Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds

Abstract

Given a compact Riemannian manifold M, we consider a warped product manifold ${\bar{M}} = I \times _h M$, where I is an open interval in ${\mathbb {R}}$. For a positive function $\psi$ defined on ${\bar{M}}$, we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287–1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535), to obtain the curvature estimates for Hessian equations $\sigma _k(\kappa )=\psi (V,\nu (V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.

1 Introduction

Assume that $\Sigma ^n$ is a hypersurface in a Riemannian manfold $\bar{M}^{n+1}$. The Weingarten curvature equation is given by

$$\begin{aligned} \sigma _k(\kappa (X))=\psi (X), \ \ \forall \, X\in \Sigma , \end{aligned}$$

where X is the position vector field of the hypersurface $\Sigma$ in $\bar{M}^{n+1}$ and $\sigma _k$ is the kth elementary symmetric function.

Finding closed hypersurfaces with prescribed Weingarten curvature in Riemannian manifolds attracts many authors’ interest. Such results were obtained for the case of prescribing mean curvature by Bakelman and Kantor [10, 11] and by Treibergs and Wei [46] in the Euclidean space, for the case of prescribing Gaussian curvature by Oliker [41], and for general Weingarten curvatures by Aleksandrov [1,2,3,4,5,6,7], Firey [19], Caffarelli et al. [17]. For Riemannian manifolds, some results have been obtained by Li and Oliker [37] for the unit sphere, Barbosa et al. [13] for space forms, Jin and Li [34] for the hyperbolic space, Andrade et al. [8] for warped product manifolds, Li and Sheng [36] for the Riemannain manifold equipped with a global normal Gaussian coordinate system.

For the hypersurface $\Sigma$ in the Euclidean space ${\mathbb {R}}^{n+1}$, the Weingarten curvature equation in general form is defined by

$$\begin{aligned} \sigma _k(\kappa (X))=\psi (X,\nu (X)), \ \ \forall \, X\in \Sigma , \end{aligned}$$

where $\nu (X)$ is the normal vector field along the hypersurface $\Sigma$. In many cases, the curvature estimates are the key part for the above prescribed curvature problems. Let us give a brief review. When $k=1$, curvature estimate comes from the theory of quasilinear PDEs. If $k=n$, curvature estimate in this case for general $\psi (X, \nu )$ was due to Caffarelli et al. [15]. Ivochkina [32, 33] considered the Dirichlet problem of the above equation on domains in ${\mathbb {R}}^n$, and obtained $C^2$ estimates there under some extra conditions on the dependence of f on $\nu$. $C^2$ estimate was also proved for equation of prescribing curvature measures problem in [25, 27]. If the function $\psi$ is convex with respect to the normal $\nu$, it is well known that the global $C^2$ estimate has been obtained by Guan [21]. Recently, Guan et al. [30] obtained global $C^2$ estimates for a closed convex hypersurface $\Sigma \subset {\mathbb {R}}^{n+1}$ and then solved the long standing problem (1.2). In the same paper [30], they also proved the estimate for starshaped 2-convex hypersurfaces by introducing some new auxiliary curvature functions. Li et al. [35] substitute the convex by $(k+1)$- convex for any k Hessian equations. In [42], Ren and the third author completely solved the case $k=n-1$, that is the global curvature estimates of $n-1$ convex solutions for $n-1$ Hessian equations. In [45], Spruck–Xiao extended 2-curvature equations in [30] to space forms and gave a simple proof if the hypersurface in the Euclidean space. We also note the recently important work on the curvature estimates and $C^2$ estimates developed by Guan [22] and Guan et al. [31].

These type of equations and estimates with generalized right hand sides appear some new geometric applications recently, which will be mentioned in detail in the following. Phong et al. [38, 39], generalized the Fu–Yau’s equations, which is a complex 2-Hessian equation depending on gradient term on the right hand side. In [39, 40], they obtained their $C^2$ estimates using the idea of [30]. Guan and Lu [28] considered the curvature estimate for isometric embedding system in general Riemannian manifolds, which is also a 2-Hessian equation depending on the normal vector field. The estimates in [30] are applied in [14, 47] too.

Let $(M^n,g^\prime )$ be a compact Riemannian manifold and I be an open interval in ${\mathbb {R}}$. The warped product manifold $\bar{M}=I\times _h M$ is endowed with the metric

$$\begin{aligned} \bar{g}^2=dt^2+h^2(t)g^\prime , \end{aligned}$$

(1.1)

where $h: I\longrightarrow {\mathbb {R}}^+$ is a positive differential function. Given a differentiable function $z:M\longrightarrow I$, its graph is defined as the hypersurface

$$\begin{aligned} \Sigma =\{X(u)=(z(u),u)|u\in M \}. \end{aligned}$$

For the Weingarten curvature equation in general form

$$\begin{aligned} \sigma _k(\kappa (V))=f(\kappa (V))=\psi (V,\nu (V)), \ \ \forall V\in \Sigma , \end{aligned}$$

(1.2)

where $V=h\,\displaystyle \frac{\partial }{\partial t}$ is the position vector field of hypersurface $\Sigma$ in ${\bar{M}}$, $\sigma _k$ is the kth elementary symmetric function, $\nu (V)$ is the inward unit normal vector field along the hypersurface $\Sigma$ and $\kappa (V)=(\kappa _1,\ldots ,\kappa _n)$ are principal curvatures of hypersurface $\Sigma$ at V. Given $t_-, t_+$ with $t_-<t_+$, we define the annulus domain $\bar{M}_-^+=\{(t,u)\in \bar{M}|t_-\le t \le t_+\}$.

In this article, we will generalize the results in [30, 42] to the hypersurfaces in warped product manifolds. The main results of this paper are the followings:

Theorem 1.1

Let $M^n$ be a compact Riemannian manifold and ${\bar{M}}$ be the warped product manifold with the metric (1.1). Assume that h is a positive differential function and $h^\prime >0$. Suppose that $\psi$ satisfies

• (a) $\psi (t,u,\nu (u)) > C^k_n(\kappa (t))^k$ for $t\le t_-$,

• (b) $\psi (t,u,\nu (u)) < C^k_n(\kappa (t))^k$ for $t\ge t_+$,

• (c) $\partial _t \big (h^k\psi (V,\nu )\big ) \le 0$ for $t_-<t<t_+$,

where $\kappa (t)=h'(t)/h(t)$ and $C^k_n$ is the combinatorial numbers. Then there exists a unique differentiable function $z:M^n\rightarrow I$ solve the Eq. (1.2) for $k=2$ and $k=n-1$ whose graph $\Sigma$ is contained in the interior of the region $\bar{M}_-^+$.

For the convex hypersurface in any warped product manifolds, we obtain the global curvature estimates.

Theorem 1.2

Suppose $\Sigma \longrightarrow \bar{M}^{n+1}$ is a convex compact hypersurface satisfying curvature Eq. (1.2) for some positive function $\psi (V, \nu )\in C^{2}(\Gamma )$, where $\Gamma$ is an open neighborhood of unit normal bundle of M in $\bar{M}^{n+1} \times {\mathbb {S}}^n$. Then there is a constant C depending only on n, k, $|z|_{C^1}$, $\inf \psi$ and $\Vert \psi \Vert _{C^2}$, such that

$$\begin{aligned} \max _{u\in M} \kappa _i(u) \le C. \end{aligned}$$

(1.3)

Since the second fundamental form does not satisfy Codazzi properties for hypersurfaces in warped product manifolds in general, the constant rank theorem is still unknown. Thus, the above estimates only can imply the existence results in the sphere.

Theorem 1.3

Let ${\bar{M}}$ be the sphere with sectional curvature $\lambda >0$ which means the metric ${\bar{g}}$ of ${\bar{M}}$ is defined by (1.1), where function h is defined by

$$\begin{aligned} h(t)=\displaystyle \frac{\sin \sqrt{\lambda } t}{\sqrt{\lambda }}. \end{aligned}$$

(1.4)

Suppose that $\psi$ satisfies

• (a) $\psi (t,u,\nu (u)) > \kappa (t)$ for $t\le t_-$,

• (b) $\psi (t,u,\nu (u))< \kappa (t)$ for $t\ge t_+$,

• (c) $(\psi ^{-1/k})_{ij}+\lambda \psi ^{-1/k}g_{ij}\ge 0,$ for any $\nu$,

where $\kappa (t)=h'(t)/h(t)=\sqrt{\lambda }\cot (\sqrt{\lambda t})$ and $t_+<\pi /2$. Then there exists a differentiable function $z:\mathbb {S}^n\rightarrow I$ solve the Eq. (1.2) for any k whose graph $\Sigma$ is a strictly convex hypersurface and is contained in the interior of the region $\bar{M}_-^+$.

The paper is organized as follows. In Sect. 2, we fix notations and recall some basic formulas for geometric and analytic preliminaries, including the detailed description of the problem. In Sect. 3, the gradient estimates of (1.2) are presented. In Sect. 4, the curvature estimates are proved for the starshaped 2-convex hypersurfaces. In Sects. 5 and 6, $C^2$ estimates are obtained for convex and $(n-1)$-convex hypersurface in the warped product manifold ${\bar{M}}$. In the last section, we derive the constant rank theorem and existence results .

2 Preliminaries

2.1 Warped product manifold ${\bar{M}}$

Let $M^n$ be a compact Riemannian manifold with the metric $g^\prime$ and I be an open interval in ${\mathbb {R}}$. Assuming $h: I\longrightarrow {\mathbb {R}}^+$ is a positive differential function and $h^\prime >0$, the manifold $\bar{M}=I\times _h M$ is called the warped product if it is endowed with the metric

$$\begin{aligned} \bar{g}^2=dt^2+h^2(t)g^\prime . \end{aligned}$$

(2.1)

In the section, we use Latin lower case letters $i,j,\ldots$ to refer to indices running from 1 to n and $a,b,\ldots$ to indices from 0 to $n-1$. The Einstein summation convention is used throughout the paper.

The metric in ${\bar{M}}$ is denoted by $\langle \cdot ,\cdot \rangle$. The corresponding Riemannian connection in ${\bar{M}}$ will be denoted by ${\bar{\nabla }}$. The usual connection in M will be denoted $\nabla '$. The curvature tensors in M and ${\bar{M}}$ will be denoted by R and ${\bar{R}}$, respectively.

Let $e_1, \ldots , e_{n-1}$ be an orthonormal frame field in M and let $\theta _1, \ldots , \theta _n$ be the associated dual frame. The connection forms $\theta _{ij}$ and curvature forms $\Theta _{ij}$ in M satisfy the structural equations

$$\begin{aligned}&\text {d}\theta _i = \sum _j\theta _{ij}\wedge \theta _j,\quad \theta _{ij}=-\theta _{ji}, \end{aligned}$$

(2.2)

$$\begin{aligned}&\text {d}\theta _{ij} - \sum _k\theta _{ik}\wedge \theta _{kj}=\Theta _{ij}=-\frac{1}{2} \sum _{k,l}R_{ijkl}\theta _k\wedge \theta _l. \end{aligned}$$

(2.3)

An orthonormal frame in ${\bar{M}}$ may be defined by $\bar{e}_i = (1/h)e_i,\, 1\le i\le n-1,$ and $\bar{e}_{0} = \partial /\partial t$. The associated dual frame is then $\bar{\theta }_i = h\theta _i$ for $1\le i\le n-1$ and $\bar{\theta }_{0}=\text {d}t$. A simple computation permits to obtain

Lemma 2.1

On the leaf $M_t$, the curvature satisfies

$$\begin{aligned} {\bar{R}}_{ijk0}=0 \end{aligned}$$

(2.4)

and the principle curvature is given by

$$\begin{aligned} \kappa (t) = h'(t)/h(t) \end{aligned}$$

(2.5)

where the inward unit normal $-{\bar{e}}_0=-\partial /\partial t$ is chosen for each leaf $M_t$.

