2 Preliminary

2.1 Semi-Riemannian manifold and its spacelike hypersurface

For the semi-Riemannian manifold $(N,\bar g)$ , we define its curvature tensor of type $(1,3)$ and that of type $(0,4),$ respectively, as

$$ \begin{align*} \bar R(X, Y)Z=\bar\nabla_X\bar\nabla_Y Z-\bar\nabla_Y\bar\nabla_X Z-\bar\nabla_{[X,Y]}Z\nonumber \end{align*} $$

and

$$ \begin{align*} \bar R(W, Z, X, Y)=\bar g \big(\bar R(X, Y)Z, W\big).\nonumber \end{align*} $$

Lengthy computation leads to the fact that $\mathbb {S}^n_1$ is of constant sectional curvature $1$ .

Let M be a closed spacelike hypersurface embedded in the semi-Riemannian manifold $(N,\bar g)$ , with its induced metric g and its Levi-Civita connection $\nabla $ . The decomposition of the covariant derivative with respect to $\bar \nabla $ is abided by the Gauss formula

$$ \begin{align*} \bar\nabla_X Y=\nabla_X Y+B(X,Y), ~ ~ \forall ~ X,Y\in TM,\nonumber \end{align*} $$

where B is the vector-valued second fundamental form. Here and in the sequel, we do not distinguish a tangential vector field to M and its push-forward. Let $\nu $ be the outward or future-directed timelike unit normal along M, then we obtain the corresponding Weingarten transformation $\mathcal {W}$ and scalar-valued second fundamental form $h,$ respectively, through

$$ \begin{align*} \bar\nabla_X \nu=\mathcal{W}(X) ~\;~ \text{and}~\;~ B(X,Y)=-\epsilon h(X,Y)\nu\nonumber \end{align*} $$

such that

$$ \begin{align*} h(X,Y)= g\big(\mathcal{W}(X) ,Y\big).\nonumber \end{align*} $$

Furthermore, we have the Gauss equation

$$ \begin{align*} \bar R(W, Z, X,Y)=R(W, Z, X, Y)-\epsilon \big[h(Y,Z)h(X,W)-h(X,Z)h(Y,W)\big]\nonumber \end{align*} $$

and the Codazzi equation

$$ \begin{align*} \bar R(\nu, Z, X, Y)=\big(\nabla_Y h\big)(X, Z)-\big(\nabla_X h\big)(Y, Z),\nonumber \end{align*} $$

where $R(\cdot ,\cdot ,\cdot ,\cdot )$ is the curvature tensor determined by g and

$$ \begin{align*} \big(\nabla_Y h\big)(X, Z)=Y(h(X, Z))-h(\nabla_Y X, Z)-h(X, \nabla_Y Z).\nonumber \end{align*} $$

Obviously, for spacelike hypersurface embedded in a semi-Riemannian space form, such as $\mathbb {S}^n_1$ , the tensor $\nabla h$ of type $(0,3)$ possesses complete symmetry.

For any symmetric $(0, 2)$ tensor, say T, on spacelike hypersurface $(M,g,\nabla )$ , the law for interchanging the order when taking covariant derivative twice is

$$ \begin{align*} \big(\nabla^2 T\big)(U, V; X, Y)-\big(\nabla^2 T\big)(U, V; Y, X)=T\big(U, R(X, Y)V\big)+T\big(R(X, Y)U, V\big),\nonumber \end{align*} $$

this is the Ricci identity.

Since every compact spacelike hypersurface embedded in $\mathbb {S}^n_1$ can be written as graph over $\mathbb {S}^n$ , i.e., we may assume that there exists some $\rho :\mathbb {S}^n\rightarrow \mathbb {R}^+$ such that

$$ \begin{align*} M=\big\{\big(\rho(\zeta), \zeta\big)|~ \zeta\in \mathbb{S}^n\big\}.\nonumber \end{align*} $$

