Remarks on Hypersurfaces in a Unit Sphere

Abstract

In 1973, D. A. Hoffman proved that a complete immersed surface in $\mathbb {S}^3$ with constant mean curvature $H\ne 0$ and Gauss curvature which does not change sign must be a sphere or a flat torus. For the n-dimensional case with $n\ge 3$, we prove that Hoffman’s theorem cannot be extended when we consider constant higher-order mean curvature $H_r$, $2\le r<n$, and Gauss–Kronecker curvature K, instead of the mean curvature H and Gauss curvature, respectively. More precisely, we show the existence of non-isoparametric hypersurfaces $\Sigma$ in $\mathbb {S}^{n+1}$, $n\ge 3$, with constant higher-order mean curvature and whose Gauss–Kronecker curvature K does not change sign. Besides, we also derive estimates for the infimum and the supremum of the principal curvatures of a hypersurface with constant higher-order mean curvature.

1 Introduction and Statement of the Results

The problem of characterizing surfaces immersed in a Riemannian 3-manifold with constant mean curvature (CMC) constitutes a classical theme into the theory of isometric immersions. There are several papers on classification results of constant mean curvature surfaces in different spaces such as space forms and, more generally, in homogeneous three-manifolds. See, for instance, [1, 6, 8] and references therein. In particular, Hoffman [11] characterized totally umbilical spheres and flat tori as the only complete surfaces immersed into the sphere ${\mathbb {S}}^3$ with constant mean curvature $H\ne 0$ and whose Gaussian curvature does not change sign (see Appendix A):

Theorem 1

Let $\Sigma$ be a complete surface immersed into the Euclidean sphere ${\mathbb {S}}^3$ with constant mean curvature $H\ne 0$. Assume that $\Sigma$ is not totally umbilical. If its Gaussian curvature K does not change sign, then $K= 0$ and $\Sigma$ is isometric to a flat torus ${\mathbb {S}}^1(\rho ) \times {\mathbb {S}}^1(\sqrt{1-\rho ^2}) \subset {\mathbb {S}}^3$, with radius $0<\rho <1$.

A (connected) hypersurface in ${\mathbb {S}}^{n+1}$ is called isoparametric if its principal curvatures are constant functions (see, for instante, Chapter 3 in [9]). In particular, an isoparametric surface in ${\mathbb {S}}^3$ must be an open subset of a great or small sphere (totally umbilic) or a standard product of two circles (flat torus), see Section 3.8 in [9] about it based on the number of distinct principal curvatures. Then as an immediate consequence of Theorem 1, we obtain:

Theorem 2

Let $\Sigma$ be a complete surface immersed into the Euclidean sphere ${\mathbb {S}}^3$ with constant mean curvature $H\ne 0$. If its Gaussian curvature K does not change sign, then $\Sigma$ is an isoparametric surface.

In higher dimension, Theorems 1 and 2 suggest a general result for the case of hypersurfaces with constant higher-order mean curvature $H_r$ in the Euclidean sphere ${\mathbb {S}}^{n+1}$. We recall that if $\Sigma$ is an oriented hypersurface in ${\mathbb {S}}^{n+1}$ with principal curvatures $\kappa _1,\dots ,\kappa _n$ and $1\le r\le n$, the r-th mean curvature $H_r$ is defined as

$$\begin{aligned} {n\atopwithdelims ()r} H_r=\sum _{1\le i_1<\cdots < i_r\le n} \kappa _{i_1}\cdots \kappa _{i_r}. \end{aligned}$$

When $r=1$, $nH_1= \sum _{i=1}^n \kappa _{i}$ is nothing but the mean curvature of $\Sigma$, which is the main extrinsic curvature of $\Sigma$. When $r=2$, we obtain $H_2=R-1$ where R is the normalized scalar curvature of $\Sigma$, and for $r=n$, we get the Gauss–Kronecker curvature, which is the product of the principal curvatures of the hypersurface and it is usually denoted by K (see Sect. 2 for more details).

In general, when r is odd, the sign of $H_r$ depends on the chosen orientation. On the other hand, when r is even, its value does not depend on the chosen orientation. Moreover, the curvature $H_r$ is intrinsic if r is even. In particular, the Gauss–Kronecker curvature K is intrinsic if n is even and intrinsic up to sign if n is odd (See [7, Exercise 1.9]). Inspired by Theorem 2, and since the Gauss curvature of a surface is an intrinsic object, we ask if it is possible to consider the Gauss–Kronecker curvature instead of the Gaussian curvature as follows.

Question 3

Let $\Sigma$ be a complete oriented hypersurface in ${\mathbb {S}}^{n+1}$ ($n \ge 3$) with constant r-th mean curvature $H_r \ne 0$ ($2\le r< n$). Suppose that its Gauss–Kronecker curvature K does not change sign. Is it true that $\Sigma$ is an isoparametric hypersurface?

Let us observe that when the ambient space is the $(n+1)$-dimensional Euclidean space ${\mathbb {R}}^{n+1}$ the answer is affirmative (see Theorem 3 in [6]) to hypersurfaces with two distinct principal curvatures, one of them being simple:

Theorem 4

Let $n \ge 3$ and $2\le r < n$. Let $\Sigma$ be a complete hypersurface immersed into the Euclidean space ${\mathbb {R}}^{n+1}$ with constant r-th mean curvature $H_r\ne 0$ and two distinct principal curvatures, one of them being simple. If its Gauss–Kronecker curvature K does not change sign, then $K=0$ and $\Sigma$ is isometric to a cylinder ${\mathbb {R}} \times {\mathbb {S}}^{n-1}(\rho ) \subset {\mathbb {R}}^{n+1}$ with radius $\rho >0$.

The approach used by authors in [6] relies on the so-called principal curvature theorem due to B. Smyth and F. Xavier (see Section 1 in [14]). In the present note we give a negative answer to the Question 3, that is, Theorem 2 does not extend to hypersurfaces with constant higher-order mean curvature $H_r$ in ${\mathbb {S}}^{n+1}$ and whose Gauss–Kronecker curvature K does not change sign. Specifically, we prove the following result.