2.2 Hypersurfaces in the warped product manifold ${\bar{M}}$

Given a differentiable function $z:M\longrightarrow I$, its graph is defined by the hypersurface

$$\begin{aligned} \Sigma =\{X(u)=(z(u),u)|u\in M \} \end{aligned}$$

(2.6)

whose tangent space is spanned at each point by the vectors

$$\begin{aligned} X_i = h\,\bar{e}_i + z_i\,\bar{e}_{0}, \end{aligned}$$

(2.7)

where $z_i$ are the components of the differential $\text {d}z= z_i\theta ^i$. The unit vector field

$$\begin{aligned} \nu = \frac{1}{\sqrt{h^2 + |\nabla ' z|^2}}\left( \sum _{i=1}^n z^i\bar{e}_i - h\bar{e}_{0}\right) \end{aligned}$$

(2.8)

is an unit inner normal vector field to $\Sigma$. Here, $|\nabla ' z|^2=z^iz_i$ is the squared norm of $\nabla 'z=z^ie_i$. The components of the induced metric in $\Sigma$ is given by

$$\begin{aligned} g_{ij} = \langle X_i, X_j\rangle = h^2\delta _{ij} + z_iz_j \end{aligned}$$

(2.9)

The second fundamental form of $\Sigma$ with components $(a_{ij})$ is determined by

$$\begin{aligned} a_{ij} =\langle {\bar{\nabla }}_{X_j}X_i,\nu \rangle = \frac{1}{\sqrt{h^2 + |\nabla ' z|^2}}\big (-hz_{ij} + 2h'z_iz_j + h^2h'\delta _{ij}\big ) \end{aligned}$$

where $z_{ij}$ are the components of the Hessian $\nabla '^2z =\nabla '\text {d}z$ of z in M.

Now we choose the coordinate systems such that $\{E_0=\nu ,E_1,\ldots ,E_n\}$ is an orthonormal frame field in some open set of $\Sigma$ and $\{\omega _0,\omega _1, \ldots , \omega _n\}$ is its associated dual frame. The connection forms $\{\omega _{ij}\}$ and curvature forms $\{\Omega _{ij}\}$ in $\Sigma$ satisfy the structural equations

$$\begin{aligned}&d\omega _i -\sum _j \omega _{ij}\wedge \omega _j=0,\quad \omega _{ij}+\omega _{ji}=0, \\&\text {d}\omega _{ij} - \sum _k\omega _{ik}\wedge \omega _{kj}=\Omega _{ij}=-\frac{1}{2} \sum _{k,l}R_{ijkl}\omega _k\wedge \omega _l. \end{aligned}$$

The coefficients $a_{ij}$ of the second fundamental form are given by Weingarten equation

$$\begin{aligned} \omega _{i0}= \sum _ja_{ij}\,\omega _j. \end{aligned}$$

The covariant derivative of the second fundamental form $a_{ij}$ in $\Sigma$ is defined by

$$\begin{aligned} \sum _k a_{ijk}\,\omega _k= & {} \text {d}a_{ij}+\sum _la_{il}\, \omega _{lj}+\sum _la_{lj}\,\omega _{li}, \\ \sum _l a_{ijkl}\,\omega _l= & {} \text {d}a_{ijk}+\sum _l a_{ljk}\,\omega _{li} +\sum _la_{ilk}\,\omega _{lj}+\sum _la_{ijl}\,\omega _{lk}. \end{aligned}$$

The Codazzi equation is a commutation formula for the first order derivative of $a_{ij}$ given by

$$\begin{aligned} a_{ijk}-a_{ikj}=-{\bar{R}}_{0ijk} \end{aligned}$$

(2.10)

and the Ricci identity is a commutation formula for the second order derivative of $a_{ij}$ given by

Lemma 2.2

Let $\bar{X}$ be a point of $\Sigma$ and $\{E_0 =\nu , E_1,\ldots , E_n\}$ be an adapted frame field such that each $E_i$ is a principal direction and $\omega ^k_i=0$ at ${\bar{X}}$. Let $(a_{ij})$ be the second quadratic form of $\Sigma$. Then, at the point $\bar{X}$, we have

$$\begin{aligned} \begin{aligned} a_{llii}=&a_{iill}-a_{lm}\left( a_{mi}a_{il}-a_{ml}a_{ii}\right) -a_{mi}\left( a_{mi}a_{ll} -a_{ml}a_{li}\right) \\&+{\bar{R}}_{0iil;l}-2a_{ml}\bar{R}_{miil}+a_{il}\bar{R}_{0i0l}+a_{ll}\bar{R}_{0ii0}\\&+\bar{R}_{0lil;i}-2a_{mi}\bar{R}_{mlil}+a_{ii}\bar{R}_{0l0l}+a_{li}\bar{R}_{0li0}. \end{aligned} \end{aligned}$$

(2.11)

In particular, we have

$$\begin{aligned} a_{ii11}-a_{11ii}=a_{11}a_{ii}^2-a_{11}^2a_{ii}+2(a_{ii}-a_{11}){\bar{R}}_{i1i1}+a_{11}\, {\bar{R}}_{i0i0}-a_{ii} \,{\bar{R}}_{1010}+{\bar{R}}_{i1i0;1}-{\bar{R}}_{1i10;i}. \end{aligned}$$

(2.12)

2.3 Two functions $\eta$ and $\tau$

Define the functions $\tau :\Sigma \rightarrow \mathbb {R}$ and $\eta :\Sigma \rightarrow \mathbb {R}$ by

$$\begin{aligned} \tau =-h \langle \nu , {\bar{e}}_0\rangle =-\langle V, \nu \rangle \quad \text {and}\quad \eta =- \int h\, \text {d}t, \end{aligned}$$

(2.13)

where $V=h{\bar{e}}_0=h\displaystyle \frac{\partial }{\partial t}$ is the position vector field and $\nu$ is the inner unit normal. Then we have

Lemma 2.3

[8] The gradient vector fields of the functions $\eta$ and $\tau$ are

$$\begin{aligned} \nabla _{E_i}\eta= & {} -h\langle \bar{e}_0, E_i \rangle E_i, \end{aligned}$$

(2.14)

$$\begin{aligned} \nabla _{E_i}\tau= & {} - \sum _j\nabla _{E_j}\eta a_{ij}, \end{aligned}$$

(2.15)

and the second order derivative of $\tau$ and $\eta$ are given by

$$\begin{aligned} \nabla ^2_{E_i,E_j}\eta= & {} \tau a_{ij}-h^{\prime } g_{ij}, \end{aligned}$$

(2.16)

$$\begin{aligned} \nabla ^2_{E_i,E_j}\tau= & {} -\sum _k\tau a_{ik}a_{kj}+h^{\prime }a_{ij}-\sum _ka_{ikj}\nabla _{E_k}\eta \nonumber \\= & {} -\tau \sum _ka_{ik}a_{kj}+h^{\prime }a_{ij}-\sum _k(a_{ijk}+{\bar{R}}_{0ijk})\nabla _{E_k}\eta . \end{aligned}$$

(2.17)

2.4 Basic formulae

Assume that $\Sigma \longrightarrow \bar{M}$ is the graph defined as the hypersurface $\Sigma$ whose points are the form $X(u)=(z(u),u)$ with $u\in M$. This graph is diffeomorphic with M and may be globally oriented by an unit normal vector field $\nu$ for which it holds that $\langle \nu ,\partial _t\rangle <0$. Let $\kappa =(\kappa _1, \ldots , \kappa _n)$ be the vector whose components $\kappa _i$ are the principal curvatures of $\Sigma$, that is, the eigenvalues of the second fundamental form $B=(\langle \bar{\nabla }_{i}E_j , \nu \rangle )$ in $\Sigma$.

The elementary symmetric function of order k ($1\le k\le n$) of $\kappa =(\kappa _1, \ldots , \kappa _n)$ is defined as following

$$\begin{aligned} \sigma _k=\sum _{i_1<\cdots < i_n}\kappa _{i_1}\ldots \kappa _{i_n}. \end{aligned}$$

(2.18)

Let $\Gamma _k$ be the connected component of $\{\kappa \in {\mathbb {R}}^n|\sigma _m>0, m=1,\ldots , k\}$ containing the positive cone $\{\kappa \in {\mathbb {R}}^n|\kappa _1,\dots ,\kappa _n>0\}$.

Definition 2.4

A positive function $z\in C^2(M^n)$ is said to be admissible for the operator $\sigma _k$ if for the corresponding hypersurface $\Sigma =\{(z(u),u)|u\in M^n\}$, at every point of $\Sigma$ with the normal as in (2.8), the principal curvatures $\kappa =(\kappa _1,\ldots ,\kappa _n)$ are in $\Gamma _k$.

Lemma 2.5

([9, 12, 16, 20, 44]) Let F be a $C^2$ symmetric function defined in some open set of Sym(n), where Sym(n) is the set of all $n\times n$ symmetric matrices. For any symmetric matrix $(b_{ij})$, there holds

$$\begin{aligned} F^{ij, kl}b_{ij}b_{kl}=\sum _{i,j} \frac{\partial ^2 f}{\partial \kappa _i \partial \kappa _j} b_{ii} b_{jj} +\sum _{i\ne j} \frac{f_i-f_j}{\kappa _i-\kappa _j} b_{ij}^2, \end{aligned}$$

where the second term on the right-hand side must be interpreted as a limit whenever $\kappa _i=\kappa _j$.

Lemma 2.6

[25, 30] Assume that $k>l$, $W=(w_{ij})$ is a Codazzi tensor which is in $\Gamma _k$. Denote $\alpha =\displaystyle \frac{1}{k-l}$. Then, for $h=1,\ldots , n$, we have the following inequality,

$$\begin{aligned}&-\displaystyle \frac{\sigma _k^{pp,qq}}{\sigma _k}(W)w_{pph}w_{qqh}+\displaystyle \frac{\sigma _l^{pp,qq}}{\sigma _l}(W) w_{pph}w_{qqh} \nonumber \\&\quad \ge \left( \displaystyle \frac{(\sigma _k(W))_h}{\sigma _k(W)}-\displaystyle \frac{(\sigma _l(W))_h}{\sigma _l(W)} \right) \left( (\alpha -1)\displaystyle \frac{(\sigma _k(W))_h}{\sigma _k(W)}-(\alpha +1)\displaystyle \frac{(\sigma _l(W))_h}{\sigma _l(W)} \right) .\qquad \end{aligned}$$

(2.19)

Furthermore, for any $\delta >0$, we have

$$\begin{aligned}&-\sigma _k^{pp,qq}(W)w_{pph}w_{qqh} +\left( 1-\alpha +\displaystyle \frac{\alpha }{\delta }\right) \displaystyle \frac{(\sigma _k(W))_h^2}{\sigma _k(W)} \nonumber \\&\quad \ge \sigma _k(W)(\alpha +1-\delta \alpha ) \left[ \displaystyle \frac{(\sigma _l(W))_h}{\sigma _l(W)} \right] ^2 -\displaystyle \frac{\sigma _k}{\sigma _l}(W)\sigma _l^{pp,qq}(W)w_{pph}w_{qqh}. \end{aligned}$$

(2.20)

3 Gradient estimates

In this section, we follow the ideas of [17, 27] to derive $C^1$ estimates for the height function z. In other words, we are looking for a lower bound of the support function $\tau$. Firstly, we need the following technical assumption:

$$\begin{aligned} \frac{\partial }{\partial t}(h(t)^k\psi (V, \nu ))\le 0,\,\,\text{ where } V=h(t)\displaystyle \frac{\partial }{\partial t}. \end{aligned}$$

(3.1)

Lemma 3.1

Let $\Sigma$ be a graph in $\bar{M}=I\times _h M$ satisfying (1.2), (3.1) and let z be the height function of $\Sigma$. If h has positive lower and upper bounds, then there is a constant C depending on the minimum and maximum values of z such that

$$\begin{aligned} |\nabla z|\le C. \end{aligned}$$

(3.2)

Proof

Set $\chi (z)= -\ln (\tau )+\gamma (-\eta (t))$, where $\gamma$ is a single variable function to be determined later. Assume that $\chi$ achieve its maximum value at point $u_0$. We claim that V is parallel to its normal $\nu$ at $u_0$ if we choose a suitable $\gamma$. We will prove it by contradiction. If not, we can choose a local orthonormal basis $\{E_i\}_{i=1}^n$ such that $\left<V, E_1\right>\ne 0,$ and $\left<V, E_i\right>=0,\, i\ge 2$. Obviously, $V=\left<V, E_1\right>E_1+\left<V, \nu \right>\nu$. At point $u_0$, by the maximum principle we have

$$\begin{aligned} 0= & {} \nabla _{E_i}\chi (z)=-\frac{\nabla _{E_i}\tau }{\tau }-\gamma ^{\prime }\nabla _{E_i}\eta , \end{aligned}$$

(3.3)

$$\begin{aligned} 0\ge & {} \nabla ^2_{E_i,E_i}\chi (z)\nonumber \\= & {} -\frac{\nabla ^2_{E_i,E_i}\tau }{\tau }+\frac{|\nabla _{E_i}\tau |^2}{\tau ^2}-\gamma ^{\prime }\nabla ^2_{E_i,E_i}\eta +\gamma ^{\prime \prime }|\nabla _{E_i}\eta |^2. \end{aligned}$$

