Existence, characterization and stability of Pansu spheres in sub-Riemannian 3-space forms

Abstract

Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda \in {\mathbb {R}}$ with $\lambda ^2+\kappa >0$, we show existence of a $C^2$ spherical surface $\mathcal {S}_\lambda (p)$ immersed in M with constant mean curvature $\lambda$. Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group (Pansu, Conference on differential geometry on homogeneous spaces (Turin, 1983), pp 159–174, 1984) and the sub-Riemannian 3-sphere (Hurtado and Rosales, Math Ann 340(3):675–708, 2008). Then, we study variational properties of $\mathcal {S}_\lambda (p)$ related to the area functional. First, we obtain uniqueness results for the spheres $\mathcal {S}_\lambda (p)$ as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian 3-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that $\mathcal {S}_\lambda (p)$ is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian 3-space forms.

Appendices

Appendix A: Proof of the second variation formula

In this section we prove Theorem 5.2. We first derive a second variation formula for admissible variations possibly moving the singular set of arbitrary CMC surfaces in Sasakian sub-Riemannian 3-manifolds.

Theorem 7.1

Let $\Sigma$ be an orientable $C^2$ surface, possibly with boundary, immersed in a Sasakian sub-Riemannian 3-manifold M. Suppose that $\Sigma -\Sigma _0$ is $C^3$ and take a variation $\varphi :I\times \Sigma \rightarrow M$ which is $C^3$ off of $\Sigma _0$. Denote by $U:=uN+Q$ the associated velocity vector field, where N is the unit normal to $\Sigma$ and $Q:=U^\top$. If $\Sigma$ has CMC H and the variation is admissible, then the functional $A+2HV$ is twice differentiable at the origin, and we have

$$\begin{aligned} (A+2HV)''(0)= & {} \int _{\Sigma }|N_{h}|^{-1}\left\{ Z(u)^2- (|B(Z)+S|^2+4\,(K-1)\,|N_{h}|^2)\,u^2\right\} d\Sigma \nonumber \\&+\int _{\Sigma }{{\mathrm{div}}}_\Sigma \{\langle N,T\rangle \,(1-\langle B(Z),S\rangle )\,u^2\,Z\}\,d\Sigma \nonumber \\&+\int _{\Sigma }{{\mathrm{div}}}_\Sigma \{\langle N,T\rangle \,(2H\,|N_{h}|\,u^2-w)\,S\}\,d\Sigma \nonumber \\&+\int _\Sigma {{\mathrm{div}}}_\Sigma (|N_{h}|\,W^\top )\,d\Sigma +\int _{\Sigma }{{\mathrm{div}}}_\Sigma (h_1 Z+h_2\,S)\,d\Sigma , \end{aligned}$$

(7.1)

provided all the terms are locally integrable with respect to $d\Sigma$. In the previous formula $\{Z,S\}$ is the tangent orthonormal basis defined in (2.13) and (2.14), B is the Riemannian shape operator, K is the Webster scalar curvature, the function w is the normal component of the acceleration vector field $W:=D_UU$, and $h_1,h_2$ are given by

$$\begin{aligned} h_1:= & {} 2\,\{H\,\langle Q,Z\rangle +\langle N,T\rangle \,\langle D_SQ,Z\rangle +|N_{h}|^{-1}\,\langle Q,S\rangle \,(\langle B(Z),S\rangle +\langle N,T\rangle ^2)\}\,u\\&+|N_{h}|\,(\langle Q,Z\rangle \,\langle D_SQ,S\rangle -\langle Q,S\rangle \,\langle D_SQ,Z\rangle )\\&+\langle N,T\rangle \,\langle Q,Z\rangle ^2\,(1-\langle B(Z),S\rangle )-\langle N,T\rangle \,\langle Q,Z\rangle \,\langle Q,S\rangle \,\langle B(S),S\rangle ,\\ h_2:= & {} -2\,\{H\,\langle Q,S\rangle +\langle N,T\rangle \,\langle D_ZQ,Z\rangle -|N_{h}|\,\langle Q,Z\rangle \}\,u\\&+|N_{h}|\,(\langle Q,S\rangle \,\langle D_ZQ,Z\rangle -\langle Q,Z\rangle \,\langle D_ZQ,S\rangle )\\&+2H\,|N_{h}|\,\langle N,T\rangle \,\langle Q,Z\rangle ^2+\langle N,T\rangle \,\langle Q,Z\rangle \,\langle Q,S\rangle \,(1+\langle B(Z),S\rangle ). \end{aligned}$$

If $U=uN$ then the last integral in (7.1) vanishes. If $\Sigma$ has empty boundary and $\varphi$ is supported on $\Sigma -\Sigma _0$, then all the divergence terms in (7.1) vanish.

The theorem was proved in [55, Thm. 5.2] for some particular variations with support in $\Sigma -\Sigma _0$, velocity vector $U=uN$ and acceleration vector $W=wN$. We will follow the arguments there, with the corresponding modifications that appear since $W^\top$ and Q need not vanish on $\Sigma$. The idea is to apply differentiation under the integral sign so that, after a long calculus, we will obtain (7.1) with the help of Lemma 7.2 below.

Proof of Theorem 7.1

We extend U along the variation by $U(\varphi _{s}(p)):=(d/dt)|_{t=s}\,\varphi _{t}(p)$. Let N be a vector field whose restriction to any $\Sigma _{s}:=\varphi _s(\Sigma )$ is a unit normal vector.

We first compute $A''(0)$. Equation (4.2) yields

$$\begin{aligned} A(s)=\int _{\Sigma -\Sigma _{0}}(|N_{h}|\circ \varphi _{s}) \,|\text {Jac}\,\varphi _{s}|\,d\Sigma . \end{aligned}$$

Since the variation is admissible we can apply Lemma 2.10 to get

$$\begin{aligned} A''(0)=\int _{\Sigma -\Sigma _{0}}\{|N_{h}|''(0)+2\,|N_{h}|'(0)\,|\text {Jac}\,\varphi _{s}|'(0)+|N_{h}|\,|\text {Jac}\,\varphi _{s}|''(0)\}\,d\Sigma , \end{aligned}$$

(7.2)

where the derivatives are taken with respect to s. Let us compute the different terms in (7.2).

The calculus of $|\text {Jac}\,\varphi _{s}|'(0)$ and $|\text {Jac}\,\varphi _{s}|''(0)$ can be found in [57, Sect. 9] and [55, Lem. 5.4]. On the one hand, we have

$$\begin{aligned} |\text {Jac}\,\varphi _{s}|'(0)&={{\mathrm{div}}}_{\Sigma }U=(-2H_{R})\,u +{{\mathrm{div}}}_\Sigma Q\nonumber \\&=-(2H\,|N_{h}|+\langle B(S),S\rangle )\,u+{{\mathrm{div}}}_\Sigma Q \end{aligned}$$