Theorem 5

Let $n\ge 3$ and $2\le r< n$. Then for each $H_r>0$, there exists a one-parameter family of complete non-isoparametric hypersurfaces $\Sigma$ immersed in ${\mathbb {S}}^{n+1}$ with constant r-th mean curvature $H_r$ and strictly negative Gauss–Kronecker curvature everywhere.

Corollary 6

For each $R>1$, there exists a one-parameter family of complete non-isoparametric hypersurfaces $\Sigma$ immersed in ${\mathbb {S}}^{n+1}$ ($n\ge 3$) with constant normalized scalar curvature R and strictly negative Gauss–Kronecker curvature everywhere.

On the other hand, it is well known that the hypersurfaces obtained from a standard product of two spheres are the only isoparametric hypersurfaces with two distinct principal curvatures $\lambda _0$ and $\mu _0$ with multiplicities $n - m$ and m, respectively. In this case, the principal curvatures satisfy $\lambda _0\mu _0+1=0$ and these hypersurfaces are also called the Clifford hypersurfaces

$$\begin{aligned} \Sigma _{n-m,m}&=\{ (x,y) \in {\mathbb {R}}^{n-m+1}\times {\mathbb {R}}^{m+1} \,\,\,:\,\, \Vert x\Vert = s \,\,\,\text {and}\,\,\, \Vert y\Vert =t \} \\&= {\mathbb {S}}^{n-m}\left( s \right) \times {\mathbb {S}}^m \left( t \right) \subset {\mathbb {S}}^{n+1} \end{aligned}$$

where $s^2+t^2 =1$ [2, 17]. Here, the value within the parentheses denotes the radius of the corresponding sphere. In this case

$$\begin{aligned} s = \frac{1}{\sqrt{1+\lambda _0^2}} \qquad \text {and} \qquad t = \frac{1}{\sqrt{1+\mu _0^2}}=\frac{|\lambda _0|}{\sqrt{1+\lambda _0^2}}. \end{aligned}$$

In particular, if $m=1$ we obtain

$$\begin{aligned} \Sigma _{n-1,1}= {\mathbb {S}}^{n-1}\left( \frac{1}{\sqrt{1+\lambda _0^2}}\right) \times {\mathbb {S}}^1 \left( \frac{|\lambda _0|}{\sqrt{1+\lambda _0^2}}\right) \subset {\mathbb {S}}^{n+1} \end{aligned}$$

(1)

where $\lambda _0$ satisfies the identity $nH_r=(n-r)\lambda _0^{r} -r \lambda _0^{r-2}$, see also equation (17) below.

As an application of a result on the principal curvatures of a hypersurface given by Smyth [22], following similar steps and ideas of the proof of Theorem 3 in [6], we also prove the following result.

Theorem 7

Let $n\ge 3$ and $2\le r< n$. For a given constant $H_r>0$, let $\lambda _0$ be the positive solution of

$$\begin{aligned} (n-r) x^{r} -r x^{r-2}-nH_r =0 \qquad \text {and let} \qquad \mu _0 = -1/\lambda _0. \end{aligned}$$

(2)

Let $\Sigma$ be a complete oriented hypersurface in ${\mathbb {S}}^{n+1}$ with constant r-th mean curvature $H_r$ and two distinct principal curvatures $\lambda$ and $\mu$ with multiplicities $(n-1)$ and 1, respectively. If r is even, the orientation of $\Sigma$ is chosen so that $\lambda >0$. Otherwise, in the case where r is odd, assume further that $\lambda >0$. Then

$$\begin{aligned} \inf _\Sigma \mu \le \mu _0<0, \qquad \sup _\Sigma \mu \le {H_r}^{1/r}\le \inf _\Sigma \lambda ,\qquad 0<\lambda _0 \le \sup _\Sigma \lambda . \end{aligned}$$

Under the hypothesis of our Theorem 7, we will see in Sect. 2 that $\lambda (p)\ne 0$ for every $p\in \Sigma$. On the other hand, when $r=2$ we have $H_2=R-1$, where R is the normalized scalar curvature. Then, the first equation of (2) reduces to

$$\begin{aligned} (n-2)x^2 = n (R-1)+2. \end{aligned}$$

It follows that the isoparametric hypersurface $\Sigma _{n-1,1}$ given by (1) has principal curvatures

$$\begin{aligned} \lambda _0 = \sqrt{\frac{n(R-1) + 2}{n-2}} \qquad \text {and} \qquad \mu _0= - \sqrt{\frac{n-2}{n(R-1) + 2}}. \end{aligned}$$

(3)

Thus, as a direct application of Theorem 7, we obtain the following consequence.

Corollary 8

Let $\Sigma ^n$ be a complete oriented hypersurface in ${\mathbb {S}}^{n+1}$ ($n\ge 3$) with constant normalized scalar curvature $R> 1$ and having two distinct principal curvatures, one of them being simple. Then

$$\begin{aligned} (n-1) \frac{n(R-1)+2}{n-2} + \frac{n-2}{n(R-1)+2} \le \sup _\Sigma |A|^2 \end{aligned}$$

(4)

where |A| is the norm of the second fundamental form of $\Sigma$.

The estimate (4) is the right inequality of (8) in [16, Theorem 1.6]. When the inequality (4) is written in terms of the total umbilicity tensor, then it agrees with the estimate given by Alías, García-Martínez and Rigoli in [4, Theorem 2]. Indeed, if $\Phi =A-HI$, with I the identity operator on $T\Sigma$. Then, $\textrm{tr}(\Phi )=0$ and $|\Phi |^2=\textrm{tr}(\Phi ^2)=|A|^2-nH^2\ge 0.$ It follows from (7) that

$$\begin{aligned} |\Phi |^2=\frac{n-1}{n}|A|^2-(n-1)(R-1).\end{aligned}$$

Hence, (4) becomes

$$\begin{aligned} \sup _\Sigma |\Phi |^2\ge \frac{n(n-1)R^2}{(n-2)(nR-(n-2))}>0. \end{aligned}$$

In Corollary 8, we assume that $\Sigma$ has two different principal curvatures, one of them simple, and the estimate in [4] holds for more general complete hypersurfaces. However, it is worth pointing out that we obtained this estimate using a completely different technique.