(3.4)

From (2.15), (2.17) and (3.3), we have

$$\begin{aligned} \begin{aligned} 0\ge&-\frac{\nabla ^2_{E_i,E_i}\tau }{\tau }+\frac{|\nabla _{E_i}\tau |^2}{\tau ^2}-\gamma ^{\prime }\nabla ^2_{E_i,E_i}\eta +\gamma ^{\prime \prime }|\nabla _{E_i}\eta |^2\\ =&-\frac{1}{\tau }\left( -\tau a_{il}a_{li}+h^{\prime }a_{ii}-(a_{iil}+{\bar{R}}_{0iil})\eta _l\right) +\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \eta _i^2-\gamma ^{\prime }\left( \tau a_{ii}-h^{\prime } g_{ii}\right) . \end{aligned} \end{aligned}$$

(3.5)

By (2.15) and (3.3), we get

$$\begin{aligned} a_{11}=\tau \gamma ^{\prime },\qquad a_{i1}=0,\qquad i\ge 2. \end{aligned}$$

(3.6)

Therefore, it is possible to rotate the coordinate system such that $\{E_i\}_{i=1}^n$ are the principal curvature directions of the second fundamental form $(a_{ij})$, i.e. $a_{ij}=a_{ii}\delta _{ij}$, which means that $(\sigma _k^{ij})$ is also diagonal. By multiplying $\sigma _k^{ii}$ in the inequality (3.5) both sides and taking sum on i from 1 to n, one gets from (3.5) and (3.6)

$$\begin{aligned} \begin{aligned} 0&\ge \sigma _k^{ii}a_{ii}^2-\frac{1}{\tau }h^{\prime }\sigma _k^{ii}a_{ii}+\frac{1}{\tau }\sigma _k^{ii}(a_{iil}+{\bar{R}}_{0iil})\eta _l+\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{ii}\eta _i^2 \\&\quad -\gamma ^{\prime }\left( \tau \sigma ^{ii}a_{ii}-h^{\prime } \sum _{i=1}^n\sigma _k^{ii}\right) \\&=\sigma _k^{ii}a_{ii}^2+\frac{1}{\tau }\sigma _k^{ii}a_{ii1}\eta _1+\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0ii1}\eta _1+\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{11}\eta _1^2 \\&\quad +\gamma ^{\prime }h^{\prime } \sum _{i=1}^n\sigma _k^{ii}-\gamma ^{\prime }\tau k\psi -\frac{1}{\tau }h^{\prime }k\psi \end{aligned} \end{aligned}$$

(3.7)

where $F^{ii}a_{ii}= k\psi$ is used. Differentiating Eq. (1.2) with respect to $E_1$ we obtain

$$\begin{aligned} \sigma _k^{ii}a_{ii1}=d_V\psi (\nabla _{E_1}V)-a_{11}d_{\nu }\psi (E_1). \end{aligned}$$

(3.8)

Putting (3.6) and (3.8) into (3.5) yields

$$\begin{aligned} 0\ge & {} \sigma _k^{ii}a_{ii}^2+\frac{1}{\tau }\Big (d_V\psi (\nabla _{E_1}V)-a_{11}d_{\nu }\psi (E_1)\Big )\eta _1+\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0ii1}\eta _1\nonumber \\&+\,\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{11}\eta _1^2+\gamma ^{\prime }h^{\prime } \sum _{i=1}^n\sigma _k^{ii}-\gamma ^{\prime }\tau k\psi -\frac{1}{\tau }h^{\prime }k\psi \nonumber \\= & {} \sigma _k^{ii}a_{ii}^2-\frac{1}{\tau }\Big (kh^{\prime }\psi +\langle V,E_1 \rangle d_V\psi (\nabla _{E_1}V)\Big )+\gamma ^{\prime } d_{\nu }\psi (E_1)\langle V,E_1 \rangle +\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0i1i}\langle V,E_1 \rangle \nonumber \\&+\,\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{11}\langle V,E_1 \rangle ^2-k\gamma ^{\prime }\tau \psi +\gamma ^{\prime }h^{\prime } \sum _{i=1}^n\sigma _k^{ii}. \end{aligned}$$

(3.9)

Since $V=\langle V,E_1\rangle E_1+\langle V, \nu \rangle \nu$, we have

$$\begin{aligned} d_V\psi (V, \nu )=\langle V, E_1\rangle d_V\psi (\nabla _{E_1}V)+\langle V, \nu \rangle d_V\psi (\nabla _{\nu }V). \end{aligned}$$

(3.10)

Putting (3.10) into (3.9) gets

$$\begin{aligned} \begin{aligned} 0&\ge \sigma _k^{ii}a_{ii}^2-\frac{1}{\tau }\left( kh^{\prime }\psi + d_V\psi (V, \nu )\right) +\gamma ^{\prime } d_{\nu }\psi (E_1)\langle V,E_1 \rangle +\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0i1i}\langle V,E_1 \rangle \\&\quad +\,\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{11}\langle V,E_1 \rangle ^2-k\gamma ^{\prime }\tau \psi +\gamma ^{\prime }h^{\prime } \sum _{i=1}^n\sigma _k^{ii}+ d_V\psi (\nabla _{\nu }V)\\&\ge \sigma _k^{ii}a_{ii}^2+\left( \gamma ^{\prime \prime }+(\gamma ^{\prime })^2\right) \sigma _k^{11}\langle V,E_1 \rangle ^2+\gamma ^{\prime }h^{\prime } \sum _{i=1}^n\sigma _k^{ii}+\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0i1i}\langle V,E_1 \rangle \\&\quad +\,\gamma ^{\prime } d_{\nu }\psi (E_1)\langle V,E_1 \rangle -k\gamma ^{\prime }\tau \psi + d_V\psi (\nabla _{\nu }V), \end{aligned} \end{aligned}$$

(3.11)

where we use the assumption (3.1). Choosing the function $\gamma (t)=\displaystyle \frac{\alpha }{t}$ for a positive constant $\alpha$, we have

$$\begin{aligned} \begin{aligned} \gamma ^{\prime }(t)=-\frac{\alpha }{t^2},\qquad \gamma ^{\prime \prime }(t)=\frac{2\alpha }{t^3}. \end{aligned} \end{aligned}$$

(3.12)

By (3.6) and the choice of function $\gamma$, we have $a_{11}\le 0$. Thus, the Newton–Maclaurin inequality implies

$$\begin{aligned} \sigma _k^{11}\ge \sigma _{k-1}\ge \frac{k}{(n-k+1)(k-1)}(C_n^k)^{\frac{1}{k}} \psi ^{\frac{k-1}{k}}. \end{aligned}$$

(3.13)

Therefore by the previous three inequalities, we have

$$\begin{aligned} \begin{aligned} 0&\ge \sigma _k^{11}a_{11}^2+\left( \frac{\alpha ^2}{t^4}+\frac{2\alpha }{t^3}\right) \sigma _k^{11}\langle V,E_1 \rangle ^2-\frac{\alpha }{t^2}h^{\prime } \sum _{i=1}^n\sigma _k^{ii}+\frac{1}{\tau } \sigma _k^{ii}{\bar{R}}_{0i1i}\langle V,E_1 \rangle \\&\quad \,-\,\frac{\alpha }{t^2} d_{\nu }\psi (E_1)\langle V,E_1 \rangle +\frac{\alpha }{t^2}k\tau \psi + d_V\psi (\nabla _{\nu }V). \end{aligned} \end{aligned}$$

(3.14)

Since $V=\left<V, E_1\right>E_1+\left<V, \nu \right>\nu$, one can find that $V \perp \text {Span}\{E_2,\ldots ,E_n\}$. On the other hand, $E_1, \nu \perp \text {Span}\{E_2,\ldots ,E_n\}$. It is possible to choose coordinate systems such that ${\bar{e}}_1\perp \text {Span}\{E_2,\ldots ,E_n\}$, which implies that the pair $\{V, {\bar{e}}_1\}$ and $\{\nu ,E_1\}$ lie in the same plane and

$$\begin{aligned} \text {Span}\{E_2,\ldots ,E_n\}=\text {Span}\{{\bar{e}}_2,\ldots ,{\bar{e}}_n\}. \end{aligned}$$

Therefore, we can choose $E_2={\bar{e}}_2,\ldots ,E_n={\bar{e}}_n$. The vector $\nu$ and $E_1$ can decompose into

$$\begin{aligned} \begin{aligned} \nu =&\langle \nu ,{\bar{e}}_0\rangle {\bar{e}}_0 +\langle \nu ,{\bar{e}}_1\rangle {\bar{e}}_1 =-\frac{\tau }{h}{\bar{e}}_0 +\langle \nu ,{\bar{e}}_1\rangle {\bar{e}}_1,\\ E_1=&\langle E_1,{\bar{e}}_0\rangle {\bar{e}}_0 +\langle E_1,{\bar{e}}_1\rangle {\bar{e}}_1. \end{aligned} \end{aligned}$$

For (2.4) and $V=\left<V, E_1\right>E_1+\left<V, \nu \right>\nu$, we obtain

$$\begin{aligned} \begin{aligned} {\bar{R}}_{0i1i}&= {\bar{R}}(\nu ,E_i,E_1,E_i)\\&=-\frac{\tau }{h} \langle E_1,{\bar{e}}_0\rangle {\bar{R}}({\bar{e}}_0,{\bar{e}}_i,{\bar{e}}_0,{\bar{e}}_i)+\langle \nu ,{\bar{e}}_1\rangle \langle E_1,{\bar{e}}_1\rangle {\bar{R}}({\bar{e}}_1,{\bar{e}}_i,{\bar{e}}_1,{\bar{e}}_i)\\&=-\frac{\tau }{h}\langle E_1,{\bar{e}}_0\rangle {\bar{R}}({\bar{e}}_0,{\bar{e}}_i,{\bar{e}}_0,{\bar{e}}_i)-\tau \frac{\langle \nu ,{\bar{e}}_1\rangle ^2}{\langle E_1,V\rangle }{\bar{R}}({\bar{e}}_1,{\bar{e}}_i,{\bar{e}}_1,{\bar{e}}_i)\\&=\tau \left( -\frac{1}{h}\langle E_1,{\bar{e}}_0\rangle {\bar{R}}({\bar{e}}_0,{\bar{e}}_i,{\bar{e}}_0,{\bar{e}}_i)- \frac{\langle \nu ,{\bar{e}}_1\rangle ^2}{\langle E_1,V\rangle }{\bar{R}}({\bar{e}}_1,{\bar{e}}_i,{\bar{e}}_1,{\bar{e}}_i)\right) . \end{aligned} \end{aligned}$$

(3.15)

The third equality comes from $0=\langle V, {\bar{e}}_1 \rangle$. From (3.6), (3.13) and (3.15), (3.14) becomes

$$\begin{aligned} 0\ge & {} \alpha ^2\sigma _k^{11}(\tau ^2(\gamma ')^2+\frac{\alpha ^2}{t^4}\langle V, E_1\rangle ^2)-C_1\alpha \sigma _{k-1}-C_2\alpha |d_{\nu }\psi (e_1)|-|d_V\psi (\nabla _{\nu }V)|\\\ge & {} C\alpha ^2|V|^2\sigma _k^{11}-C_1\alpha \sigma _k^{11}-C_2\alpha |d_{\nu }\psi (e_1)|-|d_V\psi (\nabla _{\nu }V)| \end{aligned}$$

where $C,C_1,C_2$ depends on k, n, the $C^0$ bound of h and the curvature ${\bar{R}}$. Thus, we have a contradiction when $\alpha$ is large enough. Hence, V is parallel to the normal $\nu$ which implies the lower bound of $\tau$. $\square$

4 $C^2$ estimates for $\sigma _2$

In this section, we study the solution of the following normalized equation

$$\begin{aligned} F(b)= \left( \begin{array}{c} n\\ 2 \end{array} \right) ^{(-1/2)}\sigma _2(\kappa (a))^{1/2} =f(\kappa (a_{ij}))=\overline{\psi }(V,\,\nu ). \end{aligned}$$

(4.1)

Now we can prove the $C^2$ estimate for 2-convex hypersurfaces.