(7.3)

where $-2H_{R}:={{\mathrm{div}}}_{\Sigma }N$ is the Riemannian mean curvature of $\Sigma$, and we have used that $2H_{R}=\langle B(Z),Z\rangle +\langle B(S),S\rangle =2H\,|N_{h}|+\langle B(S),S\rangle$. On the other hand, if for any $p\in \Sigma -\Sigma _0$, we choose the orthonormal basis $\{e_1,e_2\}:=\{Z_p,S_p\}$ of $T_p\Sigma$, and we take into account that $D_eU=e(u)\,N_p-u(p)\,B(e)+D_eQ$ for any $e\in T_p\Sigma$, then we deduce

$$\begin{aligned} |\text {Jac}\,\varphi _{s}|''(0)&={{\mathrm{div}}}_{\Sigma }W+({{\mathrm{div}}}_{\Sigma }U)^2 +\sum _{i=1}^2|(D_{e_{i}}U)^\bot |^2 \nonumber \\&\quad -\sum _{i=1}^2\langle R(U,e_{i})U,e_{i}\rangle -\sum _{i,j=1}^2\langle D_{e_{i}}U,e_{j}\rangle \,\langle D_{e_{j}}U,e_{i}\rangle \nonumber \\&={{\mathrm{div}}}_{\Sigma }W+|\nabla _{\Sigma }u|^2+|B(Q)|^2+2\,\langle \nabla _{\Sigma }u,B(Q)\rangle -\text {Ric}(U,U) \nonumber \\&\quad +\langle R(U,N)U,N\rangle +2\,(\langle D_Z U,Z\rangle \,\langle D_S U,S\rangle \nonumber \\&\quad -\,\langle D_Z U,S\rangle \,\langle D_S U,Z\rangle ), \end{aligned}$$

(7.4)

where $\nabla _{\Sigma }u$ is the gradient of u relative to $\Sigma$ and $\text {Ric}$ is the Ricci tensor in (M, g). In the previous formula the curvature term $\langle R(U,N)U,N\rangle -\text {Ric}(U,U)$ can be computed from (2.3), (2.4) and (2.5). After simplifying, this term equals

$$\begin{aligned} 4\,(1-K)\,(\langle N,T\rangle ^2\,|Q|^2+2\,|N_{h}|\,\langle N,T\rangle \,\langle Q,S\rangle \,u+|N_{h}|^2\,u^2)-|Q|^2-2u^2, \end{aligned}$$

which together with the fact that

$$\begin{aligned} \langle D_{e_i} U,e_j\rangle =-\langle B(e_i),e_j\rangle \,u+\langle D_{e_i}Q,e_j\rangle , \end{aligned}$$

gives us the identity

$$\begin{aligned} |\text {Jac}\,\varphi _{s}|''(0)&={{\mathrm{div}}}_{\Sigma }W+|\nabla _{\Sigma }u|^2+|B(Q)|^2+2\,\langle \nabla _\Sigma u,B(Q)\rangle -|Q|^2 \nonumber \\&\quad +2\,(2H\,|N_{h}|\,\langle B(S),S\rangle -\langle B(Z),S\rangle ^2-1)\,u^2 \nonumber \\&\quad +4\,(1-K)\,(\langle N,T\rangle ^2\,|Q|^2+2\,|N_{h}|\,\langle N,T\rangle \,\langle Q,S\rangle \,u+|N_{h}|^2\,u^2) \nonumber \\&\quad +2\,\langle B(Z),S\rangle \,(\langle D_Z Q,S\rangle +\langle D_S Q,Z\rangle )\,u \nonumber \\&\quad -2\,\langle B(S),S\rangle \,\langle D_ZQ,Z\rangle \,u-4H\,|N_{h}|\,\langle D_SQ,S\rangle \,u \nonumber \\&\quad +2\,(\langle D_ZQ,Z\rangle \,\langle D_SQ,S\rangle -\langle D_Z Q,S\rangle \,\langle D_SQ,Z\rangle ). \end{aligned}$$

(7.5)

Now we obtain expressions for $|N_{h}|'(0)$ and $|N_{h}|''(0)$. From (2.16) and (2.1) it follows that

$$\begin{aligned} |N_{h}|'(s)=U(|N_{h}|)=\langle D_{U}N,\nu _{h}\rangle +\langle N,T\rangle \,\langle U,Z\rangle . \end{aligned}$$

The second equality in (2.15) together with the fact that

$$\begin{aligned} D_{U}N=-\nabla _{\Sigma }u-B(Q), \end{aligned}$$

(7.6)

implies that

$$\begin{aligned} |N_{h}|'(0)=-\langle N,T\rangle \,(S(u)+\langle B(Q),S\rangle -\langle Q,Z\rangle ). \end{aligned}$$

(7.7)

Moreover, we have

$$\begin{aligned} |N_{h}|''(0)= & {} \langle D_{U}D_{U}N,\nu _{h}\rangle +\langle D_{U}N,D_{U}\nu _{h}\rangle \nonumber \\&+\,U(\langle N,T\rangle )\,\langle Q,Z\rangle +\langle N,T\rangle \,U(\langle U,Z\rangle ). \end{aligned}$$

(7.8)

By using (2.18), (7.6) and the third relation in (2.15), we get

$$\begin{aligned} \langle D_{U}N,D_{U}\nu _{h}\rangle&=|N_{h}|^{-1}\,Z(u)^2+\langle N,T\rangle \,Z(u)\,u\nonumber \\&\quad +|N_{h}|^{-1}\,(\langle N,T\rangle ^2\,\langle Q,S\rangle +2\,\langle B(Q),Z\rangle )\,Z(u) \nonumber \\&\quad +|N_{h}|\,\langle Q,Z\rangle \,S(u)+\langle N,T\rangle \,\langle B(Q),Z\rangle \,u +|N_{h}|^{-1}(\langle B(Q),Z\rangle ^2\nonumber \\&\quad +\langle N,T\rangle ^2\,\langle Q,S\rangle \,\langle B(Q),Z\rangle )+|N_{h}|\,\langle Q,Z\rangle \,\langle B(Q),S\rangle . \end{aligned}$$

(7.9)

It is also easy to check from (2.17) that

$$\begin{aligned} U(\langle N,T\rangle )=|N_{h}|\,(S(u)+\langle B(Q),S\rangle -\langle Q,Z\rangle ). \end{aligned}$$

(7.10)

On the other hand

$$\begin{aligned} U(\langle U,Z\rangle )&=\langle W,Z\rangle +\langle D_UZ,\nu _{h}\rangle \,\langle U,\nu _{h}\rangle +\langle D_UZ,T\rangle \,\langle U,T\rangle \nonumber \\&=\langle W,Z\rangle -\langle Z,D_U\nu _{h}\rangle \,\langle U,\nu _{h}\rangle -\langle Z,D_UT\rangle \,\langle U,T\rangle \nonumber \\&=\langle W,Z\rangle +(u+|N_{h}|^{-1}\,\langle N,T\rangle \,\langle Q,S\rangle )(Z(u)\nonumber \\&\quad +\,\langle B(Q),Z\rangle +\langle Q,S\rangle ), \end{aligned}$$

(7.11)

where the last equality comes from (2.18) and (2.15). It remains to compute $D_{U}D_{U}N$. For a fixed point $p\in \Sigma -\Sigma _{0}$ we denote $E_i(s):=e_i(\varphi _s)$. It is clear that $E_i(0)=e_i$ and $[U,E_i]=0$. Moreover, $\{E_1(s),E_2(s)\}$ is a basis of the tangent space to $\Sigma _s$ at $\varphi _s(p)$. Note that $\{e_{1},e_{2},N_p\}$ is an orthonormal basis of $T_pM$. As a consequence

$$\begin{aligned} D_{U}D_{U}N=\sum _{i=1}^2\langle D_{U}D_{U}N,E_{i}\rangle \,E_{i} +\langle D_{U}D_{U}N,N\rangle \,N. \end{aligned}$$