In general, we may assume the spacelike hypersurface embedded in Riemannian or Lorentzian warped product (1.1) can be written as graph over $\mathbb {S}^n$ . Corresponding to the natural frame $\{\partial _i\}_{i=1}^n$ on $\mathbb {S}^n$ , there is one special tangential frame

$$ \begin{align*} \{\partial_i+\rho_i\partial_r , ~ i=1, 2,\dots,n\},\nonumber \end{align*} $$

under which the induced metric on M is

(2.1) $$ \begin{align} g_{ij}=\phi^2(\rho)\sigma_{ij}+\epsilon \rho_i\rho_j, \end{align} $$

where $D\rho =(\rho _1,\dots ,\rho _n)$ is the gradient of function $\rho $ with respect to the Levi-Civita connection D on $\mathbb {S}^n$ . Since M is spacelike, the induced metric is a Riemannian one, which is equivalent to

$$ \begin{align*} 1+\epsilon\phi^{-2}|D\rho|^2>0,\nonumber \end{align*} $$

where $|D\rho |^2=\sigma ^{ij}\rho _i\rho _j$ . And the inverse of the matrix $[g_{ij}]$ is

$$ \begin{align*} g^{ij}=\phi^{-2}\bigg(\sigma^{ij}-\epsilon\frac{\rho^i\rho^j}{\phi^2\upsilon^2}\bigg).\nonumber \end{align*} $$

Here and in the sequel, Einstein’s summation convention is adopted.

It is trivial to check that

$$ \begin{align*} \nu=\frac{1}{\upsilon}\big(\partial_r-\epsilon \phi^{-2}D\rho\big)\nonumber \end{align*} $$

is the outward or the future-directed timelike unit normal to graphical hypersurface, where $\upsilon =\sqrt {1+\epsilon \phi ^{-2}|D\rho |^2}$ . And the support function

(2.2) $$ \begin{align} u=\epsilon \bar g(\phi\partial_r,\nu)=\frac{\phi}{\upsilon} \end{align} $$

depends only on $\rho $ and its derivatives of first order and is obviously larger (smaller) than $\phi $ in Lorentzian (Riemannian) setting.

Now the scalar-valued second fundamental form determines a symmetric matrix with components

(2.3) $$ \begin{align} h_{ij}=\frac{1}{\upsilon}\big(\phi\phi'\sigma_{ij}-\epsilon\rho_{, ij}+2\epsilon\phi^{-1}\phi'\rho_i\rho_j\big), \end{align} $$

where $\rho _{,\, ij}$ is the covariant derivatives with respect to $\sigma $ . From (2.1), it yields that the Christoffel symbols of $g_{ij}$ and of $\sigma _{ij}$ are interrelated, we conclude further that

(2.4) $$ \begin{align} h_{ij}=\upsilon\big(-\epsilon\rho_{;ij}+\phi\phi'\sigma_{ij}\big), \end{align} $$

where $\rho _{;\, ij}$ is the Hessian with respect to g. Notice that the second fundamental form contains the derivatives of $\rho $ up to second order.

Under any adapted frame $\{\xi _1,\dots ,\xi _n,\nu \}$ along spacelike hypersurface embedded in the semi-Riemannian manifold $(N,\bar g)$ , the Gauss and Codazzi equations obtain their local expressions as

$$ \begin{align*} R_{ijkl}=\bar R_{ijkl}+\epsilon(h_{ik}h_{jl}-h_{il}h_{jk})\nonumber \end{align*} $$

and

$$ \begin{align*} h_{ij;k}-h_{ik;j}=\bar R(\nu,\xi_i,\xi_j,\xi_k),\nonumber \end{align*} $$

respectively. Furthermore, applying the Gauss equation, the Ricci identity and the Codazzi equation, we obtain the following Simons-type identity:

(2.5) $$ \begin{align} h_{ij; kl}&=h_{kl;ij}+\epsilon(h_i^mh_{mj}h_{kl}-h_i^mh_{ml}h_{kj}+h_k^mh_{mj}h_{il}-h_k^mh_{ml}h_{ij})\nonumber\\ &\quad+h_i^m\bar R_{mkjl}+h_j^m\bar R_{mkil}+h_k^m\bar R_{mijl}+h_l^m\bar R_{mijk}\nonumber\\ &\quad+\epsilon h_{kl}\bar R(\nu,\xi_i,\nu,\xi_j)-\epsilon h_{ij}\bar R(\nu,\xi_k,\nu,\xi_l)\\ &\quad+(\bar\nabla_{\xi_l}\bar R)(\nu,\xi_i,\xi_j,\xi_k)+(\bar\nabla_{\xi_j}\bar R)(\nu,\xi_k,\xi_i,\xi_l).\nonumber \end{align} $$