Theorem 4.1

With the assumption of Theorem 1.1, there is a constant C depending only on $n,k,t_-,t_+$, the $C^1$ bound of z and $|\bar{\psi }|_{C^2}$, such that

$$\begin{aligned} \max _{u\in M}|\kappa _i(u)|\le C. \end{aligned}$$

(4.2)

Proof

Define the function

$$\begin{aligned} W(u,\xi )=e^{-\beta \eta }\frac{B(\xi ,\xi )}{\tau -a} \end{aligned}$$

(4.3)

where $\tau \ge 2a$ and $\beta$ is a large constant to be chosen, $\xi$ is a tangent vector of $\Sigma$ and B is the second fundamental form. Assume that W is achieved at $X_0=(z(u_0),\,u_0)$ along $\xi$, and we may choose a local orthonormal frame $E_1, \dots , E_n$ around $X_0$ such that $\xi =E_1$ and $a_{ij} (X_0) = \kappa _i \delta _{ij}$, where $\kappa _1\ge \kappa _2\ge \ldots \ge \kappa _n$ are the principal curvatures of $\Sigma$ at $u_0$. Thus at $u_0,\, \ln W=\ln {a_{11}}-\log {(\tau -a)}-\beta \eta$ has a local maximum. Therefore,

$$\begin{aligned} 0=\frac{a_{11i}}{a_{11}}-\frac{\nabla _i\tau }{\tau -a}-\beta \eta _i, \end{aligned}$$

(4.4)

and

$$\begin{aligned} 0\ge \frac{a_{11ii}}{a_{11}}-\left( \frac{a_{11i}}{a_{11}}\right) ^2 -\frac{\nabla _{ii}\tau }{\tau -a}+\left( \frac{\nabla _i\tau }{\tau -a}\right) ^2-\beta \eta _{ii}. \end{aligned}$$

(4.5)

Multiplying $F^{ii}$ both sides in (4.5) and using (2.14)–(2.17), we have

$$\begin{aligned} \begin{aligned} 0&\ge \frac{1}{\kappa _1} F^{ii}a_{11ii}-\frac{1}{\kappa _1^2}F^{ii}\left( a_{11i}\right) ^2- \frac{1}{\tau -a}F^{ii}\tau _{ii}+F^{ii}\left( \frac{\tau _i}{\tau -a}\right) ^2-\beta F^{ii}\eta _{ii}\\&=\frac{1}{\kappa _1} F^{ii}a_{11ii}-\frac{1}{\kappa _1^2}F^{ii}\left( a_{11i}\right) ^2 +\frac{\tau }{\tau -a}F^{ii}\kappa _i^2-\frac{h^{\prime }}{\tau -a}{\bar{\psi }} \\&\quad \, +\frac{1}{\tau -a}\sum _lF^{ii}(a_{iil}+{\bar{R}}_{0iil})\eta _l\\&\quad \,+\sum _iF^{ii}\left( \frac{\kappa _i\eta _i}{\tau -a}\right) ^2-\beta \tau {\bar{\psi }}+h^{\prime }\beta \sum _{i=1}^nF^{ii}. \end{aligned} \end{aligned}$$

(4.6)

The Ricci identity (2.11) yields

$$\begin{aligned} \begin{aligned} F^{ii}a_{ii11}-F^{ii}a_{11ii}&=a_{11}F^{ii}a_{ii}^2-a_{11}^2F^{ii}a_{ii}+2F^{ii}(a_{ii}-a_{11}){\bar{R}}_{i1i1} \\&\quad \,+a_{11}\, F^{ii} {\bar{R}}_{i0i0}-F^{ii}a_{ii} \,{\bar{R}}_{1010}\\&\quad \,+F^{ii} {\bar{R}}_{i1i0;1}-F^{ii}{\bar{R}}_{1i10;i}\\&\ge -C_1\kappa _1^2-C_2\kappa _1\sum _i F^{ii}, \end{aligned} \end{aligned}$$

(4.7)

for sufficiently large $\kappa _1$. Inserting (4.7) into (4.6), one gives

$$\begin{aligned} \begin{aligned} 0&\ge \frac{1}{\kappa _1} F^{ii}a_{ii11}-\frac{1}{\kappa _1^2}F^{ii}\left( a_{11i}\right) ^2 +\frac{\tau }{\tau -a}F^{ii}\kappa _i^2+\frac{1}{\tau -a}\sum _lF^{ii}(a_{iil}+{\bar{R}}_{0iil})\eta _l\\&\quad \,+\sum _iF^{ii}\left( \frac{\kappa _i\eta _i}{\tau -a}\right) ^2-C_1\kappa _1+(h'\beta -C_2)\sum _i F^{ii}-C_3(\beta ). \end{aligned} \end{aligned}$$

(4.8)

Taking covariant derivative with respect to the equation (4.1) yields

$$\begin{aligned} F^{ii}a_{iij}=\bar{\psi }_V(\nabla _{E_j}V)-a_{jl}\bar{\psi }_{\nu }(E_l). \end{aligned}$$

(4.9)

Taking covariant derivative with respect to the Eq. (4.9) again yields

$$\begin{aligned} \begin{aligned} F^{ii}a_{ii11}+F^{ij,kl}a_{ij1}a_{kl1}&={\bar{\psi }}_{VV} (\nabla _{E_1}V,\nabla _{E_1}V)+2a_{1l}{\bar{\psi }}_{V\nu }(\nabla _{E_1}V,E_l) \\&\quad \,-a_{1l1}{\bar{\psi }}_{\nu }(E_l)+a_{1k}a_{1l}{\bar{\psi }}_{\nu \nu }(E_l,E_l)\\&\ge -C(1+\kappa _1^2)-a_{1l1}{\bar{\psi }}_{\nu }(E_l)\\&=-C(1+\kappa _1^2)-(a_{11l}-{\bar{R}}_{01l1}){\bar{\psi }}_{\nu }(E_l)\\&\ge -C(1+\kappa _1^2+\beta \kappa _1)-a_{11l}{\bar{\psi }}_V(E_l). \end{aligned} \end{aligned}$$

(4.10)

where we have used the Codazzi equation in the last equality, (4.4) and the bound of the curvature of the ambient manifold in the last inequality.

We also have

$$\begin{aligned} \frac{1}{\kappa _1}\sum _l a_{11l}{\bar{\psi }}_V(E_l)-\sum _l\frac{\eta _l}{\tau -a}F^{ii}a_{iil}=\sum _l\beta \eta _l{\bar{\psi }}_V(E_l)-\sum _l\frac{\eta _l}{\tau -a}\bar{\psi }_V(\nabla _{E_j}V).\nonumber \\ \end{aligned}$$

(4.11)

Combining the inequality (4.10) and (4.11), (4.8) gives

$$\begin{aligned} \begin{aligned} 0\ge&\frac{1}{\kappa _1} \left( -F^{ij,kl}a_{ij1}a_{kl1} \right) -\frac{1}{\kappa _1^2}F^{ii}\left( a_{11i}\right) ^2 +\frac{\tau }{\tau -a}F^{ii}\kappa _i^2\\&+\sum _{i=1}^nF^{ii}\left( \frac{\kappa _i\eta _i}{\tau -a}\right) ^2-C_1\kappa _1+(h'\beta -C_2)\sum _i F^{ii}-C_3(\beta )\\ \end{aligned} \end{aligned}$$

(4.12)

In the following, we consider two cases.

Case 1 We suppose that $\kappa _{n}\le -\theta \kappa _{1}$ for some positive constant $\theta$ to be chosen later. In this case, using the concavity of F, we discard the term $-\frac{1}{\kappa _1}F^{ij,kl}a_{ij1}a_{kl1}$.

By Young’s inequality and (4.4), we have

$$\begin{aligned} \begin{aligned} \frac{1}{\kappa _1^2} F^{ii}|a_{11i}|^2&\le (1+\epsilon ^{-1})\beta ^2 F^{ii}|\eta _i|^2 +\frac{(1+\epsilon )}{(\tau -a)^2} F^{ii}|\tau _i|^2\\&\le C_4(1+\epsilon ^{-1})\beta ^2 \sum _iF^{ii} +\frac{(1+\epsilon )}{(\tau -a)^2} F^{ii}|\tau _i|^2 \end{aligned} \end{aligned}$$

(4.13)

for any $\epsilon >0$, where we have used $|\nabla \eta |\le C$. From (4.12) and (4.13), we obtain

$$\begin{aligned} \begin{aligned} 0\ge&-C_1\kappa _1-C_3(\beta )+\left( \frac{\tau }{\tau -a}-C_5\epsilon \right) F^{ii}\kappa _i^2 +\left( h^{\prime }\beta -C_2-C_4(1+\epsilon ^{-1})\beta ^2\right) \sum _iF^{ii}\\ \ge&-{\bar{C}}(\kappa _1+\beta )+C_6\sum _{i=1}^nF^{ii}\kappa _i^2-C_7\beta ^2 \sum _iF^{ii}. \end{aligned} \end{aligned}$$

(4.14)

Since $F^{11}\le F^{22}\le \cdots \le F^{nn}$ and $\kappa _{n}\le -\theta \kappa _{1}$, we get

$$\begin{aligned} \sum _{i=1}^n F^{ii} \kappa _i^2 \ge F^{nn} \kappa _n^2\ge \frac{1}{n}\theta ^2 \sum _iF^{ii} \kappa _1^2. \end{aligned}$$

Hence,

$$\begin{aligned} 0\ge -{\bar{C}}(\kappa _1+\beta )+\left( C_6\frac{1}{n}\theta ^2\kappa _1^2-C_7\beta ^2\right) \sum _iF^{ii}. \end{aligned}$$

(4.15)

Since $\sum _iF^{ii}\ge 1$ for sufficiently large $\kappa _1$, the inequality (4.15) clearly implies the bound of $\kappa _1$ from above.

Case 2 In this case, we assume that $\kappa _{n}\ge -\theta \kappa _{1}$. Hence, $\kappa _{i}\ge \kappa _n\ge -\theta \kappa _{1}$. We then group the indices $\{1,...,n\}$ into two sets $I=\{j:F^{jj}\le 4F^{11}\}$ and $J=\{j:F^{jj}>4F^{11}\}$. Using (4.4), we can infer

$$\begin{aligned} \begin{aligned} \frac{1}{\kappa _1^2}\sum _{i\in I} F^{ii}|a_{11i}|^2 \le&C_1(1+\epsilon ^{-1})\beta ^2 F^{11} +\frac{(1+\epsilon )}{(\tau -a)^2} F^{ii}|\tau _i|^2 \end{aligned} \end{aligned}$$

(4.16)

for any $\epsilon >0$. Therefore it follows from (4.12) that

$$\begin{aligned} \begin{aligned} 0\ge&-C_1\kappa _1-C_3(\beta )-\frac{1}{\kappa _1}F^{ij,kl}a_{ij1}a_{kl1} +\left( \frac{\tau }{\tau -a}-C_5\epsilon \right) F^{ii}\kappa _i^2\\&+\left( h^{\prime }\beta -C_2\right) \sum _iF^{ii}-\frac{1}{\kappa _1^2}\sum _{i\in J}F^{ii}\left( \nabla _ia_{11}\right) ^2- C_4(1+\epsilon ^{-1})\beta ^2 F^{11}. \end{aligned} \end{aligned}$$

(4.17)

Using Lemma (2.5) and the Codazzi’s equation, one gets

$$\begin{aligned} \begin{aligned} -\frac{1}{\kappa _{1}}F^{ij,kl}a_{ij1}a_{kl1} \ge&-\frac{2}{\kappa _{1}}\sum _{j\in J}\frac{f_{1}-f_{j}}{\kappa _{1} -\kappa _{j}}\big (a_{1j1} \big )^{2}=-\frac{2}{\kappa _{1}}\sum _{j\in J}\frac{f_{1}-f_{j}}{\kappa _{1} -\kappa _{j}}\big (a_{11j}-{\bar{R}}_{01j1}\big )^{2}. \end{aligned} \end{aligned}$$

(4.18)

Following the argument in [34], we may verify that choosing $\theta =\displaystyle \frac{1}{2}$ it holds that for all $j\in J$,

$$\begin{aligned} \begin{aligned} -\frac{2}{\kappa _{1}}\frac{f_{1}-f_{j}}{\kappa _{1} -\kappa _{j}}\ge \frac{f_{j}}{\kappa _{1}^{2}}=\frac{F^{jj}}{\kappa _1^2}. \end{aligned} \end{aligned}$$

(4.19)

Combining (4.17), (4.18) and (4.19), we obtain

$$\begin{aligned} \begin{aligned} 0\ge&-C_1\kappa _1-C_3(\beta )-2\frac{1}{\kappa _{1}^2}\sum _{j\in J}F^{jj} a_{11j}{\bar{R}}_{01j1}+\left( \frac{\tau }{\tau -a}-C_5\epsilon \right) F^{ii}\kappa _i^2\\&+\left( h^{\prime }\beta -C_2\right) \sum _iF^{ii}- C_4(1+\epsilon ^{-1})\beta ^2 F^{11}\\ \ge&-{\bar{C}}(\kappa _1+\beta )+C_6\sum _{i=1}^nF^{ii}\kappa _i^2 +\left( h^{\prime }\beta -C\right) \sum _iF^{ii}- C_7\beta ^2 F^{11}\\ \ge&(C_8(h'\beta -C_2)-C_1)\kappa _1+(C_6\kappa _1^2-C_7\beta ^2)F^{11}-{\bar{C}}_3(\beta ), \end{aligned} \end{aligned}$$

(4.20)

by choosing $\epsilon$ small and sufficiently large $\kappa _1$. Here we also used (4.4) and

$$\begin{aligned} \sum _{i=1}^n F^{ii}\ge C\kappa _1. \end{aligned}$$

For $\beta >0$ sufficiently large, we may obtain an upper bound for $\kappa _1$ by (4.20). $\square$

Remark 4.2

The similar idea also has been used in [18, 23, 43].