Since $\langle N,E_{i}\rangle =0$ and $[U,E_i]=0$, then we get

$$\begin{aligned} \langle D_{U}D_{U}N,E_{i}\rangle&=-2\,\langle D_{U}N,D_{U}E_i\rangle -\langle N,D_{U}D_{U}E_i\rangle \\&=-2\,\langle D_{U}N,D_{e_i}U\rangle -\langle N,D_{U}D_{e_i}U\rangle \nonumber \\&=-2\,\langle D_{U}N,D_{e_{i}}U\rangle +\langle R(U,E_{i})U,N\rangle -\langle D_{e_{i}}W,N\rangle \nonumber \\&=\,2\,\langle \nabla _{\Sigma }u,D_{e_{i}}U\rangle +2\,\langle B(Q),D_{e_i}U\rangle +\langle R(U,E_{i})U,N\rangle -\langle D_{e_{i}}W,N\rangle , \end{aligned}$$

where we have employed (7.6). Moreover, since $|N|^2=1$ on $\Sigma$, we deduce

$$\begin{aligned} \langle D_{U}D_{U}N,N\rangle =-|D_{U}N|^2=-|\nabla _{\Sigma }u|^2-|B(Q)|^2-2\,\langle \nabla _\Sigma u,B(Q)\rangle . \end{aligned}$$

Recall that $e_{1}=Z_{p}$ and $e_{2}=S_{p}$. Then, the previous equalities together with $\langle S,\nu _{h}\rangle =\langle N,T\rangle$ lead us to the expression

$$\begin{aligned} \langle D_{U}D_{U}N,\nu _{h}\rangle&=\langle N,T\rangle \,\langle D_U D_UN,S\rangle +|N_{h}|\,\langle D_U D_U N,N\rangle \nonumber \\&=\langle N,T\rangle \,(\!-2\,\langle \nabla _\Sigma u,B(S)\rangle \,u+2\,\langle \nabla _\Sigma u,D_S Q\rangle -2\,\langle B(Q),B(S)\rangle \,u\nonumber \\&\quad +2\,\langle B(Q),D_SQ\rangle +\langle R(U,S)U,N\rangle -\langle D_SW,N\rangle ) \nonumber \\&\quad -|N_{h}|\,(|\nabla _\Sigma u|^2+|B(Q)|^2+2\,\langle \nabla _\Sigma u,B(Q)\rangle ). \end{aligned}$$

(7.12)

The curvature term above can be derived from (2.3), (2.4) and (2.5), so that we obtain

$$\begin{aligned} \langle R(U,S)U,N)\rangle =4\,(K-1)\,|N_{h}|\,\langle N,T\rangle \,\langle Q,Z\rangle ^2-\langle Q,S\rangle \,u. \end{aligned}$$

Equations (7.12), (7.9), (7.10) and (7.11) allow us to compute $|N_{h}|''(0)$ from (7.8). We use the resulting formula together with (7.7), (7.3) and (7.5). By taking into account the equalities

and simplifying, we obtain

$$\begin{aligned}&|N_{h}|''(0)+2\,|N_{h}|'(0)\,|\text {Jac}\,\varphi _{s}|'(0)+|N_{h}|\,|\text {Jac}\,\varphi _{s}|''(0) \nonumber \\&\quad =|N_{h}|^{-1}\,Z(u)^2+2\,\langle N,T\rangle \,(1-\langle B(Z),S\rangle )\,Z(u)\,u+4H\,|N_{h}|\,\langle N,T\rangle \,S(u)\,u \nonumber \\&\quad \quad +|N_{h}|\,{{\mathrm{div}}}_\Sigma W-\langle N,T\rangle \,\langle D_SW,N\rangle +\langle N,T\rangle \,\langle W,Z\rangle \nonumber \\&\quad \quad +q_1\,u^2+q_2\,u+q_3\,Z(u)+q_4\,S(u)+q_5\,\langle D_ZQ,Z\rangle +q_6\,\langle D_SQ,S\rangle +q_7\, \langle D_SQ,Z\rangle \nonumber \\&\quad \quad +2\,|N_{h}|\,(\langle D_ZQ,Z\rangle \,\langle D_SQ,S\rangle -\langle D_ZQ,S\rangle \,\langle D_SQ,Z\rangle )+2\,|N_{h}|\,\langle B(Z),S\rangle \,\langle D_ZQ,S\rangle \,u \nonumber \\&\quad \quad +|N_{h}|^{-1}\,\langle B(Q),Z\rangle \,(\langle B(Q),Z\rangle +2\,\langle N,T\rangle ^2\,\langle Q,S\rangle )+2\,|N_{h}|\,\langle Q,Z\rangle \,\langle B(Q),S\rangle \nonumber \\&\quad \quad +|N_{h}|^{-1}\,\langle Q,S\rangle ^2-2\,|N_{h}|\,|Q|^2+4\,(1-K)\,|N_{h}|\,\langle N,T\rangle ^2\,\langle Q,S\rangle ^2, \end{aligned}$$

(7.13)

where the functions $q_i$ with $i=1,\ldots ,7$ are defined by

$$\begin{aligned} q_1&=4H\,|N_{h}|^2\,\langle B(S),S\rangle -2\,|N_{h}|\,\langle B(Z),S\rangle ^2+4\,(1-K)\,|N_{h}|^3-2\,|N_{h}|, \nonumber \\ q_2&=2\,\langle N,T\rangle \,\{\langle B(Q),Z\rangle \,(1-\langle B(Z),S\rangle )-\langle Q,Z\rangle \,\langle B(S),S\rangle \nonumber \\&\quad +2H\,|N_{h}|\,(\langle B(Q),S\rangle -\langle Q,Z\rangle )+4\,(1-K)\,|N_{h}|^2\,\langle Q,S\rangle \},\nonumber \\ q_3&=2\,(\langle N,T\rangle \,\langle D_SQ,Z\rangle +|N_{h}|^{-1}\,\langle B(Q),Z\rangle +|N_{h}|^{-1}\,\langle N,T\rangle ^2\,\langle Q,S\rangle ),\nonumber \\ q_4&=2\,(|N_{h}|\,\langle Q,Z\rangle -\langle N,T\rangle \,\langle D_ZQ,Z\rangle ),\nonumber \\ q_5&=2\,\{\langle N,T\rangle \,(\langle Q,Z\rangle -\langle B(Q),S\rangle )-|N_{h}|\,\langle B(S),S\rangle \,u\},\nonumber \\ q_6&=2\,\langle N,T\rangle \,\langle Q,Z\rangle -4H\,|N_{h}|^2\,u,\nonumber \\ q_7&=2\,(\langle N,T\rangle \,\langle B(Q),Z\rangle +|N_{h}|\,\langle B(Z),S\rangle \,u). \end{aligned}$$

Note that Eqs. (7.2) and (7.13) provide the derivative $A''(0)$. Now, we proceed to the calculus of $V''(0)$. From (4.5) we have

$$\begin{aligned} V'(s)=\int _{\Sigma _{s}}\langle U,N\rangle \,d\Sigma _{s}=\int _{\Sigma }(\langle U,N\rangle \circ \varphi _{s})\,|\text {Jac}\, \varphi _{s}|\,d\Sigma , \end{aligned}$$

which implies that

$$\begin{aligned} V''(0)=\int _\Sigma \{\langle U,N\rangle '(0)+u\,|\text {Jac}\,\varphi _s|'(0)\}\,d\Sigma . \end{aligned}$$

(7.14)

Thus, from (7.6) and (7.3) we deduce

$$\begin{aligned} \langle U,N\rangle '(0)+u\,|\text {Jac}\,\varphi _{s}|'(0)&=w-\langle \nabla _\Sigma u,Q\rangle -\langle B(Q),Q\rangle \nonumber \\&\quad +u\,{{\mathrm{div}}}_\Sigma Q-(2H\,|N_{h}|+\langle B(S),S\rangle )\,u^2. \end{aligned}$$