2 Preliminary Results

Let ${\mathbb {S}}^{n+1}$ denote the Euclidean unit sphere

$$\begin{aligned} {{\mathbb {S}}^{n+1}}=\{x=(x_1,\ldots ,x_{n+2})\in {\mathbb {R}}^{n+2}: \, |x|=1 \}\subset {\mathbb {R}}^{n+2}. \end{aligned}$$

Let $\Sigma$ be an orientable and connected hypersurface isometrically immersed into ${\mathbb {S}}^{n+1}$. Choose a unit normal field N along $\Sigma$ and we let A denote the second fundamental tensor of the immersion, i.e., ${\langle AX ,Y \rangle } = {\langle {\overline{\nabla }}_X Y ,N \rangle }$ for every $X,Y\in {\mathfrak {X}}(\Sigma ),$ where ${\overline{\nabla }}$ is the connection of ${\mathbb {S}}^{n+1}$. The relation between the curvature tensor of $\Sigma$ and the curvature of $\mathbb {S}^{n+1}$ is expressed via the Gauss equation, which can be written in terms of A in the form

$$\begin{aligned} R(X,Y)Z={\langle X ,Z \rangle }Y-{\langle Y ,Z \rangle }X+{\langle AX ,Z \rangle }AY-{\langle AY ,Z \rangle }AX \end{aligned}$$

(5)

for every $X,Y,Z\in {\mathfrak {X}}(\Sigma )$. In particular, the Ricci and the scalar curvatures of the hypersurface are given by

$$\begin{aligned} \textrm{Ric}(X,X)=(n-1)|X|^2+nH{\langle AX ,X \rangle }-|AX|^2, \end{aligned}$$

(6)

where $H=\frac{1}{n}{\text {Tr}}(A)$ is the mean curvature of $\Sigma$, and

$$\begin{aligned} {\textrm{Scal}}=n(n-1)R=n(n-1)+n^2H^2-|A|^2, \end{aligned}$$

(7)

respectively. Here, R is the normalized scalar curvature.

If $\kappa _1,\ldots , \kappa _n$ are the principal curvatures of $\Sigma$, the r-th mean curvature $H_r$ of the hypersurface is defined by

$$\begin{aligned} {n\atopwithdelims ()r} H_r=\sum _{1\le i_1<\cdots < i_r\le n} \kappa _{i_1}\cdots \kappa _{i_r}. \end{aligned}$$

In particular, when $r=n$ we obtain the Gauss–Kronecker curvature of $\Sigma$

$$\begin{aligned} K = H_n = \textrm{det}(A)= \kappa _{1}\cdots \kappa _{n}. \end{aligned}$$

Henceforth, we consider hypersurfaces with two distinct principal curvatures $\lambda$ and $\mu$, with multiplicities $n-1$ and 1, respectively. Then, the r-th mean curvature can be written as

$$\begin{aligned} nH_r = \lambda ^{r-1}( (n-r)\lambda + r\mu ). \end{aligned}$$

(8)

Remark 9

It follows from do Carmo and Dajczer [10, Theorem 4.2] that a hypersurface of ${\mathbb {S}}^{n+1}$ with constant r-th mean curvature $H_r$ ($2\le r< n$) and two distinct principal curvatures, with multiplicities $n-1$ and 1, is contained in a rotation hypersurface.

The Gauss–Kronecker curvature of $\Sigma$ is

$$\begin{aligned} K=H_n = \lambda ^{n-1}\mu . \end{aligned}$$

Note that if $H_r\ne 0$ and $r\ge 2$, then from (8) we obtain that the principal curvature $\lambda$ does not vanish on $\Sigma$ and the principal curvature $\mu$ can be written in terms of $\lambda$ as

$$\begin{aligned} \mu (\lambda ) = \frac{nH_r}{ r \lambda ^{r-1}} - \frac{n-r}{r}\lambda . \end{aligned}$$

(9)

From here on we suppose that $H_r > 0$, with $2 \le r <n$. When $r=2$, we have from (7) and (8) that $H_2=R-1$. Then, (8) and (9) reduce to

$$\begin{aligned} n(R-1) =(n-2)\lambda ^2 + 2\lambda \mu = \lambda ( (n-2)\lambda + 2 \mu ) \end{aligned}$$

and

$$\begin{aligned} \mu (\lambda ) = \frac{n(R-1)}{2\lambda } - \frac{n-2}{2}\lambda , \end{aligned}$$

respectively. For the proof of our results, we need the following auxiliar result (see Lemma 10 in [6]).

Lemma 10

Let $H_r>0$, with $r\ge 2$, $\lambda >0$. Then

• (a) The function $\mu (\lambda )$ given by (9) is decreasing on $(0,\infty )$.

• (b) The only positive root of the equation $\mu (\lambda )=0$ is given by

  $$\begin{aligned} {\hat{\lambda }}=\left( \frac{nH_r}{n-r}\right) ^{1/r}. \end{aligned}$$

  (10)

• (c) $\mu (\lambda )=\lambda$ if and only if $\lambda =\mu =H_r^{1/r}$.

In order to prove our results, we will consider connected rotational hypersurfaces in the Euclidean sphere with constant higher-order mean curvature. We first recall their definition and some basic properties.