Let $\Phi $ be a primitive of function $\phi $ , then

(2.6) $$ \begin{align} \Phi_{;ij}=\epsilon(\phi' g_{ij}-u h_{ij}). \end{align} $$

Since the vector field $\phi \partial _r$ is conformal Killing with respect to $\bar g$ , then

$$ \begin{align*} \nabla u=\mathcal{W}(\nabla\Phi),\nonumber \end{align*} $$

which is locally equivalent to

(2.7) $$ \begin{align} u_{;i}=h_i^k \Phi_{; k}, \end{align} $$

where $\nabla \Phi $ is the tangential part of $\bar \nabla \Phi =\epsilon \phi \partial _r$ , and furthermore

(2.8) $$ \begin{align} u_{; ij}=\epsilon\phi' h_{ij}+g(\nabla\Phi, \nabla h_{ij})-\epsilon uh_{ik}h^k_j+\bar R(\nu, \xi_i,\nabla\Phi,\xi_j). \end{align} $$

2.2 Elementary symmetric polynomials

For spacelike smooth hypersurface embedded in some semi-Riemannian manifold, its eigenvalues of the first Newton transformation and of the Weingarten transformation satisfy

$$ \begin{align*} \lambda_i=\sum_{j\neq i}\kappa_j, ~\;~\forall ~\;~ i=1,\dots, n.\nonumber \end{align*} $$

For $\lambda :=(\lambda _1,\dots ,\lambda _n)$ , its kth normalized elementary symmetric polynomial is defined as

$$ \begin{align*} E_k(\lambda)=\binom{n}{k}^{-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\lambda_{i_1}\dots\lambda_{i_k},\nonumber \end{align*} $$

where $\binom {n}{k}$ is the combinatorial number. Let

$$ \begin{align*} G(\lambda)=\bigg[\frac{E_k(\lambda)}{E_l(\lambda)}\bigg]^{\frac{1}{k-l}},\nonumber \end{align*} $$

then G is homogeneous of degree one such that $G(1,\dots ,1)=1$ and is also monotone increasing and concave in $\Gamma _k$ . And the function G will be considered occasionally as in symmetric matrix $\eta =[\eta _{ij}]$ such that

(2.9) $$ \begin{align} G^{ij,pq}(\eta)\mathcal{S}_{ij}\mathcal{S}_{pq}=\sum_{i, j}\frac{\partial^2 G}{\partial\lambda_i\partial\lambda_j}(\lambda)\mathcal{S}_{ii}\mathcal{S}_{jj} +\sum_{i\neq j} \frac{G^i(\lambda)-G^j(\lambda)}{\lambda_i-\lambda_j}\mathcal{S}^2_{ij}, \end{align} $$

where $(\mathcal {S}_{ij})$ is any symmetric matrix and

$$ \begin{align*} G^{ij,pq}(\eta)=\frac{\partial^2 G}{\partial\eta_{ij}\partial\eta_{pq}}(\eta)~~\;~~ \text{and} ~~\; ~~ G^i(\lambda)=\frac{\partial G}{\partial\lambda_i}(\lambda).\nonumber \end{align*} $$

For convenience, we introduce the following notations:

$$ \begin{align*} G^{ij}:=\frac{\partial G}{\partial\eta_{ij}},~\;~ F^{ii}:=\sum_{j\neq i}G^{jj}.\nonumber \end{align*} $$

Indeed, $G(\lambda )$ is monotone increasing in $\Gamma _k$ implies that $G(\eta )$ is elliptic, or the matrix $G^{ij}(\eta )$ is positive definite if the eigenvalues of matrix $\eta $ belongs to $\Gamma _k$ . Several formulae are stated in the following lemma in consideration of their great importance and interest.