5 A global $C^2$ estimate for convex hypersurfaces in the warped product space

In this section, following the arguments in [30], we can obtain $C^2$ estimates of convex solutions for the curvature Eq. (1.2) in $\Sigma$, namely, proving Theorem 1.2.

Define the following auxiliary function,

$$\begin{aligned} \quad \Psi =\frac{1}{2}\ln P(\kappa )-N \log \tau -\beta {\eta }, \end{aligned}$$

(5.1)

where $P(\kappa )= \kappa ^2_1+\cdots +\kappa _n^2=\sum _{i,j=1}^na_{ij}^2$, and $N,\beta$ are two constants to be determined later.

We assume that $\Psi$ achieves its maximum value at $X_0\in \Sigma$. By a proper rotation, we may assume that $(a_{ij})$ is a diagonal matrix at the point, and $a_{11}\ge a_{22}\ldots \ge a_{nn}$.

At $x_0$, covariant differentiate $\Psi$ twice,

$$\begin{aligned} 0=\Psi _i= \frac{\sum _{l,j}a_{lj}a_{lji}}{P}-N\frac{\tau _i}{\tau }-\beta \eta _i =\frac{\sum _l\kappa _la_{lli}}{P}+N\frac{a_{ii}\eta _i}{\tau }-\beta \eta _i\ \ = \ \ 0, \end{aligned}$$

(5.2)

and

$$\begin{aligned} 0\ge & {} \Psi _{ii}\\\ge & {} \frac{1}{P}\left( \sum _l\kappa _la_{llii}+\sum _{l}a_{lli}^2+\sum _{p\ne q}a_{pqi}^2\right) -\frac{2}{P^2}\left( \sum _l\kappa _la_{lli}\right) ^2 -N\frac{\tau _{ii}}{\tau } +N\frac{\tau _i^2}{\tau ^2}-\beta \eta _{ii}\\= & {} \frac{1}{P}\left[ \sum _{l}\kappa _l\left( a_{iill}-a_{lm}\left( a_{mi}a_{il}-a_{ml}a_{ii}\right) -a_{mi}\left( a_{mi}a_{ll} -a_{ml}a_{li}\right) \right. \right. \\&\qquad +{\bar{R}}_{0iil;l}-2a_{ml}\bar{R}_{miil}+a_{il}\bar{R}_{0i0l}+a_{ll}\bar{R}_{0ii0} \\&\qquad \left. \left. +\bar{R}_{0lil;i}-2a_{mi}\bar{R}_{mlil}+a_{ii}\bar{R}_{0l0l}+a_{li}\bar{R}_{0li0}\right) +\sum _{l}a_{lli}^2+\sum _{p\ne q}a_{pqi}^2\right] \\&-\frac{2}{P^2}\left( \sum _l\kappa _la_{lli}\ \right) ^2+\frac{N}{\tau }\sum _la_{iil}\eta _l -\frac{Nh^\prime }{\tau }\kappa _i +N\kappa _i^2+\frac{N}{\tau ^2}\kappa _{i}^2\eta _i^2 \\&+\frac{N}{\tau }\sum _l\bar{R}_{0iil}\eta _l+\beta (h^{\prime }\delta _{ii}-\tau \kappa _i). \end{aligned}$$

Multiplying $\sigma _k^{ii}$ both sides gives

$$\begin{aligned} \begin{aligned} 0\ge&\frac{1}{P}\left[ \sum _{l}\kappa _l\left( \sigma _k^{ii}a_{iill}-\sigma _k^{ii}a_{lm}\left( a_{mi}a_{il}-a_{ml}a_{ii}\right) -\sigma _k^{ii}a_{mi}\left( a_{mi}a_{ll} -a_{ml}a_{li}\right) \right. \right. \\&\qquad +\sigma _k^{ii}{\bar{R}}_{0iil;l}-2\sigma _k^{ii}a_{ml} \bar{R}_{miil}+\sigma _k^{ii}a_{il}\bar{R}_{0i0l} +\sigma _k^{ii}a_{ll}\bar{R}_{0ii0}+\sigma _k^{ii}\bar{R}_{0lil;i} \\&\qquad \left. -2\sigma _k^{ii}a_{mi}\bar{R}_{mlil}+\sigma _k^{ii}a_{ii} \bar{R}_{0l0l}+\sigma _k^{ii}a_{li}\bar{R}_{0li0}\right) \\&\qquad \left. +\sum _{l}\sigma _k^{ii}a_{lli}^2+\sum _{p\ne q}\sigma _k^{ii}a_{pqi}^2\right] \\&-\frac{2}{P^2}\sigma _k^{ii}\left( \sum _l\kappa _la_{lli}\ \right) ^2+\frac{N}{\tau }\sum _l\sigma _k^{ii}a_{iil}\eta _l -\frac{Nh^\prime }{\tau }kf +N\sigma _k^{ii}\kappa _i^2+\frac{N}{\tau ^2}\sigma _k^{ii}\kappa _{i}^2\eta _i^2\\&+\frac{N}{\tau }\sum _l\sigma _k^{ii}\bar{R}_{0iil}\eta _l+\beta \left( h^{\prime }\sum _i\sigma _k^{ii}-\tau kf\right) \\ \ge&\frac{1}{P}\left[ \sum _{l}\kappa _l\sigma _k^{ii}a_{iill}+kf\sum _l\kappa _l^3-C(1+\kappa _1^2)\sum _i\sigma _k^{ii}+\sum _{l}\sigma _k^{ii}a_{lli}^2+\sum _{p\ne q}\sigma _k^{ii}a_{pqi}^2\right] \\&-\frac{2}{P^2}\sigma _k^{ii}\left( \sum _l\kappa _la_{lli}\ \right) ^2+\frac{N}{\tau }\sum _l\sigma _k^{ii}a_{iil}\eta _l +(N-1)\sigma _k^{ii}\kappa _i^2 \\&+\left( C_1\beta -C_2N\right) \sum _i\sigma _k^{ii}-C(\beta ,N). \end{aligned} \end{aligned}$$

(5.3)

Now covariant differentiate the Eq. (1.2) twice,

$$\begin{aligned} \sigma _k^{ii}a_{iij}= & {} d_V\psi (\nabla _j V) + d_{\nu } \psi ( \nabla _j\nu )\ \ = \ \ h^\prime d_V \psi (E_j)-a_{jl}d_{\nu }\psi (E_l), \end{aligned}$$

(5.4)

and

$$\begin{aligned}&\sigma _k^{ii}a_{iijj}+\sigma _k^{pq,rs}a_{pqj}a_{rsj} \nonumber \\&\quad =d_V\psi (\nabla _{jj}V)+d^2_{V}\psi (\nabla _jV,\nabla _jV)+2d_Vd_{\nu }\psi (\nabla _jV,\nabla _j\nu )\nonumber \\&\qquad +d^2_{\nu }\psi (\nabla _j\nu ,\nabla _j\nu )+d_{\nu }\psi (\nabla _{jj}\nu ).\nonumber \\&\quad = -\frac{h^{\prime \prime }}{h}\eta _j d_V\psi (E_j)+h^\prime a_{jj}d_V\psi (\nu )+(h^\prime )^2 d^2_{V}\psi (E_j,E_j) \nonumber \\&\qquad -2h^\prime a_{jj}d_Vd_{\nu }\psi (E_j,E_j)+a_{jj}^2d^2_{\nu }\psi (E_j,E_j)\nonumber \\&\qquad -\sum _la_{ljj}d_{\nu }\psi (E_l)-a_{jj}^2d_{\nu }\psi (\nu )\nonumber \\&\quad \ge -C-C\kappa _j^2-\sum _la_{ljj}d_{\nu }\psi (E_l), \end{aligned}$$

(5.5)

where the Schwarz inequality is used in the last inequality.

Since

$$\begin{aligned} -\sigma _k^{pq,rs}a_{pql}a_{rsl}\ \ =\ \ -\sigma _k^{pp,qq}a_{ppl}a_{qql}+\sigma _k^{pp,qq}a_{pql}^2, \end{aligned}$$

(5.6)

it follows from (5.2) and (5.4), and Codazzi equation (2.10) implies

$$\begin{aligned} \begin{aligned} \frac{1}{P}\sum _{l,j}\kappa _ja_{ljj}d_{\nu }\psi (E_l)=&\frac{N}{\tau }\sum _j\sigma _k^{ii}a_{iij}\eta _j -\frac{Nh^\prime }{\tau }\sum _jd_V\psi (E_j)\eta _j+\beta \sum _j\eta _j d_\nu \psi (E_j)\\&-\frac{1}{P}\sum _{l,j}\kappa _j\bar{R}_{0jlj}d_{\nu }\psi (E_l). \end{aligned} \end{aligned}$$

(5.7)

Denote

$$\begin{aligned}&A_i= \frac{\kappa _i}{P}\left( K(\sigma _k)_i^2-\sum _{p,q}\sigma _k^{pp,qq}a_{ppi}a_{qqi}\right) , \ \ B_{i}=2\sum _j\frac{\kappa _j}{P}\sigma _k^{jj,ii}a_{jji}^2, \\&C_i=2\sum _{j\ne i}\frac{\sigma _k^{jj}}{P}a_{jji}^2, \ \ D_i=\frac{1}{P}\sum _j\sigma _k^{ii}a_{jji}^2,\ \ E_i=\frac{2\sigma _k^{ii}}{P^2}\left( \sum _j \kappa _ja_{jji}\right) ^2. \end{aligned}$$

By (5.4) and (5.7), we can infer

$$\begin{aligned} \begin{aligned} 0\ge&\frac{1}{P}\left[ \sum _{l}\kappa _l\left( -C-C\kappa _l^2-K(\sigma _k)_l^2+K(\sigma _k)_l^2-\sigma _k^{pp,qq}a_{ppl}a_{qql}+2\sum _{j\not = l}\sigma _k^{ll,jj}a_{ljl}^2\right) \right. \\&\qquad \left. +kf\sum _l\kappa _l^3+\sum _{l}\sigma _k^{ii}a_{lli}^2+2\sum _{j\ne i}\sigma _k^{ii}a_{iji}^2\right] \\&-\frac{2}{P^2}\sigma _k^{ii}\left( \sum _l\kappa _la_{lli}\ \right) ^2 +(N-1)\sigma _k^{ii}\kappa _i^2 \\&+\left( C_1\beta -C_2N-C_3\right) \sum _i\sigma _k^{ii}-C(\beta ,N)-\frac{C_4}{\kappa _{1}}.\\ \end{aligned} \end{aligned}$$

(5.8)

From the Codazzi equation $a_{iji}=a_{iij}-{\bar{R}}_{0iji}$ and the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} 2\sum _{j\not = l}\sigma _k^{ll,jj}\kappa _la_{ljl}^2=&2\sum _{j\not = l}\sigma _k^{ll,jj}\kappa _l\left( a_{llj}-{\bar{R}}_{0ljl}\right) ^2\\ \ge&(2-\delta )\sum _{j\not = l}\kappa _l\sigma _k^{ll,jj}a_{llj}^2-C_\delta \sum _{j}\sigma _k^{jj}\\ =&(2-\delta )\sum _{j\not = l}\kappa _l\sigma _k^{ll,jj}a_{llj}^2-C_\delta \sum _{j\not = l}\kappa _l\sigma _k^{ll,jj}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} 2\sum _{j\ne i}\sigma _k^{ii}a_{iji}^2=2\sum _{j\not = i}\sigma _k^{ii,jj}\left( a_{iij}-{\bar{R}}_{0iji}\right) ^2 \ge&(2-\delta )\sum _{j\not = i}\sigma _k^{ii}a_{iij}^2-C_\delta \sum _{i}\sigma _k^{ii}, \end{aligned} \end{aligned}$$

where $\delta$ is a small constant to be determined later and $C_\delta$ is a constant depending on $\delta$. Therefore, we obtain

$$\begin{aligned} \begin{aligned} 0\ge&\frac{1}{P}\left[ \sum _{l}\kappa _l\left( -C-C\kappa _l^2-K(\sigma _k)_l^2\right) +kf\sum _l\kappa _l^3\right] \\&+(N-1)\sigma _k^{ii}\kappa _i^2+\left( C_1\beta -C_2N-C_3-C_\delta \frac{1}{P}\right) \sum _i\sigma _k^{ii}-C(\beta ,N)-\frac{C_4}{\kappa _{1}}\\&+\left( 1-\frac{\delta }{2}\right) \sum _{i} (A_i+B_i+C_i+D_i-E_i)+\frac{\delta }{2}\sum _{i}\left( A_i+D_i\right) \\&-\frac{\delta }{2}\frac{2}{P^2}\sigma _k^{ii}\left( \sum _l\kappa _la_{lli}\right) ^2. \end{aligned} \end{aligned}$$

(5.9)

According to the proof of Lemma 4.2, Lemma 4.3 and Corollary 4.4 in [30], we have the following alternatives. There exist positive numbers $\delta _2,\delta _3\ldots ,\delta _{n}$ depending only on k, n, such that either

$$\begin{aligned} \kappa _i> \delta _i\kappa _1, \forall \ \ 2\le i\le n, \end{aligned}$$

or

$$\begin{aligned} A_i+B_i+C_i+D_i-E_i\ge 0, \forall \ \ 1\le i\le n. \end{aligned}$$

Thus, in the following, the proof will be divided into two cases.