It follows from [10, Section 2] that any complete rotation hypersurface $\Sigma$ with respect to the geodesic $\gamma (s)=( 0, \ldots , 0,\cos s, \sin s) \in {\mathbb {S}}^{n+1}$ is given by an immersion $\phi : {\mathbb {S}}^{n-1} \times {\mathbb {R}} \rightarrow {\mathbb {S}}^{n+1}$ defined by

$$\begin{aligned} \phi (y,t) = \left( \rho (t) \,y, \sqrt{1-\rho ^2(t)} \cos \theta (t), \sqrt{1-\rho ^2(t)} \sin \theta (t)\right) \end{aligned}$$

(11)

where $y\in {\mathbb {S}}^{n-1}$, $\rho : {\mathbb {R}}\rightarrow (0,1)$ is a differentiable function that satisfies $\rho ^2+{\dot{\rho }}^2\le 1$ and

$$\begin{aligned} \theta (t)= \int _0^t \frac{ \sqrt{1- \rho ^2(\tau ) - {\dot{\rho }}^2(\tau )} }{ 1 - \rho ^2(\tau ) } \,d\tau . \end{aligned}$$

(12)

The profile curve of $\Sigma$ is

$$\begin{aligned} \alpha (t) = \left( \rho (t), 0,\ldots ,0, \sqrt{1-\rho ^2(t)} \cos \theta (t), \sqrt{1-\rho ^2(t)} \sin \theta (t)\right) . \end{aligned}$$

In this paper we consider the projection of the profile curve over the unit disk $x_{n+1}^2 + x_{n+2}^2 \le 1$. If $\rho (t)=\sin t$ we obtain a totally geodesic sphere ${\mathbb {S}}^{n} \subset {\mathbb {S}}^{n+1}$ given by $\phi (y,t) = \left( y\sin t, \cos t, 0\right)$. However, we will omit totally umbilical spheres. See also [20] for more explicit formulas on immersions of hypersurfaces in Riemannian and semi-Riemannian manifolds. In [10, Proposition 3.2], do Carmo and Dajczer calculated the principal curvatures of $\Sigma$,

$$\begin{aligned} \lambda = \frac{ \sqrt{1- \rho ^2 - {\dot{\rho }}^2} }{\rho } \qquad \text{ and } \qquad \mu = -\frac{ \ddot{\rho } + \rho }{ \sqrt{1- \rho ^2 - {\dot{\rho }}^2 } } \end{aligned}$$

(13)

of multiplicities $(n-1)$ and 1, respectively. Let us observe that if $0<\rho _0<1$ is constant, then the profile curve is a circle centered at the origin and the principal curvatures of $\Sigma$ are constant, that is, $\Sigma$ is defined by

$$\begin{aligned} \phi (y,s) = \left( \rho _0 y, \sqrt{1-\rho ^2_0} \,\cos \theta (s), \sqrt{1-\rho ^2_0} \,\sin \theta (s) \right) , \quad {y\in {\mathbb {S}}^{n-1}, s\in {\mathbb {R}}} \end{aligned}$$

where $\displaystyle { \theta (s)= \frac{s}{\sqrt{1-\rho ^2_0}}}.$ In this case notice that $\Sigma = {\mathbb {S}}^{n-1}(\rho _0) \times {\mathbb {S}}^1 (\sqrt{1-\rho _0^2})$.

From (8) and (13), we obtain that the function $\rho$ must satisfy the differential equation

$$\begin{aligned} nH_r \rho ^r = (n-r)(1-\rho ^2 - {\dot{\rho }}^2)^{r/2}- r \rho (\ddot{\rho } + \rho ) (1-\rho ^2 - {\dot{\rho }}^2)^{(r-2)/2}. \end{aligned}$$

(14)

Taking into account that $H_r$ is constant, we may multiplying (14) by $\rho ^{n-r-1}$ and integrating to obtain the function

$$\begin{aligned} G(\rho ,{\dot{\rho }}) = \rho ^{n-r}( (1- \rho ^2 - {\dot{\rho }}^2)^{r/2}-H_r \rho ^r)=C, \end{aligned}$$

(15)

for some a constant C, see [18, Proposition 2]. Because of the form of the principal curvatures in (13), we are interested only in the behavior of G in the region $\rho ^2+{\dot{\rho }}^2< 1$, with $\rho >0$.

From Lemma 2.7, Lemma 3.1 and Proposition 3.2 of [16] we have the function G has only one critical point of the form $(\rho _0,0)$, for which

$$\begin{aligned} x_0= \frac{\rho _0}{\sqrt{1-\rho _0^2}} \qquad \text {satisfies} \qquad nH_r x^r + rx^2 - (n-r)=0. \end{aligned}$$

(16)

Moreover, $C_0:=G(\rho _0,0)>0$ is a local maximum and for each $0<C<C_0$ the level set $G(\rho ,{\dot{\rho }})=C$ is a closed curve surrounding $(\rho _0,0)$ and contained in the region interior to the zero level set of G.

3 Key Lemmas

Notice that the critical point $(\rho _0,0)$ of G corresponds to an isoparametric hypersurface

$$\begin{aligned} {\mathbb {S}}^{n-1}\left( \rho _0\right) \times {\mathbb {S}}^1 \left( \sqrt{1-\rho _0^2}\right) \subset {\mathbb {S}}^{n+1} \qquad \text {with} \qquad \rho _0^2= \frac{1}{1+\lambda _0^2}= \frac{x_0^2}{1+x_0^2}. \end{aligned}$$

Hence, $\lambda _0=\frac{1}{x_0}$ and (16) is equivalent to

$$\begin{aligned} nH_r +r \lambda _0^{r-2} - (n-r)\lambda _0^{r}=0. \end{aligned}$$

(17)

Translating the above properties of G into the corresponding rotational hypersurfaces, we have the following key lemma.

Lemma 11

Let $H_r>0$. Then for every $C\in (0,C_0]$, the level curve $G(\rho ,{{\dot{\rho }}})=C$ is totally contained in the region $\rho ^2+{\dot{\rho }}^2<1$, and gives rise to a complete hypersurface $\Sigma$.

