q)}&\le \tfrac{1}{9}\sin (d)\cos (t_p+t_q). \end{aligned}$$

The third inequality of (6.2) implies $\sin (\tfrac{c}{2})\cos (\tfrac{d}{2}) \le 2\sin (\tfrac{c}{4})\cos (\tfrac{d}{2})\le 2\sin (\tfrac{d}{2})\cos (\tfrac{d}{2})=\sin (d)$. Thus the second inequality of (3.4) indeed follows with the help of (6.3) from (6.2) and (6.4).

We shall show first that there exists a vector field $\vartheta$ obeying at $s=0$

$$\begin{aligned} \delta&\le \frac{1}{18}\sin \left( \tfrac{c}{2}\right) \cos (t_p+t_q),\quad -\frac{\mathfrak {h}(p',\,p')}{|p'|^2} \le \frac{\delta }{4}\quad \text{ and } \quad -\frac{\mathfrak {h}(q',\,q')}{|q'|^2} \le \frac{\delta }{4}. \end{aligned}$$

The Killing field $\vartheta$ is uniquely determined by two choices: firstly, the choice of a great 2-sphere $\mathbb S^2\subset \mathbb S^3$, which contains the closed geodesic from p to q, and secondly, a choice of the zeros of $\vartheta$, or equivalently a choice of the coordinates $t_p$ and $t_q$ with $t_q-t_p=d\mod \pi$. We start with $t_q=-t_p=\frac{d}{2}$ and set $\delta =\frac{1}{18}\sin (\frac{c}{2})$.

Now we choose the 2-sphere $\mathbb S^2\subset \mathbb S^3$ which contains the geodesic which connects p and q. This 2-sphere intersects M along curves at p and q. It is uniquely determined either by the line in $T_pM$ tangent to $\mathbb S^2$ or by a line in $T_qM$, which is tangent to $\mathbb S^2$. Since $\mathrm {f}$ is a mean-convex Alexandrov embedding and both principal curvatures are uniformly bounded by $\kappa _{\text{ max }}$, the cone angles of the double cones $\{X\in T_pM\mid \mathfrak {h}(X,X)\ge -\frac{1}{4}\delta |X|^2\}$ and $\{X\in T_qM\mid \mathfrak {h}(X,X)\ge -\frac{1}{4}\delta |X|^2\}$ are not smaller than $\tfrac{\pi }{2}+\mathrm{O}(\delta )$. For sufficiently small $\epsilon \ge (\sin ^2(\chi _p)+\sin ^2(\chi _q))^{\frac{1}{2}}$ the tangent direction in the plane orthogonal to $\dot{\gamma }(p)$ in $T_pN$, and in the plane orthogonal to $\dot{\gamma }(q)$ in $T_qN$ of the corresponding spheres build two double cones with cone angles not smaller than $\tfrac{\pi }{2}$. Hence the intersection of both double cones is non-empty and there exists a 2-sphere where the intersecting curves of $\mathbb S^2 \cap M$ at p and q has tangent vectors which satisfy both second and third conditions of (6.4).

Secondly we shall show that the inequalities (6.2) and (6.4) are satisfied for $s\in [0,\,s_0]$ with some $s_0>0$. Since the curvature is bounded by $\cot (c)$ and due to the assumption $(\sin ^2(\chi _p)+\sin ^2(\chi _q))^{\frac{1}{2}}\le \epsilon$ there exists $s_0$ such that $t_p$ and $t_q$ do not reach the roots of $\vartheta$ for $s\in [0,\,s_0]$. Since the derivatives of $\cos (\chi _p), \cos (\chi _q), t_p$ and $t_q$ with respect to s are uniformly bounded, there exists $s_0>0$ such that the inequalities (6.2) and the first inequality of (6.4) are satisfied for $s\in [0,\,s_0]$. Due to (1.2) also the derivatives of $\mathfrak {h}(p',p')$ and $\mathfrak {h}(q',q')$ are uniformly bounded. Hence there exists $s_0>0$ only depending on $c$ and $C$, such that the second and the third inequality of (3.4) are satisfied for $s\in [0,\,s_0]$. We choose $s_0>0$ small such that $|d(s)-d(s_0)| \le c/2$.

Finally we have to satisfy the first inequality of (3.4). At the start point $s=0$ this is always the case for one choice of the sign of $\vartheta$. Now the second inequality of (3.4) implies the first. $\square$