(7.15)

We conclude from (7.2), (7.13), (7.14) and (7.15) that

$$\begin{aligned} (A+2HV)''(0)=\int _{\Sigma -\Sigma _0}\beta \,d\Sigma , \end{aligned}$$

(7.16)

where the function $\beta$ has the expression

$$\begin{aligned} \beta&:=|N_{h}|^{-1}\,Z(u)^2+2\,\langle N,T\rangle \,(1-\langle B(Z),S\rangle )\,Z(u)\,u\nonumber \\&\quad +4H\,|N_{h}|\,\langle N,T\rangle \,S(u)\,u+(q_1-4H^2\,|N_{h}|-2H\,\langle B(S),S\rangle )\,u^2\nonumber \\&\quad +2Hw+|N_{h}|\,{{\mathrm{div}}}_\Sigma W-\langle N,T\rangle \,\langle D_SW,N\rangle +\langle N,T\rangle \,\langle W,Z\rangle \nonumber \\&\quad +q_2\,u+(q_3-4H\,\langle Q,Z\rangle )\,Z(u) +(q_4-4H\,\langle Q,S\rangle )\,S(u)\nonumber \\&\quad +{{\mathrm{div}}}_\Sigma (2Hu\,Q)-2H\,\langle B(Q),Q\rangle +q_5\,\langle D_ZQ,Z\rangle +q_6\,\langle D_SQ,S\rangle +q_7\,\langle D_SQ,Z\rangle \nonumber \\&\quad +2\,|N_{h}|\,(\langle D_ZQ,Z\rangle \,\langle D_SQ,S\rangle \!-\!\langle D_ZQ,S\rangle \,\langle D_SQ,Z\rangle )\!+\!2\,|N_{h}|\,\langle B(Z),S\rangle \,\langle D_ZQ,S\rangle \,u\nonumber \\&\quad +|N_{h}|^{-1}\,\langle B(Q),Z\rangle \,(\langle B(Q),Z\rangle +2\,\langle N,T\rangle ^2\,\langle Q,S\rangle )+2\,|N_{h}|\,\langle Q,Z\rangle \,\langle B(Q),S\rangle \nonumber \\&\quad +|N_{h}|^{-1}\,\langle Q,S\rangle ^2-2\,|N_{h}|\,|Q|^2+4\,(1-K)\,|N_{h}|\,\langle N,T\rangle ^2\,\langle Q,S\rangle ^2. \end{aligned}$$

(7.17)

Let us simplify the terms containing W. From (2.16) and the fact that $\nu _{h}^\top =\langle N,T\rangle \,S$, it is easy to check that

$$\begin{aligned} {{\mathrm{div}}}_\Sigma (|N_{h}|\,W^\top )=|N_{h}|\,{{\mathrm{div}}}_\Sigma W^\top -\langle N,T\rangle \,\langle B(S),W\rangle +\langle N,T\rangle \,\langle W,Z\rangle , \end{aligned}$$

and so

$$\begin{aligned}&2Hw+|N_{h}|\,{{\mathrm{div}}}_\Sigma W-\langle N,T\rangle \,\langle D_SW,N\rangle +\langle N,T\rangle \,\langle W,Z\rangle \nonumber \\&\quad ={{\mathrm{div}}}_\Sigma (|N_{h}|\,W^\top )-\langle N,T\rangle \,S(w)+(2H\,\langle N,T\rangle ^2-|N_{h}|\,\langle B(S),S\rangle )\,w. \end{aligned}$$

Now, we can use the computations below [55, Eq. (5.20)] to infer that the first three lines of (7.17) equal

$$\begin{aligned}&|N_{h}|^{-1}\{Z(u)^2-(|B(Z)+S|^2+4\,(K-1)\,|N_{h}|^2)\,u^2\} \nonumber \\&\quad +{{\mathrm{div}}}_\Sigma \{\langle N,T\rangle \,(1-\langle B(Z),S\rangle )\,u^2\,Z\} \nonumber \\&\quad +{{\mathrm{div}}}_\Sigma \{\langle N,T\rangle \,(2H\,|N_{h}|\,u^2-w)\,S\}+{{\mathrm{div}}}_\Sigma (|N_{h}|\,W^\top ). \end{aligned}$$

(7.18)

To prove the statement, it suffices to show that the remainder summands in (7.17) equal ${{\mathrm{div}}}_\Sigma \,(h_1\,Z+h_2\,S)$. This is a long but straightforward calculus similar to the one at the end of [55, Proof of Thm. 5.2]. Indeed, by using Lemma 7.2 below and the formulas

$$\begin{aligned} S(\langle N,T\rangle )&=|N_{h}|\,\langle B(S),S\rangle , \quad \quad \quad \quad \quad \quad \quad \quad \,\,\ \quad \quad S(|N_{h}|)=-\langle N,T\rangle \,\langle B(S),S\rangle , \nonumber \\ Z(\langle N,T\rangle )&=|N_{h}|\,(\langle B(Z),S\rangle -1), \quad \quad \quad \quad \quad \quad \quad \, Z(|N_{h}|)=\langle N,T\rangle \,(1-\langle B(Z),S\rangle ), \nonumber \\ \langle D_SZ,S\rangle&=|N_{h}|^{-1}\,\langle N,T\rangle \,(1+\langle B(Z),S\rangle ),\quad \ \!\langle D_ZS,Z\rangle =-2H\,\langle N,T\rangle , \nonumber \\ Z(\langle B(Z),S\rangle )&=4\,|N_{h}|\,\langle N,T\rangle \,(1\!-\!K\!-\!H^2)\!-\!2\,|N_{h}|^{-1}\,\langle N,T\rangle \,\langle B(Z),S\rangle \,(1\!+\!\langle B(Z),S\rangle ), \end{aligned}$$

(7.19)

we can deduce (7.1) from (7.16), (7.17) and (7.18). This finishes the proof. $\square$

Lemma 7.2

([55, Lem. 5.5]) Let $\Sigma$ be an orientable $C^2$ surface immersed in a Sasakian sub-Riemannian 3-manifold M. For any $\phi \in C^1(\Sigma )$ we have the following equalities in $\Sigma -\Sigma _{0}$

$$\begin{aligned} {{\mathrm{div}}}_{\Sigma }(\phi \,Z)&=Z(\phi )+|N_{h}|^{-1}\,\langle N,T\rangle \,(1+\langle B(Z),S\rangle )\,\phi , \\ {{\mathrm{div}}}_{\Sigma }(\phi \,S)&=S(\phi )-2H\,\langle N,T\rangle \,\phi . \end{aligned}$$

Remarks 7.3

• 1. The regularity hypotheses in Theorem 7.1 are necessary to compute the different terms in the proof for an arbitrary variation $\varphi$. However, it is possible to derive the second variation for some particular variations of a surface $\Sigma$ with less regularity and $\Sigma _0=\emptyset$, see for instance [18, Thm. 14.5], [34, Thm. 3.7], [55, Thm. 5.2], [24, Thm. 7.3] and [27, Thm. 4.1]. In our case the $C^3$ regularity of $\Sigma -\Sigma _0$ does not suppose loss of generality since we are only interested in stability properties of the spherical surfaces $\mathcal {S}_\lambda (p)$.

• 2. The fact that the variation $\varphi$ is admissible is only used to differentiate under the integral sign in (4.2). If a variation is not admissible then $A''(0)$ may exist or not and, in case of existence, it is not easy to compute it. Some examples of these situations were discussed in the Heisenberg group [34, Prop. 3.11] and in pseudo-Hermitian 3-manifolds, see [12, Ex. 4.3] and [24, Lem. 7.7].

