We now consider the stability of the constant mean curvature hypersurfaces described in the previous sections. We prove the instability of the O(*n*) × O(*m*)-invariant noncompact constant mean curvature hypersurfaces considered in Theorem 1.1 and Theorem 1.2. The arguments for instability go along the lines of the ones given by Pedrosa and Ritoré in [14], see also [18].

One says that an immersion *j*:*Σ*^{k − 1} → *M*^{k} of a hypersurface with constant mean curvature is *stable* if and only if *Q*_{Σ}(*u*) ≥ 0 for all differentiable functions *u*:*Σ*^{k − 1} → ℝ with compact support such that
∫_{Σ} *u dA* = 0; see [4]. Here the index form *Q*_{Σ}(*u*) is given by
$$\begin{array}{}{\displaystyle {Q}_{\mathrm{\Sigma}}(u)=\underset{\mathit{\Sigma}}{\int}\{\parallel \mathrm{\nabla}u{\parallel}^{2}-(\text{Ric}(N)+|B{|}^{2}){u}^{2}\}\phantom{\rule{thinmathspace}{0ex}}d\mathit{\Sigma},}\end{array}$$(9)

where *N* is a unit vector normal to *Σ*, *dΣ* is the volume element on *Σ*, Ric(*N*) is the Ricci curvature of *N*, and |*B*| is the norm of the second fundamental form *B* of *Σ*. This is also written as
$$\begin{array}{}{\displaystyle {Q}_{\mathit{\Sigma}}(u)=-\underset{\mathit{\Sigma}}{\int}uLu\phantom{\rule{thinmathspace}{0ex}}d\mathit{\Sigma},}\end{array}$$(10)

where *L* is the Jacobi operator *L*(*u*) = *Δ u* +(Ric (*N*) + |*B*|^{2}) *u*.

Let *Σ* ⊂ ℝ^{n} × **S**^{m} be a hypersurface invariant by the O(*n*) × O(*m*) action and generated by a curve *φ* (*t*) as in Section 2 which satisfies
$$\begin{array}{}{x}^{\prime}(t)=\mathrm{cos}(\sigma (t))\\ {y}^{\prime}(t)=\mathrm{sin}(\sigma (t))\\ {\displaystyle {\sigma}^{\prime}(t)=(m-1)\mathrm{cot}(y(t))\mathrm{cos}(\sigma (t))-(n-1)\frac{\mathrm{sin}(\sigma (t))}{x(t)}-h.}\end{array}$$(11)

If *h* is constant, then direct computation gives the following:
$$\begin{array}{}\text{Ric}(N)=(m-1){\mathrm{cos}}^{2}(\sigma ),\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}|B{|}^{2}={\sigma}^{\prime 2}+(m-1){\mathrm{cot}}^{2}(y){\mathrm{cos}}^{2}(\sigma )+{\displaystyle \frac{(n-1)}{{x}^{2}}}{\mathrm{sin}}^{2}(\sigma ),\\ \phantom{\rule{2em}{0ex}}d\mathit{\Sigma}={x}^{n-1}{\mathrm{sin}}^{m-1}(y)d{\omega}_{m-1}d{\omega}_{n-1}dt,\end{array}$$(12)

where *d* *ω*_{m} is the volume element of the *m*-sphere and
$$\begin{array}{}\mathit{\Delta}u={u}^{\u2033}+((n-1){\displaystyle \frac{{x}^{\prime}(t)}{x(t)}}+(m-1)\mathrm{cot}(y(t)){y}^{\prime}(t)){u}^{\prime}\end{array}$$

for an invariant function *u*(*t*). Hence we can rewrite the index form for hypersurfaces generated by solutions of (11) on an invariant function *u*(*t*), *t* ∈ (*t*_{0}, *t*_{1}), as
$$\begin{array}{}{\displaystyle {Q}_{\mathit{\Sigma}}(u)=-{\omega}_{m-1}{\omega}_{n-1}\underset{{t}_{0}}{\overset{{t}_{1}}{\int}}uLu{x}^{n-1}{\mathrm{sin}}^{m-1}(y)\phantom{\rule{thinmathspace}{0ex}}dt,}\end{array}$$

with
$$\begin{array}{}{\displaystyle Lu\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& ={u}^{\u2033}+({\displaystyle \frac{(n-1)}{x}}{x}^{\prime}(t)+(m-1)\mathrm{cot}(y){y}^{\prime}(t)){u}^{\prime}\\ & \phantom{=}+((m-1){\mathrm{cos}}^{2}(\sigma )+{\sigma}^{\prime 2}+(m-1){\mathrm{cot}}^{2}(y){\mathrm{cos}}^{2}(\sigma )+{\displaystyle \frac{(n-1)}{{x}^{2}}}{\mathrm{sin}}^{2}(\sigma ))u.\end{array}$$(13)