Theorem 7 will be an application of a result given by Smyth in [22]. This result gives geometric information on the principal curvatures of a complete hypersurface immersed in the unit sphere. For each complete and orientable hypersurface $\Sigma$ in ${\mathbb {S}}^{n+1}$ oriented by a normal field N to $\Sigma$ and with associated second fundamental form A, we set

$$\begin{aligned}\Lambda =\{\kappa \in {\mathbb {R}} \,|\, \kappa \; \text {is an eigenvalue of }A_p\text { for some }p \in \Sigma \},\end{aligned}$$

calling it the principal curvature set of $\Sigma$. Let $\Lambda ^{\pm } = \Lambda \cap {\mathbb {R}}^{\pm }$. When $\Lambda ^{\pm }$ are both non-empty we set

$$\begin{aligned} \alpha = \inf \Lambda ^-, \qquad a=\sup \Lambda ^-, \qquad b=\inf \Lambda ^+, \qquad \beta =\sup \Lambda ^+. \end{aligned}$$

For the sake of completeness and for the readers’ convenience, we state Smyth’s result.

Lemma 12

(B. Smyth in [22]). Let $f: \Sigma ^n \rightarrow {\mathbb {S}}^{n+1}$ be a complete immersed orientable hypersurface in the unit sphere which is not totally geodesic.

• (i) If $\Lambda ^+$ or $\Lambda ^-$ is empty, then $\Lambda$ is connected, $\Sigma ^n$ is diffeomorphic to ${\mathbb {S}}^n$ and f embeds $\Sigma ^n$ as the boundary of a convex domain in a closed hemisphere.

• (ii) If $\Lambda ^{\pm }$ are both non-empty, then in the double inequality

  $$\begin{aligned} \alpha \beta \le -1 \le ab, \end{aligned}$$

  the left-hand inequality fails if and only if f is parallel to such a hypersurface as described in i), while the right-hand inequality fails only if f immerses $\Sigma ^n$ as a totally geodesic sphere bundle over a complete even-dimensional submanifold of ${\mathbb {S}}^{n+1}$, in which case $\# \Lambda$ is odd $\ge 3$.

Here, $\# \Lambda$ is the number of connected components of $\Lambda$. There is a similar result for complete orientable hypersurfaces in the Euclidean space [14] and some applications in [3, 5, 13].

4 Proof of Theorem 7

The proof follows similar steps and ideas of the proof of Theorem 3 in [6]. First of all note that $\lambda (p)\ne {H_r}^{1/r}$ for every $p\in \Sigma$. In fact, if $\lambda (p_0)= {H_r}^{1/r}$ at a point $p_0\in \Sigma$ then we would have from Lemma 10 that

$$\begin{aligned} \mu (p_0)= {H_r}^{1/r}=\lambda (p_0), \end{aligned}$$

which contradicts that $\Sigma$ has two distinct principal curvatures. From continuity of $\lambda$ and connectedness of $\Sigma$, it follows that either

$$\begin{aligned} 0< \lambda (p) <{H_r}^{1/r} \quad \text { for every } p\in \Sigma \end{aligned}$$

(18)

or

$$\begin{aligned} {H_r}^{1/r} < \lambda (p) \quad \text { for every } p\in \Sigma . \end{aligned}$$

(19)

Let us first see that case (18) cannot happen. In this case, it follows from Lemma 10 that

$$\begin{aligned} 0<\lambda (p)<{H_r}^{1/r}<\mu (p) \quad \text { for every } p\in \Sigma . \end{aligned}$$

(20)

Hence, $\Lambda ^- =\emptyset$. Thus, Lemma 12 (i) implies that $\Lambda$ is connected and $\Sigma$ is diffeomorphic to a sphere ${\mathbb {S}}^n$. In particular, $\Sigma$ is compact and we can consider

$$\begin{aligned} \lambda _*=\min _\Sigma \lambda , \quad \lambda ^*=\max _\Sigma \lambda , \quad \mu _*=\min _\Sigma \mu , \quad \mu ^*=\max _\Sigma \mu . \end{aligned}$$

From the compactness of $\Sigma$ and from (20), we obtain that $[\lambda _*, \lambda ^*]$ and $[\mu _*, \mu ^*]$ are disjoint. Thus,

$$\begin{aligned} \Lambda =[\lambda _*, \lambda ^*]\cup [\mu _*, \mu ^*] \end{aligned}$$

is disconnected, with $\Lambda ^{-}=\emptyset$. But this contradicts Lemma 12 (i). Hence, the case (18) cannot happen. As a consequence, we have that ${H_r}^{1/r} < \lambda (p)$ for every $p\in \Sigma$. It follows that

$$\begin{aligned} \mu (p)< {H_r}^{1/r}< \lambda (p) \quad \text {for every }p \in \Sigma . \end{aligned}$$

(21)

Hence, by taking the infimum and supremum of $\lambda$ and $\mu$, respectively, we get

$$\begin{aligned} \sup _\Sigma \mu \le {H_r}^{1/r}\le \inf _\Sigma \lambda . \end{aligned}$$

We claim that ${\hat{\lambda }} =\left( \frac{nH_r}{n-r}\right) ^{1/r}< \sup _\Sigma \lambda$. Otherwise, we suppose that

$$\begin{aligned} {H_r}^{1/r} < \lambda (p) \le {\hat{\lambda }} \quad \text { for every } p\in \Sigma . \end{aligned}$$

From (9), it follows that

$$\begin{aligned} 0 \le \mu (p)<{H_r}^{1/r} \quad \text { for every } p\in \Sigma . \end{aligned}$$

Thus, we have that $\Lambda ^- =\emptyset$. Again by Lemma 12 (i) implies that $\Lambda$ is connected and $\Sigma$ is compact. Then

$$\begin{aligned} 0\le \mu _*\le \mu ^{*}<{H_r}^{1/r} <\lambda _*\le \lambda ^{*}\le {\hat{\lambda }}. \end{aligned}$$

Therefore, the principal curvature set $\Lambda =[\mu _*, \mu ^*]\cup [\lambda _*, \lambda ^*]$ is disconnected, which is again a contradiction. Hence, ${\hat{\lambda }} < \sup _\Sigma \lambda$. It implies that

$$\begin{aligned} \inf _\Sigma \mu = \mu \left( \sup _\Sigma \lambda \right) < 0. \end{aligned}$$