Lemma 2.1

$$ \begin{align*} \sum_{i=1}^nF^{ii}&=(n-1)\sum_{i=1}^n G^{ii},\nonumber\\\sum_{i=1}^n F^{ii}h_{ii}&=\sum_{i=1}^n G^{ii}\eta_{ii},\nonumber \end{align*} $$

and for every $p\in \{1,\dots ,n\}$ ,

$$ \begin{align*} \sum_{i=1}^n F^{ii}h_{ii;p}=\sum_{i=1}^n G^{ii}\eta_{ii; p}.\nonumber \end{align*} $$

From now on, we will assume that k is an integer larger than or equal to $2$ and M is a compact spacelike $(\eta ,k)$ -convex hypersurface embedded in $\mathbb {S}^n_1$ , such that along which (1.7) holds, then

(2.10) $$ \begin{align} H(\kappa)=\frac{n}{n-1}E_1(\lambda)>0. \end{align} $$

3 $C^0$ and gradient estimates

Suppose the function $\rho :\mathbb {S}^n\rightarrow \mathbb {R^+}$ , corresponding to M, attains its maximum $\rho _{\max }$ at $\zeta _0\in \mathbb {S}^n$ , then

$$ \begin{align*} D\rho(\zeta_0)=0, ~\;~ D^2\rho (\zeta_0)\leq 0.\nonumber \end{align*} $$

From (2.1) and (2.4), it yields consequently that at the considered point, Weingarten transformation $\mathcal {W}$ satisfies

(3.1) $$ \begin{align} h^i_j\leq \tanh(\rho_{\max})\delta^i_j. \end{align} $$

In other words, the principal curvatures there are less than or equal to $\tanh (\rho _{\max })$ , which implies further that

(3.2) $$ \begin{align} \lambda_i\leq(n-1)\tanh(\rho_{\max}),~\;~ \forall~\;~ i=1,\dots, n. \end{align} $$

Since G is now monotone increasing and homogeneous of degree one, then

$$ \begin{align*} G(\lambda)\leq (n-1)\tanh(\rho_{\max}).\nonumber \end{align*} $$

Similarly, at the point where $\rho $ attains its minimum $\rho _{\min }$ ,

$$ \begin{align*} G(\lambda)\geq (n-1) \tanh(\rho_{\min}).\nonumber \end{align*} $$

From the first item in Assumption 1.1, we can conclude that

(3.3) $$ \begin{align} r_1\leq\rho\leq r_2. \end{align} $$

Now, due to (2.2) and (3.3), the gradient estimate for function $\rho $ follows directly from the upper bound for support function u. Noticing also that the support function u is now naturally bounded from below by $\phi (r_1)>1$ . To obtain its upper bound, we apply the maximum principle to the auxiliary function

$$ \begin{align*} \varphi=\ln u-\gamma(\Phi),\nonumber \end{align*} $$

where $\gamma (\Phi )=\beta \ln \Phi $ with $\beta>1$ a constant to be determined. Without loss of generality, we may assume that $\Phi>1$ .

At the point where $\varphi $ attains its maximum, there holds

(3.4) $$ \begin{align} \nabla \ln u=\gamma'(\Phi)\nabla \Phi, \end{align} $$

while the Hessian

$$ \begin{align*} \varphi_{;\, ij}=-(\ln u)_{;\, i} (\ln u)_{;\, j}+\frac{1}{u} u_{;\, ij}-\gamma"(\Phi)\Phi_{; i}\Phi_{; j}-\gamma'(\Phi)\Phi_{; ij}\nonumber \end{align*} $$

is nonpositive definite.

We may assume further that $\phi \partial _r=-\bar \nabla \Phi $ is not parallel to $\nu $ at the considered point, since otherwise the proof is done. Hence, we may choose the particular local orthonormal frame $\{\xi _i\}_{i=1}^n$ such that

$$ \begin{align*} g(\nabla\Phi, \xi_1 )=|\nabla\Phi|\neq 0, ~\;~ g(\nabla\Phi, \xi_j)=0, ~\;~ \forall ~\;~ j=2,\dots, n,\nonumber \end{align*} $$

where the norm $|\cdot |$ is with respect to g. From (3.4) and (2.7), it follows then

$$ \begin{align*} h_1^1=u \gamma'(\Phi), ~\;~ h_j^1=0, ~\;~ \forall~\;~ j=2, \dots, n.\nonumber \end{align*} $$

