## 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.

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## Acknowledgments

We would like to thank Harold Rosenberg for discussions on the maximum principle at infinity that lead to the proof of Proposition 2.3.

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## Appendix: Proof of Lemma 3.2

### 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

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

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

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

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

Hence the second derivative is equal to

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

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

Furthermore, the last two terms of \(d''\) are bounded by

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

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

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

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

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 \(s=0\)

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. \(\square \)

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Hauswirth, L., Kilian, M. & Schmidt, M.U. On mean-convex Alexandrov embedded surfaces in the 3-sphere.
*Math. Z.* **281**, 483–499 (2015). https://doi.org/10.1007/s00209-015-1497-5

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DOI: https://doi.org/10.1007/s00209-015-1497-5