See Fig. 2. So far we have

$$\begin{aligned} \alpha = \inf \Lambda ^- = \inf _\Sigma \mu <0,\qquad \beta =\sup \Lambda ^+ =\sup _\Sigma \lambda >0. \end{aligned}$$

(22)

We claim that

$$\begin{aligned} \alpha \beta \le -1. \end{aligned}$$

(23)

Let us assume that $\alpha \beta > -1$ and derive a contradiction. If the inequality of (23) fails, by Lemma 12, $\Sigma$ is parallel to a hypersurface ${\overline{M}}$ such that the set $\Lambda ^{{\overline{M}}}$ is connected. Then, since the number of connected components is the same for parallel hypersurfaces, $\Lambda ^\Sigma$ would also be connected. But this is a contradiction, because from (21) we see that $\Lambda ^\Sigma$ has 2 connected components. It follows from (9) and (22) that

$$\begin{aligned} \alpha \beta = \frac{nH_r}{ r (\lambda ^*)^{r-2}} - \frac{n-r}{r}(\lambda ^*)^2\le -1 \end{aligned}$$

where $\lambda ^*=\sup _\Sigma \lambda$. Hence

$$\begin{aligned} nH_r +r(\lambda ^*)^{r-2} - (n-r)(\lambda ^*)^{r} \le 0. \end{aligned}$$

From (16) and (17), we get

$$\begin{aligned} 0< \lambda _0 \le \lambda ^*=\sup _\Sigma \lambda . \end{aligned}$$

Finally, using Lemma 10 we obtain $\inf _\Sigma \mu = \mu _* \le \mu _0<0$. This finishes the proof of Theorem 7.

5 Proof of Theorem 5

Let $H_r>0$. Let $\Sigma$ be a rotational hypersurface with constant r-th mean curvature $H_r$ corresponding to the level curve

$$\begin{aligned} G(\rho ,{\dot{\rho }}) = \rho ^{n-r}( (1- \rho ^2 - {\dot{\rho }}^2)^{r/2}-H_r \rho ^r)=C \qquad \text {with }\; 0<C<C_0 \end{aligned}$$

where $C_0=G(\rho _0, 0)$ and $\rho _0$ is given by (16). By Lemma 11, $\Sigma$ is a complete hypersurface and its corresponding level curve surrounds the critical point $(\rho _0,0)$. Notice that the principal curvatures of the isoparametric hypersurface associated to the critical point $(\rho _0,0)$ are given by

$$\begin{aligned} \lambda _0 = \frac{\sqrt{1-\rho ^2_0}}{\rho _0} \qquad \text {and} \qquad \mu _0=- \frac{\rho _0}{ \sqrt{1-\rho _0^2}} \end{aligned}$$

(24)

of multiplicities $(n - 1)$ and 1, respectively.

For the level curve $G(\rho ,{\dot{\rho }})=C$, the coordinate $\rho$ varies in the compact interval $[\rho _{\min }, \rho _{\max }]$ where

$$\begin{aligned} \rho _{\min } = \min _\Sigma \rho , \qquad \rho _{\max } = \max _\Sigma \rho , \qquad \rho _0 \in [\rho _{\min }, \rho _{\max }] \subset {\mathbb {R}}^+. \end{aligned}$$

(25)

See Fig. 1. Therefore, for all C sufficiently close to $C_0$, the level curve $G=C$ cross the $\rho$-axis at the points $(\rho _{\max },0)$ and $(\rho _{\min },0)$. On the other hand, using (13) and (15), we can write the principal curvature $\lambda$ as

$$\begin{aligned} \lambda = \frac{ \sqrt{1-\rho ^2 - {\dot{\rho }}^2 } }{\rho } =\left( \frac{C}{\rho ^n}+ H_r\right) ^{1/r} = \lambda (\rho , C), \quad \qquad \rho >0. \end{aligned}$$

(26)

Fig. 1

Level curves of G in the $\rho {\dot{\rho }}$-plane

Fig. 2

Graph of the function $\mu (\lambda ) = \frac{nH_r}{r \lambda ^{r-1}} - \frac{n-r}{r}\lambda$ for $H_r>0$

Thus, by the monotony and continuity of $\lambda$ relative to $\rho$, $\lambda (\rho , C)$ attains its extreme values at the leftmost and rightmost point of the level curve, that is,

$$\begin{aligned} \lambda ^* \equiv \max _\Sigma \lambda = \lambda (\rho _{\min },C ) \qquad \text {and}\qquad \lambda _*\equiv \min _\Sigma \lambda = \lambda (\rho _{\max },C ). \end{aligned}$$

From (13) the values of $\lambda$ at these points are

$$\begin{aligned} \lambda ^* = \frac{\sqrt{1-\rho _{\min }^2}}{\rho _{\min }}, \qquad \qquad \lambda _* = \frac{\sqrt{1-\rho _{\max }^2} }{\rho _{\max }}. \end{aligned}$$

On the other hand, by (25), we have that $\lambda _0 \in [\lambda _*, \lambda ^*]$. It is worth pointing out that ${\hat{\lambda }} < \lambda _0$. Indeed, from (16) we find out that $nH_r x_0^r -(n-r) <0$. This implies that

$$\begin{aligned} x_0 = \frac{\rho _0}{\sqrt{1-\rho _0^2}} <\left( \frac{n-r}{nH_r} \right) ^{1/r}. \end{aligned}$$

From (10) and (24), this is equivalent to ${\hat{\lambda }} < \lambda _0$. Hence by continuity of $\lambda =\lambda (\rho , C)$, for C close enough to $C_0$ with $0<C<C_0$, the associated hypersurface satisfies

$$\begin{aligned} 0<{\hat{\lambda }}< \lambda _*< \lambda _0 < \lambda _*. \end{aligned}$$

Then $0<{\hat{\lambda }} < \lambda (p)$ for every $p \in \Sigma$. Using Lemma 10, we obtain $\mu < 0$ on $\Sigma$. Therefore, the Gauss–Kronecker curvature $K= \lambda ^{n-1} \mu$ is also negative on $\Sigma$. Moreover, since the associated level curve is not reduced to a point, the principal curvatures are not constant and hence the hypersurface is not isoparametric.