Case (A): There exists some $2\le i\le k-1$, such that $\kappa _i\ge \delta _i\kappa _1$ and $\kappa _{i+1}\le \delta _{i+1}\kappa _1$. Choosing K sufficiently large, we have $A_i$ is positive by Lemma 2.6 . By the above alternatives, we can infer

$$\begin{aligned} \sum _{i} (A_i+B_i+C_i+D_i-E_i)\ge 0. \end{aligned}$$

From (5.2) and Cauchy–Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} \sigma _k^{ii}\left( \frac{1}{P^2}\sum _l\kappa _la_{lli}\right) ^2=&\sigma _k^{ii}\left( \frac{N}{\tau }\kappa _i-\beta \right) ^2\eta _i^2 \le C_5\left( N^2\sigma _k^{ii}\kappa _i^2+\beta ^2\sum _{i}\sigma _k^{ii}\right) . \end{aligned} \end{aligned}$$

(5.10)

Inserting (5.10) into (5.9), we get

$$\begin{aligned} 0&\ge \frac{1}{P}\left[ \sum _{l}\kappa _l\left( -C-C\kappa _l^2-K(\sigma _k)_l^2\right) +kf\sum _l\kappa _l^3\right] \nonumber \\&\quad +\,N\left( \frac{1}{2}- C_5\delta N\right) \sigma _k^{ii}\kappa _i^2+ \left( \frac{N}{2}-1\right) \sigma _k^{ii}\kappa _i^2\nonumber \\&\quad +\,\left( C_1\beta -C_2N-C_3-C_\delta \frac{1}{P}-C_5\delta \beta ^2\right) \sum _i\sigma _k^{ii}-C(\beta ,N)-\frac{C_4}{\kappa _{1}}\nonumber \\&\ge -\frac{1}{P}\left( C(K)+C(K)\kappa _1^3\right) +C_6\left( \frac{N}{2}-1\right) \kappa _1\nonumber \\&\quad +\,N\left( \frac{1}{2}- C_5\delta N\right) \sigma _k^{ii}\kappa _i^2+ \left( C_1\beta -C_2N-C_3-C_\delta \frac{1}{P}-C_5\delta \beta ^2\right) \nonumber \\&\quad \times \sum _i\sigma _k^{ii}-C(\beta ,N)-\frac{C_4}{\kappa _{1}}\nonumber \\&\ge \left( \frac{1}{2} C_6N-C(K)\right) \kappa _1+N\left( \frac{1}{2}- C_5\delta N\right) \sigma _k^{ii}\kappa _i^2\nonumber \\&\quad +\, \left( C_1\beta -C_2N-C_3-C_\delta \frac{1}{P}-C_5\delta \beta ^2\right) \sum _i\sigma _k^{ii}-C(\beta ,N)-\frac{C_4}{\kappa _{1}}, \end{aligned}$$

(5.11)

where we have used $\sigma _k^{ii}\kappa _i^2\ge c_0\kappa _1$. Now let us choose these constants carefully. Firstly, choose N such that

$$\begin{aligned} C(K)+1\le \frac{1}{2} C_5N, \text { and } N\ge 4. \end{aligned}$$

Secondly, choose $\beta$ such that

$$\begin{aligned} C_1\beta -C_2N-C_3-3\ge 0 . \end{aligned}$$

Thirdly, choose the constant $\delta$ satisfying

$$\begin{aligned} \max \{N^2,\beta ^2\}\le (2C_5\delta )^{-1}. \end{aligned}$$

At last, take sufficiently large $\kappa _1$ satisfying

$$\begin{aligned} \frac{C_\delta }{P}\le 1. \end{aligned}$$

Otherwise we are done. Finally, $\kappa _1$ has upper bound by (5.11).

Case(B): If the Case(A) does not hold. That means $\kappa _k\ge \delta _k\kappa _1$. Since $\kappa _l\ge 0$, we have,

$$\begin{aligned} \sigma _k\ge \kappa _1\kappa _2\ldots \kappa _k\ge \delta _k^{k-1}\kappa _1^k. \end{aligned}$$

This implies the bound of $\kappa _1$.

6 A global curvature estimate for $(n-1)$ convex hypersurfaces

For the functions $\tau$ and $\eta$, we employ the following auxiliary function which is introduced firstly in [30],

$$\begin{aligned} \Psi =\log \log P-N\ln (\tau )-\beta \eta , \end{aligned}$$

where $P=\displaystyle \sum \nolimits _le^{\kappa _l}$ and $\{\kappa _{l}\}_{l=1}^n$ are the eigenvalues of the second fundamental form.

We may assume that the maximum of $\Psi$ is achieved at some point $X_0\in \Sigma$. After rotating the coordinates, we may assume the matrix $(a_{ij})$ is diagonal at that point, and we can further assume that $a_{11}\ge a_{22}\ldots \ge a_{nn}$. Denote $\kappa _i=a_{ii}$.

Covariant differentiating the function $\Psi$ twice at $X_0$, we have

$$\begin{aligned} \begin{aligned} 0=\Psi _i=&\displaystyle \frac{P_i}{P\log P}- N\frac{\tau _i}{\tau }-\beta \eta _i =\displaystyle \frac{1}{P\log P}\sum _le^{\kappa _l}a_{lli}+N\frac{a_{ij}\eta _j}{\tau }-\beta \eta _i, \end{aligned} \end{aligned}$$

(6.1)

and

$$\begin{aligned} 0\ge & {} \Psi _{ii}\nonumber \\= & {} \frac{P_{ii}}{P\log P}-\frac{P_i^2}{P^2\log P}-\frac{P_i^2}{(P\log P)^2}-N\frac{\tau _{ii}}{\tau }+N\frac{\tau _i^2}{\tau ^2}-\beta \eta _{ii}\nonumber \\= & {} \frac{1}{P\log P}\left[ \sum _le^{\kappa _l}a_{llii}+\sum _le^{\kappa _l}a_{lli}^2+\sum _{\alpha \ne \gamma }\frac{e^{\kappa _{\alpha }}-e^{\kappa _{\gamma }}}{\kappa _{\alpha }-\kappa _{\gamma }}a_{\alpha \gamma i}^2-\left( \frac{1}{P}+\frac{1}{P\log P}\right) P_i^2\right] \nonumber \\&+\frac{N}{\tau }\sum _la_{iil}\eta _l-\frac{ Nh^\prime }{\tau }\kappa _i+N\kappa _i^2+\frac{N}{\tau ^2}\kappa _i^2\eta _i^2 +\frac{N}{\tau }\sum _l{\bar{R}}_{0iil}\eta _l-\beta (\tau \kappa _i-h^\prime \delta _{ii})\nonumber \\= & {} \frac{1}{P\log P}\left[ \sum _le^{\kappa _l}\left( a_{iill}-a_{lm}\left( a_{mi}a_{il}-a_{ml}a_{ii}\right) -a_{mi}\left( a_{mi}a_{ll} -a_{ml}a_{li}\right) +{\bar{R}}_{0iil;l}\right. \right. \nonumber \\&\left. -2a_{ml}\bar{R}_{miil}+a_{il}\bar{R}_{0i0l}+a_{ll}\bar{R}_{0ii0}+\bar{R}_{0lil;i}-2a_{mi}\bar{R}_{mlil}+a_{ii}\bar{R}_{0l0l}+a_{li}\bar{R}_{0li0}\right) \nonumber \\&\left. +\sum _le^{\kappa _l}a_{lli}^2+\sum _{\alpha \ne \gamma }\frac{e^{\kappa _{\alpha }}-e^{\kappa _{\gamma }}}{\kappa _{\alpha }-\kappa _{\gamma }}a_{\alpha \gamma i}^2-\left( \frac{1}{P}+\frac{1}{P\log P}\right) P_i^2\right] \nonumber \\&+\frac{N}{\tau }\sum _la_{iil}\eta _l-\frac{ Nh^\prime }{\tau }\kappa _i+N\kappa _i^2+\frac{N}{\tau ^2}\kappa _i^2\eta _i^2 +\frac{N}{\tau }\sum _l{\bar{R}}_{0iil}\eta _l-\beta (\tau \kappa _i-h^\prime \delta _{ii}). \end{aligned}$$

Contract with $\sigma _{n-1}^{ii}$,

$$\begin{aligned} \begin{aligned} 0&\ge \sigma _{n-1}^{ii}\Psi _{ii}\\&=\frac{1}{P\log P}\left[ \sum _le^{\kappa _l}\sigma _{n-1}^{ii}a_{iill}+(n-1)\psi \sum _le^{\kappa _l}\kappa _l^2-\sigma _{n-1}^{ii}\kappa _{i}^2\sum _le^{\kappa _l}\kappa _{l} \right. \\&\qquad \qquad \qquad \quad -C(1+\kappa _{1})P\sum _i\sigma _{n-1}^{ii} +\sum _l\sigma _{n-1}^{ii}e^{\kappa _l}a_{lli}^2 \\&\qquad \qquad \qquad \quad \left. +\sum _{\alpha \ne \gamma }\frac{e^{\kappa _{\alpha }}-e^{\kappa _{\gamma }}}{\kappa _{\alpha }-\kappa _{\gamma }}\sigma _{n-1}^{ii}a_{\alpha \gamma i}^2-\left( \frac{1}{P}+\frac{1}{P\log P}\right) \sigma _{n-1}^{ii}P_i^2\right] \\&+\frac{N}{\tau }\sum _l\sigma _{n-1}^{ii}a_{iil}\eta _l-\frac{ 1}{\tau }(n-1)Nh^\prime \psi +N\sigma _{n-1}^{ii}\kappa _i^2\\&+\frac{N}{\tau ^2}\sigma _{n-1}^{ii}\kappa _i^2\eta _i^2+\frac{N}{\tau }\sum _l\sigma _{n-1}^{ii}{\bar{R}}_{0iil}\eta _l-(n-1)\beta \tau \psi +h^\prime \beta \sum _i\sigma _{n-1}^{ii}. \end{aligned} \end{aligned}$$

(6.2)