And after rotating $\xi _2,\dots ,\xi _n$ if necessary, we may assume further that $[h_{ij}]$ and then $[\eta _{ij}]$ are diagonal. Consequently at the considered point, there holds

$$ \begin{align*} 0&\geq F^{ii}\varphi_{;ii}\nonumber\\ &\geq F^{11}\big[\gamma"(\Phi)+\gamma^{\prime2}(\Phi)\big]\phi^2-\phi'\frac{\Psi}{u}+\frac{1}{u} g(\nabla\Phi ,\nabla\Psi)+\sum_{j=2}^nF^{jj}h^2_{jj}\nonumber\\ &-\gamma'(\Phi) u\Psi+(n-1)\phi'\gamma'(\Phi)\sum_{i=1}^n G^{ii}-\gamma"(\Phi)F^{11} u^2,\nonumber \end{align*} $$

where we also used (2.6), (2.7), (2.8) and the basic fact

(3.5) $$ \begin{align} |\nabla\Phi|^2=u^2-\phi^2. \end{align} $$

Since

$$ \begin{align*} \nabla\Psi=\nabla^{\mathbb{X}}\Psi+\Psi_u\nabla u,\nonumber \end{align*} $$

where $\nabla ^{\mathbb {X}}\Psi $ stands for the tangential part of the ambient vector field

$$ \begin{align*} \bar\nabla\Psi:=\phi^{-2}D\Psi+\epsilon\Psi_r \partial_r\nonumber \end{align*} $$

and $\Psi _u$ stands for the partial derivative of $\Psi $ with respect to u, then

$$ \begin{align*} \frac{1}{u}g(\nabla\Phi, \nabla\Psi)=\frac{1}{u} g (\nabla\Phi, \nabla^{\mathbb{X}}\Psi)+\Psi_u \gamma'(\Phi)|\nabla\Phi|^2,\nonumber \end{align*} $$

where we used the critical point condition (3.4) again. On the other hand, the Cauchy–Schwarz inequality implies

$$ \begin{align*} |\nabla^{\mathbb{X}}\Psi|_g^2\leq \frac{2}{\upsilon^{2}}\big[\phi^{-2}|D\Psi|^2_{\sigma}+\Psi_r^2\big]=\frac{2}{\upsilon^{2}}||\bar\nabla\Psi||^2.\nonumber \end{align*} $$

Finally, we can conclude that

(3.6) $$ \begin{align} 0&\geq \big[\gamma"(\Phi)+\gamma^{\prime2}(\Phi)\big]F^{11}\phi^2-\gamma"(\Phi)F^{11} u^2-\sqrt{2}\phi^{-1}\,||\bar\nabla \Psi|| \sqrt{u^2-\phi^2}\nonumber\\ &+\gamma'(\Phi) u\bigg(1-\frac{\phi^2}{u^2}\bigg)u\Psi_u-u\Psi\bigg(\gamma'(\Phi)+\frac{\phi'}{u^2}\bigg)+(n-1)\phi'\gamma'(\Phi)\sum_{i=1}^n G^{ii}. \end{align} $$

Now we may choose $\beta $ large enough such that $\gamma "(\Phi )+\gamma ^{\prime 2}(\Phi )> 0$ . Furthermore, thanks to the second item in Assumption 1.1, contradiction will emerge in (3.6) once the support function u is permitted to approach to $+\infty $ .

4 Curvature estimate

In this section, we derive a priori estimate for the squared norm of the second fundamental form

$$ \begin{align*} \kappa_1^2+\cdots+\kappa_n^2.\nonumber \end{align*} $$

Due to (2.10), it suffices to prove that the largest principal curvature is bounded from above along M.

We may introduce the auxiliary function

$$ \begin{align*} \omega=\ln \kappa_{\max}- 2A\Phi, \nonumber \end{align*} $$

where $\kappa _{\max }$ stands for the largest principal curvature and $A>1$ is a constant to be determined. In general, $\kappa _{\max }$ is merely a continuous function along M, for which the maximum principle is not available.