There are two canonical examples of invariant hypersurfaces of constant mean curvature *h* in ℝ^{n} × **S**^{m}: the product
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ = **S**^{n − 1} × **S**^{m} of a sphere of constant mean curvature *h* in ℝ^{n} with **S**^{m} and the product
$\begin{array}{}{\mathit{\Sigma}}_{h}^{2}\end{array}$ = ℝ^{n} × **S**^{m − 1} of ℝ^{n} and a hypersphere of mean curvature *h* in **S**^{m}. In terms of Equation (11) they are given by the constant solutions *φ* (*t*) = ((*n* − 1)/*h*, *t*) and *φ*(*t*) = (*t*, *x*_{h}), respectively. We first consider these two cases:

#### Lemma 5.1

*For m*, *n* > 1, *the hypersurface*
$\begin{array}{}{\mathit{\Sigma}}_{h}^{2}\end{array}$ *given by the constant solution of Equation* (11) (*with h constant*), *φ*(*t*) = (*t*, *x*_{h}), *is unstable*.

#### Proof

Let *m*, *n* > 1. An invariant function *u*:
$\begin{array}{}{\mathit{\Sigma}}_{h}^{2}\end{array}$ → ℝ is a radial function on ℝ^{n}. We have
$$\begin{array}{}{\displaystyle {Q}_{{\mathit{\Sigma}}_{h}^{2}}(u)={\omega}_{m-1}{\mathrm{sin}}^{m-1}({x}_{h})\underset{{\mathbb{R}}^{n}}{\int}\parallel \mathrm{\nabla}u{\parallel}^{2}-k{u}^{2}\phantom{\rule{thinmathspace}{0ex}}dx}\end{array}$$

with
$\begin{array}{}k=((\frac{h}{m-1}{)}^{2}+1)(m-1).\end{array}$ Choose any *u* ≠ 0 with compact support such that ∫_{ℝn} *u* = 0. Then for each *α* > 0 the functions *u*_{α} (*x*) = *u*(*α x*) all have mean 0 and one can pick *α* so that
$\begin{array}{}{Q}_{{\mathit{\Sigma}}_{h}^{2}}({u}_{\alpha})<0;\end{array}$ this is just the fact that the *bottom of the spectrum* of ℝ^{n} is 0. □

#### Lemma 5.2

*For m*, *n* > 1, *the hypersurface*
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ *given by the constant solution of Equation* (11) (*with h constant*), *φ*(*t*) = ((*n* − 1)/*h*, *t*), *is unstable if and only if*
$\begin{array}{}h>\sqrt{m(n-1)}.\end{array}$

#### Proof

Note that
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ = **S**^{n − 1}(*r*) × **S**^{m}, where **S**^{n − 1}(*r*) denotes the sphere of radius *r* and *r*^{2} = (*n* − 1)^{2}/*h*^{2}. Thus for a function *u*:
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ → ℝ with mean 0, we get
$$\begin{array}{}{\displaystyle {Q}_{{\mathit{\Sigma}}_{h}^{1}}(u)=\underset{{\mathit{\Sigma}}_{h}^{1}}{\int}(\parallel \mathrm{\nabla}u{\parallel}^{2}-\frac{{h}^{2}}{n-1}{u}^{2})\phantom{\rule{thinmathspace}{0ex}}d{\mathit{\Sigma}}_{h}^{1}.}\end{array}$$