Now, we proceed to obtain the second variation formula in Theorem 5.2. For the proof we will show that all the divergence terms in (7.1) vanish for variations of a spherical surface $\mathcal {S}_\lambda (p)$. The main ingredients are the integrability of $|N_{h}|^{-1}$ in Corollary 3.7 and a divergence theorem which extends the classical situation valid for vector fields supported off of the poles.

Lemma 7.4

Let X be a bounded and tangent $C^1$ vector field on $\mathcal {S}_\lambda (p)$ minus the poles such that ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X$ is integrable with respect to $d\mathcal {S}_\lambda (p)$. Then, we have

$$\begin{aligned} \int _{\mathcal {S}_\lambda (p)}{{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X\,d\mathcal {S}_\lambda (p)=0. \end{aligned}$$

Proof

For any $\varepsilon >0$ small enough, let $\Sigma _\varepsilon :=\mathcal {S}_\lambda (p)-(D_1(\varepsilon )\cup D_2(\varepsilon ))$, where $D_i(\varepsilon )$, $i=1,2$, are small spherical caps centered at the poles. From the divergence theorem we obtain

$$\begin{aligned} \int _{\Sigma _\varepsilon }{{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X\,d\mathcal {S}_\lambda (p)=-\sum _{i=1}^2\,\int _{\partial D_i(\varepsilon )}\langle X,\eta _i\rangle \,dl, \end{aligned}$$

(7.20)

where $\eta _i$ is the inner conormal to $\partial D_i(\varepsilon )$ in $\mathcal {S}_\lambda (p)$, and dl in the Riemannian element of length. Note that the left hand side term above tends to $\int _{\mathcal {S}_\lambda (p)}{{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X\,d\mathcal {S}_\lambda (p)$ when $\varepsilon \rightarrow 0$ by the dominated convergence theorem. On the other hand, since X is bounded, the right hand side term goes to zero when $\varepsilon \rightarrow 0$. This proves the claim.

Proof of Theorem 5.2

We know that $\mathcal {S}_\lambda (p)$ is a $C^\infty$ surface off of the poles with CMC $\lambda$. Hence, for an admissible variation $\varphi$ which is also $C^3$ off of the poles, it is possible to apply Theorem 7.1 to compute $(A+2\lambda V)''(0)$. To prove the claim it suffices to use (5.3), and to see that the divergence terms in (7.1) are all equal to zero.

Consider the $C^1$ tangent vector field $X:=\langle N,T\rangle \,(1-\langle B(Z),S\rangle )\,u^2\,Z$ defined on $\mathcal {S}_\lambda (p)$ minus the poles. Recall that $\langle B(Z),S\rangle =(1-\tau ^2)\,|N_{h}|^2$ by Lemma 3.6 (iii) and so, X is bounded. Moreover, by using Lemma 7.2 together with some of the equalities in (7.19) and the integrability of $|N_{h}|^{-1}$ in Corollary 3.7, we deduce that ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X$ is integrable with respect to $d\mathcal {S}_\lambda (p)$. So, the integral of ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X$ vanishes as a consequence of Lemma 7.4. The same argument holds for the vector field $X:=\langle N,T\rangle \,(2H\,|N_{h}|\,u^2-w)\,S$. Note that the integral of ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}\,(|N_{h}|\,W^\top )$ also vanishes by the divergence theorem since $|N_{h}|\,W^\top$ is a Lipschitz tangent vector field on $\mathcal {S}_\lambda (p)$. Hence, to finish the proof we must show that

$$\begin{aligned} \int _{\mathcal {S}_\lambda (p)}{{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X\,d\mathcal {S}_\lambda (p)=0, \end{aligned}$$

(7.21)

where $X:=h_1\,Z+h_2\,S$, and the functions $h_1$, $h_2$ are defined below equation (7.1). At the end of the proof of Theorem 7.1 we obtained that ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X$ is equal to the function $\beta$ in (7.17) by removing the first three lines. This long expression only contains bounded functions and the term $|N_{h}|^{-1}$, which is integrable on $\mathcal {S}_\lambda (p)$. From here, it follows that ${{\mathrm{div}}}_{\mathcal {S}_\lambda (p)}X$ is integrable with respect to $d\mathcal {S}_\lambda (p)$. Furthermore, from the definitions of $h_1$ and $h_2$ we deduce that all the terms contained in X are bounded vector fields off of the poles, with the exception of

$$\begin{aligned} 2\,|N_{h}|^{-1}\,\langle Q,S\rangle \,(\langle B(Z),S\rangle +\langle N,T\rangle ^2)\,u\,Z. \end{aligned}$$

Since $\langle B(Z),S\rangle =(1-\tau ^2)\,|N_{h}|^2$ and $|N_{h}|^2+\langle N,T\rangle ^2=1$, then the unique unbounded vector in the previous formula is $2\,|N_{h}|^{-1}\,\langle Q,S\rangle \,u\,Z$. Hence, we cannot apply directly Lemma 7.4 to infer equality (7.21). However, the conclusion of Lemma 7.4 also holds provided the right hand side term in (7.20) tends to zero as $\varepsilon \rightarrow 0$. So, to prove (7.21) we only have to see that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\int _{\partial D_i(\varepsilon )}|N_{h}|^{-1}\,\langle Q,S\rangle \,\langle Z,\eta _i\rangle \,u\,dl=0, \quad i=1,2, \end{aligned}$$

(7.22)

where $D_i(\varepsilon )$, $i=1,2$, are small spherical caps centered at the poles, and $\eta _i$ is a unit vector tangent to $\mathcal {S}_\lambda (p)$ and normal to $\partial D_i(\varepsilon )$.

Fix a positive orthonormal basis $\{e_1,e_2\}$ in the contact plane $\mathcal {H}_p$. Let $F(\theta ,s):=\gamma _\theta (s)$ be the flow of CC-geodesics of curvature $\lambda$ with $\gamma _\theta (0)=p$ and $\dot{\gamma }_\theta (0)=(\cos \theta )\,e_1+(\sin \theta )\,e_2$. As in Lemma 3.6 we consider the unit normal along $\mathcal {S}_\lambda (p)$ such that $Z=\dot{\gamma }_\theta$ off of the poles. For $\varepsilon >0$ small enough, the curve $\partial D_1(\varepsilon )$ is parameterized by the curve $\alpha _\varepsilon :[0,2\pi ]\rightarrow \mathcal {S}_\lambda (p)$ given by $\alpha _\varepsilon (\theta ):=F(\theta ,\varepsilon )$. From equations (3.7) and (3.6), we get

$$\begin{aligned} \dot{\alpha }_\varepsilon (\theta )=\frac{\partial F}{\partial \theta }(\theta ,s)=V_\theta (\varepsilon )=-(\lambda \,v(\varepsilon ))\,Z-\sqrt{v(\varepsilon )^2+(v'(\varepsilon )/2)^2}\,S, \end{aligned}$$

where $v(\varepsilon ):=\sin ^2(\tau \varepsilon )/\tau ^2$ and $\tau :=\sqrt{\lambda ^2+\kappa }$. The previous formula implies that

$$\begin{aligned} \eta _1:=\frac{\sqrt{v(\varepsilon )^2+(v'(\varepsilon )/2)^2}\,Z-(\lambda \,v(\varepsilon ))\,S}{|V_\theta (\varepsilon )|} \end{aligned}$$

is a unit normal to $\partial D_1(\varepsilon )$ tangent to $\mathcal {S}_\lambda (p)$. Thus, we have

$$\begin{aligned} \int _{\partial D_1(\varepsilon )}|N_{h}|^{-1}\,\langle Q,S\rangle \,\langle Z,\eta _1\rangle \,u\,dl=\frac{v(\varepsilon )^2+(v'(\varepsilon )/2)^2}{v(\varepsilon )}\int _0^{2\pi }(\langle Q,S\rangle \,u)(\alpha _\varepsilon (\theta ))\,d\theta . \end{aligned}$$