Assume that $\omega $ achieves its maximum at $P_0\in M$ . We may choose the local orthonormal frame $\{ e_1,\dots , e_n\}$ such that $h_{ij}=h_{ii}\delta _{ij}$ and $h_{11}\geq h_{22}\geq \cdots \geq h_{nn}$ at $P_0$ . Since $h_{11}(P_0)=\kappa _{\max }(P_0)$ and $h_{11}\leq \kappa _{\max }$ around $P_0$ , then with slight abuse of notation, the new auxiliary function

$$ \begin{align*} \omega=\ln h_{11}-2A \Phi\nonumber \end{align*} $$

achieves its maximum at $P_0$ as well. Now, the new auxiliary function $\omega $ is of higher regularity around $P_0$ and without loss of generality, we may assume that $h_{11}(P_0)$ approaches to $+\infty $ . It follows then, at $P_0$ ,

(4.1) $$ \begin{align} \nabla \ln h_{11}=2A \nabla\Phi, \end{align} $$

and

(4.2) $$ \begin{align} 0\geq F^{ii} \big(\ln h_{11}\big)_{; ii}- 2A F^{ii} (u h_{ii}-\phi'g_{ii}), \end{align} $$

where we used equation (2.6).

Step 1: Applying (2.5) to the first term in (4.2), it yields that

$$ \begin{align*} F^{ii}\big(\ln h_{11}\big)_{; ii}&=\frac{1}{h_{11}}F^{ii}h_{ii; 11}+\sum_{i=1}^nF^{ii}+F^{ii}h^2_{ii}\nonumber\\ &\quad -\frac{\Psi}{h_{11}}-\Psi h_{11}-F^{ii}(\ln h_{11})^2_{;i}.\nonumber \end{align*} $$

Noticing also that

$$ \begin{align*} \frac{1}{h_{11}}F^{ii} h_{ii; 11}&=\frac{1}{h_{11}}G^{ii} \eta_{ii; 11}=\frac{1}{h_{11}}\big(\Psi_{;11}-G^{pq,rs}\eta_{pq; 1} \eta_{rs; 1}\big)\nonumber\\ &=u\Psi_uh_{11}+\Phi_{;1}^2\Psi_{uu} h_{11}+\frac{d_{\mathbb{X}}\Psi(\nabla_{e_1}\nabla_{e_1}\mathbb{X})}{h_{11}}\nonumber\\ &\quad +2\Phi_{;1} d^2_{\mathbb{X} u}\Psi (\nabla_{e_1}\mathbb{X})-\phi'\Psi_u+\frac{d^2_{\mathbb{X}\mathbb{X}}\Psi(\nabla_{e_1}\mathbb{X}, \nabla_{e_1}\mathbb{X})}{h_{11}}\nonumber\\ &\quad +g (\nabla\Phi,\nabla \ln h_{11})\Psi_u-\frac{1}{h_{11}}G^{pq,rs}\eta_{pq; 1} \eta_{rs; 1},\nonumber \end{align*} $$

where we used (2.7), (2.8) and $d_{\mathbb {X}}\Psi $ , $d^2_{\mathbb {X}u}\Psi $ , and $d^2_{\mathbb {X}\mathbb {X}}\Psi $ can be viewed as matrices of different sizes. Now, when restricted to M, the prescribed function $\Psi $ is smoothly defined on some compact subset in $\mathbb {S}^n_1\times [1,+\infty )$ . Therefore,

$$ \begin{align*} F^{ii}\big(\ln h_{11}\big)_{; ii}&\geq-C_0h_{11}-\frac{C_0}{h_{11}}-C_0+g\big(\nabla\Phi,\nabla\ln h_{11}\big)\Psi_u+F^{ii}h^2_{ii}+\sum_{i=1}^nF^{ii}\nonumber\\ &\quad -\frac{2}{h_{11}}\sum_{j=2}^n G^{1j,j1}h^2_{11;j}-F^{ii}(\ln h_{11})^2_{;i},\nonumber \end{align*} $$

where $C_0$ is a sufficiently large constant depending on $\inf \rho $ , $|\rho |_{C^1}$ and $||\Psi ||_{C^2}$ , and we used (2.9) to deduce that

$$ \begin{align*} -G^{pq,rs}\eta_{pq; 1} \eta_{rs; 1}\geq -2\sum_{j=2}^n G^{1j,j1}\eta_{1j;1}^2=-2\sum_{j=2}^n G^{1j,j1}h_{11;j}^2.\nonumber \end{align*} $$