Inserting (5.4), (5.5) into (6.2), we obtain

$$\begin{aligned} 0&\ge \sigma _{n-1}^{ii}\Psi _{ii}\nonumber \\&\ge \frac{1}{P\log P}\left[ \sum _le^{\kappa _l}\left( -C-C\kappa _1^2-K(\sigma _{n-1})_l^2+K(\sigma _{n-1})_l^2-\sigma _{n-1}^{pq,rs}a_{pql}a_{rsl}\right) \right. \nonumber \\&\quad \qquad \qquad \quad -\sum _{l,j}e^{\kappa _l}a_{jll}d_{\nu }\psi (E_j)+(n-1)\psi \sum _le^{\kappa _l}\kappa _l^2-\sigma _{n-1}^{ii}\kappa _{i}^2\sum _le^{\kappa _l}\kappa _{l} \nonumber \\&\qquad \qquad \quad -C(1+\kappa _{1})P\sum _i\sigma _{n-1}^{ii} +\sum _l\sigma _{n-1}^{ii}e^{\kappa _l}a_{lli}^2 \nonumber \\&\qquad \qquad \quad \left. +\sum _{\alpha \ne \gamma }\frac{e^{\kappa _{\alpha }}-e^{\kappa _{\gamma }}}{\kappa _{\alpha }-\kappa _{\gamma }}\sigma _{n-1}^{ii}a_{\alpha \gamma i}^2-\left( \frac{1}{P}+\frac{1}{P\log P}\right) \sigma _{n-1}^{ii}P_i^2\right] \nonumber \\&\quad \quad +\frac{N}{\tau }\sum _l\sigma _{n-1}^{ii}a_{iil}\eta _l+N\sigma _{n-1}^{ii}\kappa _i^2+\frac{N}{\tau ^2}\sigma _{n-1}^{ii}\kappa _i^2\eta _i^2 +(h^\prime \beta -C)\sum _i\sigma _{n-1}^{ii}-C(\beta , N).\nonumber \\ \end{aligned}$$

(6.3)

By (6.1) and (5.4), and the Codazzi equation (2.10), we have

$$\begin{aligned} \begin{aligned} \frac{1}{P\log P}\sum _{l,j}e^{\kappa _l}a_{jll} d_{\nu }\psi (E_j) =&\frac{N}{\tau }\sum _l\sigma _{n-1}^{ii}a_{iil}\eta _l -\frac{Nh^\prime }{\tau }\sum _ld_V\psi (E_l)\eta _l\\&+\beta \sum _j\eta _jd_{\nu }\psi (E_j)-\frac{1}{P\log P}\sum _{l,j}e^{\kappa _l}\bar{R}_{0ljl} d_{\nu }\psi (E_j). \end{aligned} \end{aligned}$$

(6.4)

By using (5.6) and (6.3), we get

$$\begin{aligned} \begin{aligned} 0\ge&\frac{1}{P\log P}\left[ \sum _le^{\kappa _l}\left( -C-C\kappa _1^2-K(\sigma _{n-1})_l^2\right) +\sum _le^{\kappa _i}\left( K(\sigma _{n-1})_i^2-\sigma _{n-1}^{pp,qq}a_{ppi}a_{qqi}\right) \right. \\&\qquad \qquad \quad +2\displaystyle \sum _{l\ne i}\sigma _{n-1}^{ii,ll}e^{\kappa _l}a_{lil}^2 +(n-1)\psi \sum _le^{\kappa _l}\kappa _l^2-\sigma _{n-1}^{ii}\kappa _{i}^2\sum _le^{\kappa _l}\kappa _{l} \\&\qquad \qquad \quad -C(1+\kappa _{1})P\sum _i\sigma _{n-1}^{ii} +\sum _l\sigma _{n-1}^{ii}e^{\kappa _l}a_{lli}^2\\&\qquad \qquad \quad \left. +\,2\sum _{l\ne i }\sigma _{n-1}^{ii}\frac{e^{\kappa _{l}}-e^{\kappa _{i}}}{\kappa _{l}-\kappa _{i}}a_{li l}^2-\left( \frac{1}{P}+\frac{1}{P\log P}\right) \sigma _{n-1}^{ii}P_i^2\right] \\&+\,N\sigma _{n-1}^{ii}\kappa _i^2+\frac{N}{\tau ^2}\sigma _{n-1}^{ii}\kappa _i^2\eta _i^2 +(\beta h^\prime -C)\sum _i\sigma _{n-1}^{ii}-C(\beta ,N)-\frac{C}{\kappa _1}. \end{aligned} \end{aligned}$$

(6.5)

From the Codazzi equation (2.10) and Cauchy–Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} 2(a_{lil})^2=&2(a_{lli}-\bar{R}_{0lil})^2\ge (2-\delta )a_{lli}^2-C_\delta , \end{aligned} \end{aligned}$$

where $\delta$ is a small constant to be determined later. Denoting

$$\begin{aligned}&A_i=e^{\kappa _i}\left( K(\sigma _{n-1})_i^2-\sum _{p\ne q}\sigma _{n-1}^{pp,qq}a_{ppi}a_{qqi}\right) , \ \ B_i=2\sum _{l\ne i}\sigma _{n-1}^{ii,ll}e^{\kappa _l}a_{lli}^2, \\&C_i=\sigma _{n-1}^{ii}\sum _le^{\kappa _l}a_{lli}^2; \ \ D_i=2\sum _{l\ne i}\sigma _{n-1}^{ll}\frac{e^{\kappa _l}-e^{\kappa _i}}{\kappa _l-\kappa _i}a_{lli}^2, \ \ E_i=\frac{1+\log P}{P\log P}\sigma _{n-1}^{ii}P_i^2, \end{aligned}$$

we have, by (6.5),

$$\begin{aligned} \begin{aligned} 0\ge&\left( 1-\frac{1}{2}\delta \right) \frac{1}{P\log P}\Big [A_i+B_i+C_i+D_i-E_i\Big ]+\frac{\delta }{2}\frac{1}{P\log P}\sum _i(A_i+C_i)\\&-\frac{\delta }{2} \frac{1+\log P}{(P\log P)^2}\sigma _{n-1}^{ii}P_i^2 -\frac{C_\delta }{P\ln P}\displaystyle \sum _{l\ne i}\sigma _{n-1}^{ii,ll}e^{\kappa _l}-\frac{C_\delta }{P\ln P}\sum _{l\ne i }\sigma _{n-1}^{ii}\frac{e^{\kappa _{l}}-e^{\kappa _{i}}}{\kappa _{l}-\kappa _{i}}\\&+\left( N-1\right) \sigma _{n-1}^{ii}\kappa _i^2+\frac{N}{\tau ^2}\sigma _{n-1}^{ii}\kappa _i^2\eta _i^2 +\left( \beta h^\prime - C \right) \sum _i\sigma _{n-1}^{ii}-C\kappa _1-C(\beta ,N,K)-\frac{C}{\kappa _1}. \end{aligned} \end{aligned}$$

(6.6)

By Schwarz inequality, we always have

$$\begin{aligned} P_i^2=\left( \sum _le^{\kappa _l}a_{lli}\right) ^2\le P\sum _le^{\kappa _l}a^2_{lli} , \end{aligned}$$

which implies

$$\begin{aligned} \frac{\delta }{2}\frac{1}{P\log P}C_i\ge \frac{\delta }{2}\frac{\log P }{(P\log P)^2}\sigma _{n-1}^{ii}P_i^2. \end{aligned}$$

(6.7)

We also have

$$\begin{aligned} \displaystyle \sum _{l\ne i}\sigma _{n-1}^{ii,ll}e^{\kappa _l} \le P\sum _{l\ne i}\sigma _{n-1}^{ii,ll}=2P\sum _i\sigma _{n-2}^{ii}=6P\sigma _{n-3}. \end{aligned}$$

(6.8)

We divided into several cases to compare with $\sigma _{n-2}$.

Case (A) If $\sigma _{n-2}\ge \sigma _{n-3}$, by (6.8), we have, for $n\ge 3$,

$$\begin{aligned} \frac{C_{\delta }}{P\log P}\displaystyle \sum _{l\ne i}\sigma _{n-1}^{ii,ll}e^{\kappa _l} \le 3n^2 \left( \frac{C_{\delta }}{\log P}\sum _i\sigma _{n-1}^{ii}+1\right) . \end{aligned}$$

(6.9)

Case (B) If $\sigma _{n-2}\le \sigma _{n-3}$, in $\Gamma _{n-1}$ cone, since $|\kappa _n|\le \kappa _1/(n-1)$ by the argument in [42], we have

$$\begin{aligned} \kappa _1\ldots \kappa _{n-2}\le C_0\kappa _1\ldots \kappa _{n-3}, \end{aligned}$$

which implies $\kappa _{n-2}\le C_0$. We further divide into two sub-cases to discuss for index $l=1,\ldots ,n$.

Subcase (B1) If $2|\kappa _l|\le \kappa _1$, we have

$$\begin{aligned} \frac{e^{\kappa _l}}{P}\le e^{\kappa _l-\kappa _1}\le e^{-\frac{\kappa _1}{2}}\le \left[ \displaystyle \frac{1}{(n-3)!}\left( \displaystyle \frac{\kappa _1}{2}\right) ^{n-3}\right] ^{-1}. \end{aligned}$$

The last inequality comes from Taylor expansion. Thus, we have

$$\begin{aligned} \frac{C_{\delta }}{P\log P}\sigma _{n-1}^{ii,ll}e^{\kappa _l}\le C_1\frac{C_{\delta }}{\kappa _1}\le 1, \end{aligned}$$

for sufficiently large $\kappa _1$.

Subcase (B2) For sufficiently large $\kappa _1$, if $2|\kappa _l|\ge \kappa _1$, by $\kappa _{n-2}\le C_0$, we have $1\le l\le n-3$. In this case, we have

$$\begin{aligned} \sigma _{n-1}^{ii,ll}\le C_1\kappa _1\ldots \kappa _{l-1}\kappa _{l+1}\ldots \kappa _{n-2}\le \kappa _1\ldots \kappa _{l-1}\kappa _l\kappa _{l+1}\ldots \kappa _{n-2}\le \sigma _{n-2}. \end{aligned}$$

The middle inequality comes from $2\kappa _l\ge \kappa _1\ge 2C_1$ for sufficiently large $\kappa _1$. Thus, we have

$$\begin{aligned} \frac{C_{\delta }}{P\log P}\sigma _{n-1}^{ii,ll}e^{\kappa _l}\le \frac{C_{\delta }}{\log P}\sigma _{n-2}. \end{aligned}$$

Combining cases (B1) and (B2), we also have (6.9).

By mean value theorem, we have some $\xi$ between in $\kappa _i$ and $\kappa _l$ satisfying

$$\begin{aligned} \sum _{l\ne i}\sigma _{n-1}^{ii}\frac{e^{\kappa _{l}}-e^{\kappa _{i}}}{\kappa _{l}-\kappa _{i}}=\sum _{l\ne i}\sigma _{n-1}^{ii}e^{\xi }\le (n-1)P\sum _i\sigma _{n-1}^{ii}. \end{aligned}$$

(6.10)

Hence, using the discussion in [42], we have

$$\begin{aligned} A_i+B_i+C_i+D_i-E_i\ge 0. \end{aligned}$$

Thus, by (6.6), (7.6), (6.9), (6.10), we have

$$\begin{aligned} 0\ge & {} -\frac{\delta }{2} \frac{1}{(P\log P)^2}\sigma _{n-1}^{ii}P_i^2+\frac{\delta }{2}\frac{1}{P\log P}\sum _iA_i \nonumber \\&+\left( N-1\right) \sigma _{n-1}^{ii}\kappa _i^2+\frac{N}{\tau ^2}\sigma _{n-1}^{ii}\kappa _i^2\eta _i^2 \nonumber \\&+\left( \beta h^\prime - C-\frac{C_{\delta }}{\log P} \right) \sum _i\sigma _{n-1}^{ii}-C(\beta ,N,K)-\frac{C}{\kappa _1}. \end{aligned}$$

(6.11)

From (6.1) and the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \begin{aligned} \frac{\delta }{2} \sigma _{n-1}^{ii}\frac{P_i^2}{(P\log P)^2}=&\frac{\delta }{2} \sigma _{n-1}^{ii}\left( \frac{N}{\tau }\kappa _i-\beta \right) ^2\eta _i^2 \le C\delta \left( N^2\sigma _{n-1}^{ii}\kappa _i^2+\beta ^2\sum _i\sigma _{n-1}^{ii}\right) . \end{aligned} \end{aligned}$$

(6.12)

Therefore, by Lemma 2.6, (6.11), (6.12), we obtain

$$\begin{aligned} \begin{aligned} 0\ge&\left( N-1- C\delta N^2\right) \sigma _{n-1}^{ii}\kappa _i^2+\left( \beta h^\prime - C-\frac{C_{\delta }}{\log P}- C\delta \beta ^2\right) \\&\sum _i\sigma _{n-1}^{ii}-C(\beta ,N,K)-\frac{C}{\kappa _1}. \end{aligned} \end{aligned}$$

(6.13)

Since $\sigma _{n-1}^{ii}\kappa _i^2\ge C_1\kappa _1$, we only need to choose the constants $N,\beta ,\delta$ carefully. At first, we take constant N satisfying

$$\begin{aligned} (N-n-1)C_1-C(K)\ge 1. \end{aligned}$$

Secondly, we choose constant $\beta$ satisfying

$$\begin{aligned} \beta h^\prime -2C-2\ge C_2\beta -2C-2\ge 0. \end{aligned}$$

Thirdly, we let constant $\delta$ satisfying

$$\begin{aligned} \max \{CC_1\delta N^2,C\delta \beta ^2\}\le 1. \end{aligned}$$

At last, we take sufficiently large $\kappa _1$ satisfying

$$\begin{aligned} \frac{C_{\delta }}{\log P}\le \frac{C_{\delta }}{\kappa _1}\le 1. \end{aligned}$$

Finally, by (6.13), we obtain the upper bound of $\kappa _{1}$.