On the one hand, it is straightforward to check that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{v(\varepsilon )^2+(v'(\varepsilon )/2)^2}{v(\varepsilon )}=1. \end{aligned}$$

On the other hand, if we denote $h_\varepsilon (\theta ):=(\langle Q,S\rangle \,u)(\alpha _\varepsilon (\theta ))$, then the continuity of Q and u in $\mathcal {S}_\lambda (p)$ together with equality $S=\langle N,T\rangle \,\nu _{h}-|N_{h}|\,T=-\langle N,T\rangle \,J(\dot{\gamma }_\theta )-|N_{h}|\,T$, gives us $|h_\varepsilon (\theta )|\leqslant c$, for some $c>0$ not depending on $\varepsilon$, and $\lim _{\varepsilon \rightarrow 0}h_\varepsilon (\theta )=h(\theta )$, where

$$\begin{aligned} h(\theta ):=u(p)\,(\langle Q_p,e_1\rangle \sin \theta -\langle Q_p,e_2\rangle \cos \theta ). \end{aligned}$$

Thus, we can apply the dominated convergence theorem to conclude that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\int _0^{2\pi }(\langle Q,S\rangle \,u)(\alpha _\varepsilon (\theta ))\,d\theta =\int _0^{2\pi }h(\theta )\,d\theta =0. \end{aligned}$$

This yields Eq. (7.22) for $i=1$. The case $i=2$ is similar. This proves (7.21) and finishes the proof of the theorem. $\square$

Appendix B: Examples of admissible variations

The notion of admissible variation was introduced in Definition 5.1. For such variations we can apply Lemma 2.10 to differentiate under the integral sign twice in (4.2). In this appendix we will construct several admissible variations. We start with some immediate examples.

Example 8.1

Let U be a sub-Riemannian Killing field on M. This means that any diffeomorphism $\varphi _s$ of the associated one-parameter group is an isometry of M. So, $|\text {Jac}\,\varphi _s|_p=1$ and $|N_{h}|_p(s)=|N_{h}|(p)$ for any $s\in I$ and any $p\in \Sigma$. From here it is immediate that the resulting variation of $\Sigma$ is admissible and the area functional is constant. Observe also that these variations need not fix the singular set $\Sigma _0$.

Example 8.2

Suppose that $\Sigma$ has $C^3$ regular set $\Sigma -\Sigma _0$. Consider a variation $\varphi$ which is $C^3$ on $\Sigma -\Sigma _0$, and such that $|N_{h}|_p(s)>0$ for any $p\in \Sigma -\Sigma _0$ and any $s\in I$. Geometrically, this means that any $\varphi _s$ preserves the regular set, i.e., $\varphi _s(\Sigma -\Sigma _0)\subseteq \Sigma _s-(\Sigma _s)_0$. In particular, the function $s\mapsto f(s,p):=|N_{h}|_p(s)\,|\text {Jac}\,\varphi _s|_p$ is $C^2$ for any $p\in \Sigma -\Sigma _0$, which implies conditions (i), (ii) and (iv) in Definition 5.1. Moreover, we have

$$\begin{aligned} \frac{\partial ^2 f}{\partial s^2}(s,p)=|N_{h}|_p''(s)\,|\text {Jac}\,\varphi _s|_p+2\,|N_{h}|_p'(s)\,|\text {Jac}\,\varphi _s|_p'(s)+|N_{h}|_p(s)\,|\text {Jac}\,\varphi _s|_p''(s). \end{aligned}$$

So, a possible way of constructing admissible variations is to bound the different terms in the previous formula by locally integrable functions not depending on s. This is clear for $|\text {Jac}\,\varphi _s|'_p$ and $|\text {Jac}\,\varphi _s|''_p$ since they are continuous on $I\times \Sigma$. On the other hand, note that

$$\begin{aligned} |N_{h}|_p'(s)=U_{\varphi _s(p)}(|N_{h}|)=\langle D_UN_h,\nu _{h}\rangle (\varphi _s(p)), \end{aligned}$$

whereas

$$\begin{aligned} |N_{h}|_p''(s)=\langle D_UD_UN_h,\nu _{h}\rangle (\varphi _s(p))+ \langle D_UN_h,D_U\nu _{h}\rangle (\varphi _s(p)). \end{aligned}$$

From the expression for $D_U\nu _{h}$ in (2.18) we infer that, if there is $h:\Sigma \rightarrow {\mathbb {R}}$ locally integrable with respect to $d\Sigma$, and such that $|N_{h}|^{-1}_p(s)\leqslant h(p)$ for any $p\in \Sigma -\Sigma _0$ and any $s\in I$, then the restriction of $\varphi$ to $I'\times \Sigma$ is admissible for any $I'\subset \subset I$. In particular, if $\varphi$ is $C^3$ on $\Sigma -\Sigma _0$ and compactly supported on $\Sigma -\Sigma _0$, then there is $I'\subset \subset I$ such that the restriction of $\varphi$ to $I'\times \Sigma$ is admissible.

The analytic condition $|N_{h}|^{-1}_p(s)\leqslant h(p)$, which is sufficient to get an admissible variation, was found by Montefalcone [43, Eqs. (25) and (26)] in the setting of Carnot groups. Since this condition is not easy to verify in practice, we are led to produce admissible variation in more geometric ways. In precise terms we will focus on normal and vertical admissible variations moving $\Sigma _0$. We need some notation and facts that will be useful in the sequel.

Let $\Sigma$ be an oriented $C^2$ surface immersed in M. For any $C^1$ vector field U with compact support on $\Sigma$ we consider the $C^1$ variation $\varphi :I\times \Sigma \rightarrow M$ given by $\varphi _s(p):=\exp _p(s\,U_p)$, where I is an open interval containing 0 and $\exp _p$ denotes the Riemannian exponential map of M at p. Take a point $p\in \Sigma$ and a vector $e\in T_p\Sigma$. Let $\alpha :(-\varepsilon _0,\varepsilon _0)\rightarrow \Sigma$ be a $C^1$ curve with $\alpha (0)=p$ and $\dot{\alpha }(0)=e$. We define the map $F:(-\varepsilon _0,\varepsilon _0)\times I\rightarrow M$ by $F(\varepsilon ,s):=\varphi _s(\alpha (\varepsilon ))$. By Remark 2.7 we have that $E(s):=(\partial F/\partial \varepsilon )(0,s)=e(\varphi _s)$ is a $C^\infty$ vector field along the Riemannian geodesic $\gamma _p(s):=\exp _p(s\,U_p)$ satisfying $[\dot{\gamma }_p,E]=0$ and the Jacobi equation (2.11). Note also that $E(0)=e$ and $E'(0)=D_eU$. Fix a basis $\{e_1,e_2\}$ in $T_p\Sigma$ and let $E_i(s)=e_i(\varphi _s)$, $i=1,2$, be the associated Jacobi fields. Then $\{E_1(s),E_2(s)\}$ is a basis of $T_{\gamma _p(s)}\Sigma$ and

$$\begin{aligned} |\text {Jac}\,\varphi _s|_p=|E_1\times E_2|(s) =(|E_1|^2\,|E_2|^2-\langle E_1,E_2\rangle )^{1/2}(s), \end{aligned}$$

(8.1)

which is a $C^\infty$ function along $\gamma _p$. Moreover, the unit normal of $\Sigma _s:=\varphi _s(\Sigma )$ at $\varphi _s(p)$ is

$$\begin{aligned} N_p(s)=\frac{E_1\times E_2}{|E_1\times E_2|}\,(s), \end{aligned}$$

(8.2)

which is also $C^\infty$ along $\gamma _p$. In the two previous formulas the cross product $\times$ is taken with respect to an orthonormal basis of the tangent space to M along $\gamma _p(s)$. From the $C^\infty$ regularity of $|\text {Jac}\,\varphi _s|_p$ and $N_p(s)$ it follows that, if $|N_{h}|_p(s)>0$ for any $p\in \Sigma -\Sigma _0$ and any $s\in I$, then the conditions (i), (ii) and (iv) in Definition 5.1 hold.