Fig. 3

On the left we have the profile curve of $\Sigma$. On the right we have the graph of its principal curvature $\lambda (t)$

6 Numerical Experiments

In this short section, we will pick some values for C to show the trace of the profile curve of some explicit examples of rotational hypersurfaces with constant scalar curvature in ${\mathbb {S}}^4$. We took $R=2$ ($H_2=1$). See also [19] and [21, Section 4] for other numerical examples.

Example 13

By the intermediate value theorem, there is a value of C near ${\tilde{C}}=0.15458$ such that the solution $\rho (t)=\rho _C(t)$ of the differential equation (15) has period $T=T_C$ near to ${\tilde{T}}=2.48753$. Here,

$$\begin{aligned} \rho _C(t+T)=\rho _C(t) \qquad \text {and} \qquad \rho _{{\tilde{C}}}(t+{\tilde{T}}) \approx \rho _{{\tilde{C}}}(t). \end{aligned}$$

In fact, using Mathematica software we find the values $0<C_1,C_2< C_0$ such that

$$\begin{aligned} \rho _{C_1}(t+T_{C_1}) \approx \rho _{C_1}(t), \qquad \rho _{C_2}(t+T_{C_2}) \approx \rho _{C_2}(t) \end{aligned}$$

and satisfy

$$\begin{aligned} \theta (T_{C_1})< 2\pi \left( \frac{2}{5}\right) < \theta (T_{C_2}) \end{aligned}$$

for some constants $T_{C_1}$, $T_{C_2}$. Hence, the hypersurface $\Sigma$ is defined by the value of C such that $C_1<C<C_2$ and the function $\theta (t)$ satisfies

$$\begin{aligned} \theta (T)=\theta (T_C) =2\pi \frac{l}{m} \qquad \text {with }l=2\text { and }m=5\text {.} \end{aligned}$$

(27)

Observe that the condition on C in (27) guarantees that the associated immersion $\phi$ satisfies

$$\begin{aligned} \phi (y, t + mT ) = \phi (y, t). \end{aligned}$$

The profile curve $(\sqrt{1-\rho (t)^2} \, \cos \theta (t), \sqrt{1-\rho (t)^2} \, \sin \theta (t) )$ is given in Fig. 3. The red piece represent the curve when t moves from 0 to T. Notice that the profile curve is closed and the immersion $\phi : {\mathbb {S}}^{2} \times {\mathbb {R}} \rightarrow {\mathbb {S}}^{4}$ given by

$$\begin{aligned} \phi (y,t) = \left( \rho (t) y, \sqrt{1-\rho ^2(t)} \cos \theta (t), \sqrt{1-\rho ^2(t)} \sin \theta (t)\right) \end{aligned}$$

(28)

has period 5T. It follows that the hypersurface defined by (28) is compact.

Fig. 4

The first picture shows the profile curve of $\Sigma \subset {\mathbb {S}}^{4}$ when $H_2=1$ and ${\tilde{C}}=0.0531$. The second picture shows the profile curve when $H_2=1$ and ${\tilde{C}}=0.2213$

Fig. 5

Profile curve of $\Sigma \subset {\mathbb {S}}^{4}$ when $H_2=1$ and ${\tilde{C}}=0.26345201$

In Fig. 3, we also have the graph of the principal curvature $\lambda$. The horizontal green line is

$$\begin{aligned} {\hat{\lambda }}=\left( \frac{nH_r}{n-r}\right) ^{1/r}=\sqrt{3}. \end{aligned}$$

From the function $\mu (\lambda ) = \frac{3}{2\lambda } - \frac{\lambda }{2}$, it follows that the Gauss–Kronecker curvature $K=\lambda ^{n-1}\mu$ takes positive and negative values.

Other similar examples are given in Figure  4. On the other hand, Fig. 5 shows a profile curve with $K <0$ on $\Sigma$. The last hypersurface belongs to the one-parameter family given in Theorem 5. In this case note that the value of ${\tilde{C}}$ is close to

$$\begin{aligned} C_0 = G(\rho _0,0) = \frac{r}{n} \left( \frac{n-r}{n(H_r+1)}\right) ^{(n-r)/{r}} \approx 0.272166. \end{aligned}$$

About Theorem 1

In this appendix, we obtain Theorem 1 from [11, Theorem 4.1]. Let $\Sigma ^n$ and ${\overline{M}}^{n+k}$ be Riemannian manifolds of dimension n and $n + k$, respectively. Throughout the Appendix we use $\Sigma ^n \hookrightarrow {\overline{M}}^{n+k}$ to denote an isometric immersion of $\Sigma$ in ${\overline{M}}.$

Remember that the Gauss formula of $\Sigma ^n \hookrightarrow {\overline{M}}^{n+k}$ is

$$\begin{aligned} {\overline{\nabla }}_X Y=\nabla _X Y+B(X,Y), \end{aligned}$$

which is valid for all $X,Y \in {\mathfrak {X}}(\Sigma )$. Here, $\nabla , {\overline{\nabla }}$ are the Riemannian connections of $\Sigma ^n$ and ${\overline{M}}^{n+k}$, respectively, and B is the second fundamental form of $\Sigma$. The mean curvature vector field $\textbf{H}\in {\mathfrak {X}}^\perp (\Sigma )$ of $\Sigma ^n$ in ${\overline{M}}$ is given by

$$\begin{aligned} \textbf{H}=\frac{1}{n}\sum _{i=1}^n B(e_i,e_i), \end{aligned}$$

where $\{e_1,\dots ,e_n\}$ is a local orthonormal frame on $\Sigma$.