7 The existence results

We are in the position to give the proof of the existence Theorems.

Proof of Theorem 1.1

We use continuity method to solve the existence result. For parameter $0\le s\le 1$, according to [8, 17], we consider the following family of functions

$$\begin{aligned} \psi ^s(V,\nu )=s\psi (t,u)+(1-s)\varphi (t)\sigma _k(\kappa (t)). \end{aligned}$$

where $\kappa (t)=h^\prime /h$ and $\varphi$ is a positive function defined on I satisfying (i) $\varphi >0$; (ii) $\varphi (t)>1$ for $t\le t_{-}$; (iii) $\varphi (t) < 1$ for $t\ge t_+$; and (iv) $\varphi ^\prime (t)<0$. It is obvious that there exists a unique point $t_0\in (t_-,t_+)$ such that $\varphi (t_0)=1$. By [8], $z=t_0$ is the unique hypersurface satisfying problem (1.2) and one can check directly that $\psi ^s$ also satisfies (a), (b), (c) in Theorem 1.1. The height estimate can be easily obtained by comparison principle.

The openness and uniqueness are also similar to [17, 30]. In view of Evans–Krylov theory, we only need height, gradient and $C^2$ estimates to complete the closeness part which has been done in Sects. 3, 5 and 6. We complete our proof. $\square$

In what following, we discuss the constant rank theorem in space forms according to [24, 26, 29]. We rewrite our Eq. (1.2) to be

$$\begin{aligned} F(a)=-\sigma _k^{-1/k}(a)=-\psi ^{-1/k}(X,\nu ). \end{aligned}$$

(7.1)

Proposition 7.1

Suppose the ambient space $({\bar{M}},{\bar{g}})=({\mathbb {S}}^{n+1}, {\bar{g}})$ is the sphere with the metric defined by (1.1) and h(t) is given by (1.4). Suppose some compact hypersurface $\Sigma$ satisfies (7.1) and its second fundamental form is non-negative definite. Let X, Y be two vector fields in the ambient space and ${\bar{\nabla }}$ be the covariant derivative of the ambient space. If the function $\psi$ locally satisfies

$$\begin{aligned} {\bar{\nabla }}_{X}{\bar{\nabla }}_Y(\psi ^{-1/k})+\lambda \psi ^{-1/k}{\bar{g}}_{X,Y}\ge 0, \end{aligned}$$

at any $(u,z)\in \Sigma$, then the hypersurface $\Sigma$ is of constant rank.

Proof

According to [24], suppose $P_0$ is the point where the second fundamental form is of the minimal rank l. Let O be some open neighborhood of $P_0$. If O is sufficiently small, we can pick some constant A as in [24]. Then we use the auxiliary function $\varphi =\sigma _{l+1}(a)+A\sigma _{l+2}(a)$ to establish a differential inequality.

Now we choose a local orthonormal frame $\{e_1\ldots ,e_n\}$ in the hypersurface $\Sigma$. Since ${\bar{M}}$ is the sphere with sectional curvature $\lambda$, we obviously have

$$\begin{aligned} \bar{R}_{abcd}=\lambda (\delta _{ac}\delta _{bd}-\delta _{ad}\delta _{bc}). \end{aligned}$$

By Lemma 2.2, we have

$$\begin{aligned} \varphi _j= & {} (\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})a_{iij},\nonumber \\ \varphi _{jj}= & {} (\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})a_{iijj}+(\sigma _{l+1}^{pq,rs}+A\sigma _{l+2}^{pq,rs})a_{pqj}a_{rsj}\nonumber \\= & {} (\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})[a_{jjii}-a_{im}(a_{mj}a_{ji}-a_{mi}a_{jj})-a_{mj}(a_{mj}a_{ii}-a_{mi}a_{ij})\nonumber \\&-2a_{mi}\lambda (\delta _{mj}\delta _{ij}-\delta _{mi}\delta _{jj})+a_{ji}\lambda \delta _{00}\delta _{ji}+a_{ii}\lambda (-\delta _{00}\delta _{jj})\nonumber \\&-2a_{mj}\lambda (\delta _{mj}\delta _{ii}-\delta _{mi}\delta _{ij})+a_{jj}\lambda \delta _{00}\delta _{ii}-a_{ij}\lambda \delta _{00}\delta _{ij} ]+(\sigma _{l+1}^{pq,rs}+A\sigma _{l+2}^{pq,rs})a_{pqj}a_{rsj}.\nonumber \\ \end{aligned}$$

(7.2)

Thus, we have

$$\begin{aligned} F^{jj}\varphi _{jj}= & {} F^{jj}(\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})[a_{jjii}+a^2_{ii}a_{jj}-a^2_{jj}a_{ii}+\lambda a_{ii}\delta _{jj}-\lambda a_{jj}\delta _{ii} ] \nonumber \\&+F^{jj}(\sigma _{l+1}^{pq,rs}+A\sigma _{l+2}^{pq,rs})a_{pqj}a_{rsj}\nonumber \\= & {} (\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})[(-\psi ^{-1/k})_{ii}-\lambda \psi ^{-1/k}\delta _{ii}]-(\sigma _{l+1}^{ii}+A\sigma _{l+2}^{ii})F^{pq,rs}a_{pqi}a_{rsi}\nonumber \\&+F^{jj}(\sigma _{l+1}^{pq,rs}+A\sigma _{l+2}^{pq,rs})a_{pqj}a_{rsj}. \end{aligned}$$

(7.3)

Since the second fundamental form still satisfies Codazzi property in space forms, the process of dealing with the third order terms is same as [24], We also have

$$\begin{aligned} (\psi ^{-1/k})_{,ii}=(\psi ^{-1/k})_{ii}-a_{ii}(\psi ^{-1/k})_{\nu }, \end{aligned}$$

where the comma in the first term means taking covariant derivative with respect to the metric of the ambient space. Thus, since the index i is a bad index, the third term is useless. We have our results. $\square$

Now, we can prove Theorem 1.3.

Proof of Theorem 1.3

The proof also use the degree theory by modifying the proof in [30]. We consider the auxiliary equation

$$\begin{aligned} \sigma _k(\kappa (X))=\psi ^s=\big (s\psi ^{-1/k}(X,\nu )+(1-s){\bar{\varphi }}^{-1/k}\big )^{-k}, \end{aligned}$$

(7.4)

where ${\bar{\varphi }}$ is defined by ${\bar{\varphi }} =C^k_n\varphi \kappa ^k(t)$. We claim that, for the sphere,

$$\begin{aligned} ({\bar{\varphi }}^{-1/k})_{,ij}+\lambda {\bar{\varphi }}^{-1/k}{\bar{g}}_{ij}\ge 0. \end{aligned}$$

(7.5)

where $\{{\bar{e}}_0, \ldots , {\bar{e}}_n\}$ is the local orthonormal frame on ${\bar{M}}$ . If the claim holds, by our condition, it is obvious that the $\psi ^s$ satisfies condition (c) for parameter $0\le s\le 1$. By Proposition 7.1, the strictly convexity is preserved along the flow $\psi ^s$.

Now, let’s discuss Claim (7.5). Define $\alpha (t)=(C^k_n\varphi )^{1/k}$. Since ${\bar{\varphi }}$ is some constant on every slice, we have, for $i,j=1,\ldots , n$

$$\begin{aligned} ({\bar{\varphi }}^{-1/k})_{,ij}= & {} ({\bar{\varphi }}^{-1/k})_{ij}-a_{ij} ({\bar{\varphi }}^{-1/k})_t = -h^2(t)\kappa (t)\frac{\alpha '(t)\kappa (t)+\alpha (t)\kappa '(t)}{\alpha ^2(t) \kappa ^2(t)}g'_{ij} \\= & {} -h^2(t)\frac{\alpha '(t)\kappa (t)+\alpha (t)\kappa '(t)}{\alpha ^2(t) \kappa (t)}g'_{ij}. \end{aligned}$$

Thus, in space forms, we have

$$\begin{aligned} \frac{h''(t)}{h(t)}=\kappa ^2(t)-\frac{1}{h^2(t)}, \kappa '(t)=\frac{h''(t)}{h(t)}-\kappa ^2(t)=-\frac{1}{h^2(t)}. \end{aligned}$$

Then, we have

$$\begin{aligned} ({\bar{\varphi }}^{-1/k})_{,ij}=-h^2(t)\frac{\alpha '(t)\kappa (t)+\alpha (t)\kappa '(t)}{\alpha ^2(t) \kappa (t)}g'_{ij}=\frac{\alpha (t)-\alpha '(t)h^2(t)\kappa (t)}{\alpha ^2(t)\kappa (t)}g'_{ij}>0, \end{aligned}$$

since $\alpha '<0$.

For the unit (namely, $\lambda =1$) sphere, it is easy to see that

$$\begin{aligned} ({\bar{\varphi }}^{-1/k})_{tt}=\frac{\partial ^2 ({\bar{\varphi }}^{-1/k})}{\partial t^2}=-\frac{\partial }{\partial t} \frac{\alpha '(t)\kappa (t)+\alpha (t)\kappa '(t)}{\alpha ^2(t) \kappa ^2(t)}= \frac{\partial }{\partial t}\frac{\alpha -\alpha ' \sin t\cos t}{\alpha ^2\cos ^2 t}. \end{aligned}$$

Thus, we have

$$\begin{aligned} ({\bar{\varphi }}^{-1/k})_{tt}= & {} \frac{2\alpha '\sin ^2 t-\alpha ''\sin t\cos t}{\alpha ^2\cos ^2 t}-\frac{\alpha -\alpha '\sin t \cos t}{\alpha ^4\cos ^4 t}\\&\left[ (-2\cos t\sin t)\alpha ^2+2\alpha \alpha '\cos ^2 t\right] \\\ge & {} \frac{2\alpha '\sin ^2 t}{\alpha ^2 \cos ^2 t}+2\frac{\alpha -\alpha '\sin t \cos t}{\alpha ^2 \cos ^4 t}\cos t\sin t \\= & {} \frac{2\alpha \cos t\sin t}{\alpha ^2 \cos ^4 t}\\> & {} 0, \end{aligned}$$

if we require $\alpha '<0$ and $\alpha '' >0$. Thus, Claim (7.5) holds for unit sphere. Since it is rescaling invariant, then (7.5) holds for any $\lambda >0$.

Now we can give the requirements for functions $\varphi (t)$ to satisfying $\alpha '<0$ and $\alpha ''>0$. It is a straightforward calculation that

$$\begin{aligned} k\alpha ^{k-1}\alpha '=C^k_n\varphi '\quad \text { and } \quad k\alpha ^{k-1}\alpha ''+k(k-1)\alpha ^{k-2}(\alpha ')^2=C^k_n\varphi '', \end{aligned}$$

which implies that

$$\begin{aligned} \varphi '<0\quad \text { and } \quad \varphi \varphi ''>\frac{k-1}{k}(\varphi ')^2. \end{aligned}$$

(7.6)

We further need that (i) $\varphi >0$; (ii) $\varphi (t)>1$ for $t\le t_{-}$; (iii) $\varphi (t) < 1$ for $t\ge t_+$. There is a lot of functions satisfying (i) (ii) (iii) and (7.6), for example

$$\begin{aligned} \varphi (t)=\exp (\frac{t_-+t_+}{2}-t). \end{aligned}$$

Thus, the initial surfaces satisfy the condition of constant rank theorem and the height estimate comes from comparison principle. The curvature estimate has been obtained in Sect. 5. The rest part of this proof is similar to convex case in the Euclidean space, where we only need to replace the constant rank theorem in [30] by Proposition 8.1 here. $\square$

Remark 7.2

In hyperbolic space, the problem is that the slice spheres do not satisfy the constant rank theorem: Proposition 7.1. It may be an interesting problem to find some other nontrivial initial family of hypersurfaces to satisfy Proposition 7.1.