We are now ready to prove that the deformation of a compact surface $\Sigma$ by means of Riemannian parallel surfaces is an admissible variation.

Lemma 8.3

Let $\Sigma$ be a compact, oriented $C^2$ surface immersed inside a Sasakian sub-Riemannian 3-manifold M. Then, there is an open interval $I'\subset {\mathbb {R}}$ such that the variation $\varphi :I'\times \Sigma \rightarrow M$ defined by $\varphi _s(p):=\exp _p(sN_p)$ is admissible.

Proof

Take a point $p\in \Sigma$ and a vector $e\in T_p\Sigma$. Let $E(s):=e(\varphi _s)$ be the associated Jacobi field along $\gamma _p(s):=\exp _p(sN_p)$. From equation (2.11) we get that $\langle E,\dot{\gamma }_p\rangle$ is an affine function along $\gamma _p$. Thus equalities $E(0)=e$ and $E'(0)=D_eN$ give us $\langle E,\dot{\gamma }_p\rangle =0$ along $\gamma _p$. As e is an arbitrary vector in $T_p\Sigma$, the unit normal to $\Sigma _s$ at $\varphi _s(p)$ is $N_p(s)=\dot{\gamma }_p(s)$. It follows that $|N_{h}|^2_p(s)=1-\langle \dot{\gamma }_p,T\rangle ^2(s)=|N_{h}|^2(p)$ since $\langle \dot{\gamma }_p,T\rangle$ is constant along $\gamma _p$. So, we have $|N_{h}|_p(s)\,|\text {Jac}\,\varphi _s|_p=|N_{h}|(p)\,|\text {Jac}\,\varphi _s|_p$. Hence conditions (i), (ii) and (iv) in Definition 5.1 are satisfied. On the other hand, from equation (7.4) and the compactness of $\Sigma$, the derivative $|\text {Jac}\,\varphi _s|_p''(s)$ is uniformly bounded as a function of $s\in I'$ and $p\in \Sigma$ provided $I'$ is a small open interval containing 0. This finishes the proof. $\square$

The previous lemma together with Example 8.2 allows us to construct more general admissible variations based on parallel surfaces near the singular set.

Corollary 8.4

Let M be a Sasakian sub-Riemannian 3-manifold and $\Sigma$ an oriented $C^2$ surface immersed in M with $C^3$ regular set $\Sigma -\Sigma _0$. Consider a variation $\varphi :I\times \Sigma \rightarrow M$ of $\Sigma$ satisfying:

• (i) there is $O\subset \subset \Sigma$ with $\Sigma _0\subset O$ such that $\varphi _s(p)=\exp _p(sN_p)$ for $s\in I$ and $p\in O$,

• (ii) the restriction of $\varphi$ to $(\Sigma -O)\times I$ is of class $C^3$.

Then, there is an interval $I'\subset \subset I$ such that the restriction of $\varphi$ to $I'\times \Sigma$ is admissible.

By a vertical variation of $\Sigma$ we mean one of the form $\varphi _s(p):=\exp _p(s\,u(p)\,T_p)$, where $u\in C_0^1(\Sigma )$. This means that the surfaces $\Sigma _s:=\varphi _s(\Sigma )$ are all vertical graphs over $\Sigma$. For the case $u=1$ the variation is admissible by Example 8.1 since T is a sub-Riemannian Killing field and $\varphi _s$ coincides with the one-parameter group of T. In the next result we provide more examples of admissible vertical variations. Recall that $\{Z,S\}$ is the tangent basis to $\Sigma -\Sigma _0$ defined in (2.13) and (2.14).

Lemma 8.5

Let $\Sigma$ be an oriented $C^2$ surface immersed inside a Sasakian sub-Riemannian 3-manifold M. Suppose that $\langle N,T\rangle$ does not vanish along $\Sigma$ and that $|N_{h}|^{-1}$ is locally integrable with respect to $d\Sigma$. Let $u\in C^1_0(\Sigma )$ such that $|S(u)|\leqslant h\,|Z(u)|$ in $\Sigma -\Sigma _0$, for some bounded function h. Then, the vertical variation $\varphi _s(p):=\exp _p(s\,u(p)\,T_p)$ is admissible.

Proof

Take a point $p\in \Sigma$. The Riemannian geodesic $\gamma _p(s):=\exp _p(s\,u(p)\,T_p)$ satisfies the equality $\gamma _p(s)=\alpha _p(s\,u(p))$, where $\alpha _p$ is the integral curve of T through p. In particular, we have $\dot{\gamma }_p(s)=u(p)\,T_{\gamma _p(s)}$. Let $\{X(s),Y(s)\}$ be a positive orthonormal basis of the horizontal plane at $\gamma _p(s)$ obtained by parallel transport of a similar basis $\{X_p,Y_p\}$ of $\mathcal {H}_p$. For any vector $e\in T_p\Sigma$, we know that the associated vector field $E(s):=e(\varphi _s)$ satisfies the Jacobi equation (2.11). By using (2.3) this equation reads $E''(s)+u(p)^2\,E(s)_h=0$. Hence, an easy integration together with equalities $E(0)=e$ and $E'(0)=e(u)\,T_p+u(p)\,J(e)$ implies that

$$\begin{aligned} E(s)=x(s)\,X(s)+y(s)\,Y(s)+t(s)\,T(s), \end{aligned}$$

(8.3)

where $T(s):=T_{\gamma _{p}(s)}$, and the functions x(s), y(s), t(s) are given by

$$\begin{aligned} x(s)&:=\langle e,X_p\rangle \,\cos (u(p)\,s)-\langle e,Y_p\rangle \,\sin (u(p)\,s), \nonumber \\ y(s)&:=\langle e,Y_p\rangle \,\cos (u(p)\,s)+\langle e,X_p\rangle \,\sin (u(p)\,s), \nonumber \\ t(s)&:=e(u)\,s+\langle e,T_p\rangle . \end{aligned}$$

(8.4)

Now suppose that $p\in \Sigma -\Sigma _0$ and consider the orthonormal basis $\{e_1,e_2\}$ of $T_p\Sigma$ defined by $e_1=Z_p$ and $e_2=S_p$. Let $E_i(s):=e_i(\varphi _s)$, $i=1,2$, be the associated Jacobi fields along $\gamma _p$. From Eqs. (8.1) and (8.2), we get

$$\begin{aligned} f(s,p):=|N_{h}|_p(s)\,|\text {Jac}\,\varphi _s|_p=|(E_1(s)\times E_2(s))_h|, \end{aligned}$$

where the cross product is taken with respect to the basis $\{X(s),Y(s),T(s)\}$. Hence, a straightforward computation from (8.3) and (8.4) shows that

$$\begin{aligned} f(s,p)=Q_p(s)^{1/2}, \quad Q_p(s):=a_p\,s^2+b_p\,s+c_p, \end{aligned}$$

where

$$\begin{aligned} a_p:=\langle N_p,T_p\rangle ^2\,Z_p(u)^2+S_p(u)^2, \quad b_p:=-2\,|N_{h}|(p)\,S_p(u),\quad c_p:=|N_{h}|^2(p). \end{aligned}$$