Definition 14

An isometric immersion $\Sigma ^n \hookrightarrow {\overline{M}}^{n+k}$ is said to have parallel mean curvature if $\textbf{H}$ is parallel in the normal bundle, i.e., ${\overline{\nabla }}^\perp \textbf{H} =0$.

Remark 15

The following properties follow directly from the definition:

• 1. If $\textbf{H}$ is parallel, then $\vert \textbf{H} \vert$ is constant.

• 2. If $\textbf{H}\ne 0$, $\textbf{H}$ is parallel if and only if $\vert \textbf{H}\vert$ is constant and $\textbf{H}/\vert \textbf{H}\vert$ is parallel.

• 3. If the codimension $k=1$, $\textbf{H}$ is parallel if and only if $\vert \textbf{H} \vert$ is constant.

Now we consider a surface $\Sigma ^2$ isometrically immersed in ${\overline{M}}^4(c)$, a 4-manifold with constant sectional cuavature c.

Theorem 16

(Theorem 4.1 in [11]). A complete immersed surface $\Sigma ^2 \hookrightarrow {\overline{M}}^4(c)$, $c\ge 0$, with parallel mean curvature and Gauss curvature K which does not change sign must be minimal ($\textbf{H}\equiv 0$), a sphere of radius $(\vert \textbf{H} \vert ^2 + c)^{-\frac{1}{2}}$ or a product of circles ${\mathbb {S}}^1(s) \times {\mathbb {S}}^1(t)$, $0<s \le \infty$, $0< t < \infty$, with the standard product immersion.

Remark 17

In particular, consider the immersion $\Sigma ^2 \hookrightarrow {\mathbb {S}}^4$ with parallel mean curvature $\textbf{H} \ne 0$. If $\Sigma$ is not totally umbilical and its Gaussian curvature K does not change sign, then $K=0$ and $\Sigma$ must be a product of circles ${\mathbb {S}}^1(s) \times {\mathbb {S}}^1(t)$, $s^2+t^2=1$.

We first look at a surface $\Sigma ^2$ isometrically immersed in${\mathbb {S}}^3$. Then, we use the canonical embedding of ${\mathbb {S}}^3$ in ${\mathbb {S}}^4$ to obtain the immersion of $\Sigma$ in ${\mathbb {S}}^4$. It is well known that ${\mathbb {S}}^3$ is totally geodesic submanifold of ${\mathbb {S}}^4$, that is, their Riemannian connections coincide. Let $A =\{e_1, e_2, e_3, e_4\}$ be an adapted frame to $\Sigma$ (in ${\mathbb {S}}^4$) such that

$$\begin{aligned} e_1, e_2 \in T\Sigma , \qquad \quad e_3 \in T{\mathbb {S}}^3\subset T{\mathbb {S}}^4 \qquad \text {and} \qquad e_4 \in T^\perp {\mathbb {S}}^3 \subset T{\mathbb {S}}^4. \end{aligned}$$

Then, the mean curvature vector field of $\Sigma \hookrightarrow {\mathbb {S}}^4$ is given by

$$\begin{aligned} \textbf{H}= \frac{1}{2} (\lambda _{11}^3+\lambda _{22}^3) e_3 + \frac{1}{2} (\lambda _{11}^4+\lambda _{22}^4) e_4 \end{aligned}$$

(29)

where $\lambda _{ij}^\alpha = \langle B(e_i,e_j), e_\alpha \rangle$, $\alpha =3,4$, and B is the second fundamental form of the immersion $\Sigma \hookrightarrow {\mathbb {S}}^4$.

Now suppose that the immersion $\Sigma \hookrightarrow {\mathbb {S}}^3$ has constant nonzero mean curvature. In view of Theorem 16 (see also Remark 17), it is sufficient to show that $\textbf{H}$ is parallel (see also Section 3 in [15] for more general cases). Let

$$\begin{aligned} \mu _{ij}^{3} = \langle B_\Sigma ^{{\mathbb {S}}^3} (e_i,e_j), e_3 \rangle \end{aligned}$$

where $B_\Sigma ^{{\mathbb {S}}^3}$ is the second fundamental form of $\Sigma \hookrightarrow {\mathbb {S}}^3$. One can check directly that

for $X,Y \in T\Sigma$. It follows that

$$\begin{aligned} \lambda _{ij}^{3} = \langle B(e_i,e_j), e_3 \rangle = \langle B_\Sigma ^{{\mathbb {S}}^3} (e_i,e_j), e_3 \rangle = \mu _{ij}^{3}. \end{aligned}$$

On the other hand, since $B_\Sigma ^{{\mathbb {S}}^3} \in T {\mathbb {S}}^3$ and $e_4 \in T^\perp {\mathbb {S}}^3$ we get that

$$\begin{aligned} \lambda _{ij}^{4} = \langle B(e_i,e_j), e_4 \rangle = \langle B_\Sigma ^{{\mathbb {S}}^3} (e_i,e_j), e_4 \rangle =0. \end{aligned}$$

Thus, (29) reduces to

$$\begin{aligned} \textbf{H}= \frac{1}{2} \left( \lambda _{11}^3+\lambda _{22}^3\right) e_3 =H e_3, \qquad \text {where }H=\frac{1}{2} \left( \lambda _{11}^3+\lambda _{22}^3\right) \text { is constant.} \end{aligned}$$

From this we notice that $\vert \textbf{H}\vert = \vert H\vert$ is also constant. Furthermore, since $\textbf{H} \in T {\mathbb {S}}^4$ and ${\mathbb {S}}^3$ is a totally geodesic submanifold of ${\mathbb {S}}^4$,

$$\begin{aligned} {\overline{\nabla }}_{X}^{\perp }\, \textbf{H} = \langle {\overline{\nabla }}_X \, \textbf{H}, e_3\rangle e_3 = H \langle {\overline{\nabla }}_X \, e_3, e_3\rangle e_3 =0. \end{aligned}$$

Thus, the immersion $\Sigma ^2 \hookrightarrow {\mathbb {S}}^{4}$ has parallel mean curvature $\textbf{H}\ne 0$.