Hence the instability condition is equivalent to
$\begin{array}{}{\int}_{{\mathit{\Sigma}}_{h}^{1}}||\mathrm{\nabla}u|{|}^{2}\phantom{\rule{thinmathspace}{0ex}}d{\mathit{\Sigma}}_{h}^{1}\phantom{\rule{thickmathspace}{0ex}}/\phantom{\rule{thickmathspace}{0ex}}{\int}_{{\mathit{\Sigma}}_{h}^{1}}{u}^{2}\phantom{\rule{thinmathspace}{0ex}}d{\mathit{\Sigma}}_{h}^{1}<\frac{{h}^{2}}{n-1}\end{array}$ for some *u* with mean 0. We recall that the first eigenvalue *λ*_{1} of the positive Laplacian on
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ satisfies
$$\begin{array}{}{\displaystyle {\lambda}_{1}=\underset{\{u:\int u=0\}}{inf}\{\frac{{\int}_{{\mathit{\Sigma}}_{h}^{1}}||\mathrm{\nabla}u|{|}^{2}\phantom{\rule{thinmathspace}{0ex}}d{\mathit{\Sigma}}_{h}^{1}}{{\int}_{{\mathit{\Sigma}}_{h}^{1}}{u}^{2}\phantom{\rule{thinmathspace}{0ex}}d{\mathit{\Sigma}}_{h}^{1}}\}.}\end{array}$$

Thus the instability condition is equivalent to
$\begin{array}{}{\lambda}_{1}<\frac{{h}^{2}}{n-1}.\end{array}$

For
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ = **S**^{n − 1}(*r*) × **S**^{m} we have
$\begin{array}{}{\lambda}_{1}=min\{m,\frac{n-1}{{r}^{2}}\}=min\{m,\frac{{h}^{2}}{n-1}\}.\end{array}$ We conclude that
$\begin{array}{}{\mathit{\Sigma}}_{h}^{1}\end{array}$ is unstable if and only if
$\begin{array}{}m<\frac{{h}^{2}}{n-1}.\end{array}$

We are now ready to prove Theorem 1.3.

#### Proof of Theorem 1.3

Consider a hypersurface *Σ* that belongs to one of the families of noncompact constant mean curvature hypersurfaces described in Theorem 1.1 or Theorem 1.2. Let *f*(*s*) = (*x*(*s*), *y* (*s*), *σ* (*s*)) be the solution of Equation (11) that generates *Σ*. We have seen in the previous sections that in all the cases considered *Σ* is described by a curve which has at least one end which is given by the graph of a function *p* satisfying Equation (2). We have seen that *p* has a sequence of maxima and minima as *x* → ∞.

Let {(*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), (*x*_{3}, *y*_{3}), …} be the set of alternating maxima and minima of *f*(*s*), *p*(*x*_{i}) = *y*_{i}. Consider the function *u* = sin(*σ*). Direct computation yields
$$\begin{array}{}{\displaystyle uLu=(n-1)\frac{{y}^{\prime 2}}{{x}^{2}},}\end{array}$$

where *L* is the operator given by Equation (13). Of course, *u* can be extended by symmetry to a field on all of *Σ*. It follows that
$$\begin{array}{}{\displaystyle {Q}_{\mathit{\Sigma}}(u)=-\underset{\mathit{\Sigma}}{\int}uLu\phantom{\rule{thinmathspace}{0ex}}d\mathit{\Sigma}<0.}\end{array}$$

We next note that *u* vanishes at the set of alternating maxima and minima, and consider
$$\begin{array}{}{u}_{1}(x)=\left\{\begin{array}{ll}u(x)& \text{if\hspace{0.17em}}x\in [{x}_{1},{x}_{2}]\\ 0& \text{otherwise}\end{array}\right.\end{array}$$(14)

and similarly,
$$\begin{array}{}{u}_{2}(x)=\left\{\begin{array}{ll}u(x)& \text{if\hspace{0.17em}}x\in [{x}_{2},{x}_{3}]\\ 0& \text{otherwise}\end{array}\right.\end{array}$$(15)

These two functions have disjoint supports and satisfy *Q*_{Σ}(*u*_{i}) < 0, *i* = 1, 2. It follows that by taking a linear combination of the two, *u* = *C*_{1}*u*_{1}+*C*_{2}*u*_{2}, we can construct a function such that ∫_{Σ}*u* = 0 and *Q*_{Σ}(*u*) < 0.

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