Note that the discriminant $\text {disc}(Q_p)$ equals

$$\begin{aligned} -4\,|N_{h}|^2(p)\,\langle N_p,T_p\rangle ^2\,Z_p(u)^2, \end{aligned}$$

which is less than or equal to 0. Moreover, the fact that $\langle N,T\rangle$ never vanishes on $\Sigma$ together with inequality $|S_p(u)|\leqslant h(p)\,|Z_p(u)|$ implies that $\text {disc}(Q_p)=0$ if and only if $Z_p(u)=S_p(u)=0$. From here we deduce that $Q_p(s)>0$ for any $p\in \Sigma -\Sigma _0$ and any $s\in {\mathbb {R}}$. In particular, for any $p\in \Sigma -\Sigma _0$, the function $s\mapsto f(s,p)$ is $C^\infty$ and so, the conditions (i), (ii) and (iv) in Definition 5.1 are satisfied.

Finally, take $p\in \text {supp}(u)\cap (\Sigma -\Sigma _0)$ and suppose $a_p\ne 0$. Since $Q_p(s)\geqslant Q_p(-b_p/(2a_p))=-\text {disc}(Q_p)/(4a_p)$ and $|S_p(u)|\leqslant h(p)\,|Z_p(u)|$ with h bounded, we obtain

$$\begin{aligned} \left| \frac{\partial ^2 f}{\partial s^2}(s,p)\right|&=-\frac{\text {disc}(Q_p)}{4\,Q_p(s)^{3/2}} \leqslant \frac{2\,a_p^{3/2}}{(-\text {disc}(Q_p))^{1/2}} =\frac{\,(\langle N_p,T_p\rangle ^2\,Z_p(u)^2+S_p(u)^2)^{3/2}}{|N_{h}|(p)\,|\langle N_p,T_p\rangle |\,|Z_p(u)|} \\&\leqslant \frac{\,(\langle N_p,T_p\rangle ^2+h(p)^2)^{3/2}\,Z_p(u)^2 }{|N_{h}|(p)\,|\langle N_p,T_p\rangle |} \leqslant \frac{C\,\langle (\nabla _\Sigma u)_p,Z_p\rangle ^2}{|N_{h}|(p)}\leqslant C'\,|N_{h}|^{-1}(p), \end{aligned}$$

for some constants $C,C'>0$. The previous inequality also holds if $a_p=0$ since, in that case, $b_p=0$ and $(\partial ^2 f/\partial s^2)(s,p)=0$ for any $s\in {\mathbb {R}}$. From the hypothesis that $|N_{h}|^{-1}$ is locally integrable with respect to $d\Sigma$ we conclude that condition (iii) in Definition 5.1 holds, proving the claim. $\square$

Note that $\langle N,T\rangle \ne 0$ in small neighborhoods of $\Sigma _0$. Hence we can combine Lemma 8.5 with Example 8.2 to construct admissible variations based on suitable vertical deformations near $\Sigma _0$.

Corollary 8.6

Let M be a Sasakian sub-Riemannian 3-manifold and $\Sigma$ an oriented $C^2$ surface immersed in M. Suppose that $\Sigma -\Sigma _0$ is $C^3$ and the function $|N_{h}|^{-1}$ is locally integrable with respect to $d\Sigma$. Consider a variation $\varphi :I\times \Sigma \rightarrow M$ satisfying:

• (i) there is a small open neighborhood O of $\Sigma _0$, a function $u\in C^1(O)$ with bounded gradient, and a function h bounded on $O-\Sigma _0$, such that $|S_p(u)|\leqslant h(p)\,|Z_p(u)|$ for any $p\in O-\Sigma _0$, and $\varphi _s(p)=\exp _p(s\,u(p)\,T_p)$ for any $s\in I$ and any $p\in O$,

• (ii) the restriction of $\varphi$ to $(\Sigma -O)\times I$ is of class $C^3$.

Then, there is an interval $I'\subset \subset I$ such that the restriction of $\varphi$ to $I'\times \Sigma$ is admissible.

The previous results provide admissible variations of a $C^2$ volume-preserving area-stationary surface $\Sigma$ with $\Sigma _0\ne \emptyset$ inside a 3-dimensional space form. As we proved in Theorems 4.9 and 4.13, such a surface is either a plane $\mathcal {L}_\lambda (p)$ as in (3.4), a spherical surface $\mathcal {S}_\lambda (p)$ as in (3.3), or a surface $\mathcal {C}_{\mu ,\lambda }(\Gamma )$ as in Example 4.14. Since all these surface are $C^\infty$ off of the singular set, then Corollary 8.4 implies that $C^3$ perturbations of the deformation of $\Sigma$ by Riemannian parallels are admissible variations. On the other hand, the construction in Corollary 8.6 only applies for $\mathcal {L}_\lambda (p)$ and $\mathcal {S}_\lambda (p)$ since the function $|N_{h}|^{-1}$ is locally integrable on these surfaces, whereas it is not on $\mathcal {C}_{\mu ,\lambda }(\Gamma )$. In Example 8.7 below we give a particular but important family of functions in $\mathcal {S}_\lambda (p)$ satisfying the condition $|S(u)|\leqslant h\,|Z(u)|$, thus providing examples of admissible vertical variations. We conclude that the class of variations of a spherical surface $\mathcal {S}_\lambda (p)$ for which the second variation formula and the stability results in Sect. 5 are valid is very large.

Example 8.7

Let $\mathcal {S}_\lambda (p)$ be a spherical surface inside a 3-dimensional space form of curvature $\kappa$. We consider the immersion $F(\theta ,s):=\gamma _\theta (s)$ introduced in Lemma 3.6, and the associated unit normal N. We know from Corollary 3.7 that $|N_{h}|^{-1}$ is integrable with respect to $d\mathcal {S}_\lambda (p)$. For any $u\in C^1(\mathcal {S}_\lambda (p))$ we denote $\overline{u}:=u\circ F$. By Lemma 3.6 (ii) and Eq. (3.7), we have

$$\begin{aligned} Z(u)&=\frac{\partial \overline{u}}{\partial s}(\theta ,s), \quad S(u)=\frac{-1}{\sqrt{v(s)^2+(v'(s)/2)^2}}\,\frac{\partial \overline{u}}{\partial \theta }(\theta ,s)-\lambda \,|N_{h}|(s)\,\frac{\partial \overline{u}}{\partial s}(\theta ,s), \end{aligned}$$

where $v(s):=\sin ^2(\tau s)/\tau ^2$ and $\tau :=\sqrt{\lambda ^2+\kappa }$. From here we deduce that, if $\partial \overline{u}/\partial \theta =0$, then $S(u)=h\,Z(u)$ for some function h bounded off of the poles. In particular, any variation $\varphi$ of $\mathcal {S}_\lambda (p)$ whose restriction to small neighborhoods of the poles is of the form $\varphi _s(p)=\exp _p(s\,u(p)\,T_p)$ will be admissible by Corollary 8.6. Note that in a model space $\mathbb {M}(\kappa )$ the hypothesis $\partial \overline{u}/\partial \theta =0$ means that the function u is radially symmetric, i.e., invariant under Euclidean vertical rotations.