Using the critical point condition (4.1), we obtain finally that

(4.3) $$ \begin{align} 0&\geq -C_0h_{11}+F^{ii}h^2_{ii}+(1+2A\phi') \sum_{i=1}^nF^{ii}\nonumber\\ &-\frac{C_0}{h_{11}}-C_0-2AC_0 -\frac{2}{h_{11}}\sum_{j=2}^n G^{1j,j1}h^2_{11;j}-F^{ii}(\ln h_{11})^2_{;i}. \end{align} $$

Step 2 : Let

$$ \begin{align*} A=\max\bigg\{\frac{2nC_0}{(n-1)(n-k+1)}\frac{||\Psi||^{k-l-1}_{C^0}}{\inf \phi'}, 1\bigg\}.\nonumber \end{align*} $$

We claim that there exists a constant

$$ \begin{align*} B=\max\{ 2n\delta^{-2} C_0, 1\}\nonumber \end{align*} $$

such that if $h_{11}\geq B$ , then

(4.4) $$ \begin{align} F^{ii}h^2_{ii}+2A\phi'\sum_{i=1}^n F^{ii}\geq 2C_0 h_{11}, \end{align} $$

where $0<\delta <1$ is some constant such that

(4.5) $$ \begin{align} \frac{\binom{n-1}{k-1}[1-(n-2)\delta]^{k-1}-(n-1) \binom{n-1}{k-2}\delta[1+(n-2)\delta]^{k-2}}{[1+(n-2)\delta]^l}\geq\frac{\binom{n-1}{k-1}}{2}. \end{align} $$

We split its proof into two cases.

Case 1: $|h_{jj}|\leq \delta h_{11}$ for all $j\geq 2$ .

In this case, we have

$$ \begin{align*} |\eta_{11}|\leq (n-1)\delta h_{11}, ~\;~ [1-(n-2)\delta]h_{11}\leq \eta_{22}\leq\cdots\leq \eta_{nn}\leq [1+(n-2)\delta]h_{11}.\nonumber \end{align*} $$

By the definition of $G^{ii}$ and $F^{ii}$ , we obtain

$$ \begin{align*} \sum_{i=1}^n F^{ii}&=(n-1)\sum_{i=1}^nG^{ii}\nonumber\\ &=\frac{n-1}{k-l}\bigg[\frac{E_k(\eta)}{E_l(\eta)}\bigg]^{\frac{1}{k-l}-1}\frac{kE_l(\eta)E_{k-1}(\eta)-l E_{k}(\eta)E_{l-1}(\eta)}{E_l^2(\eta)}\nonumber\\ &\geq\frac{n-1}{\binom{n}{k-1}}\bigg[\frac{E_k(\eta)}{E_l(\eta)}\bigg]^{\frac{1}{k-l}-1}\frac{\sigma_{k-1}(\eta|1)+\eta_{11}\sigma_{k-2}(\eta|1)}{E_l(\eta)}\nonumber\\ &\geq\frac{(n-1)(n-k+1)}{2n}\Psi^{1-k+l}h^{k-l-1}_{11},\nonumber \end{align*} $$

where we used the Newton–MacLaurin inequalities and (4.5). Hence, in this case,

$$ \begin{align*} 2A\phi'\sum_{i=1}^nF^{ii}\nonumber\geq 2C_0 h_{11}, \end{align*} $$

under the assumption that $0\leq l\leq k-2$ .

Case 2: Either $h_{22}>\delta h_{11}$ or $h_{nn}<-\delta h_{11}$ .

In this case, we have

$$ \begin{align*} F^{ii}h^2_{ii}\geq F^{22}h^2_{22}+F^{nn}h^2_{nn}\geq \delta^2F^{22}h^2_{11}.\nonumber \end{align*} $$

Since $\eta _{ii}=H-h_{ii}$ , then at the considered point

$$ \begin{align*} \eta_{11}\leq\eta_{22}\leq \cdots\leq \eta_{nn},\nonumber \end{align*} $$

from which it follows directly that

$$ \begin{align*} G^{11}\geq G^{22}\geq \cdots\geq G^{nn}~\;~\text{and}~\;~F^{11}\leq F^{22}\leq \cdots\leq F^{nn}.\nonumber \end{align*} $$

And there holds further that

$$ \begin{align*} F^{22}\geq\frac{1}{n}\sum_{i=1}^n G^{ii}\geq \frac{1}{n},\nonumber \end{align*} $$

since otherwise

$$ \begin{align*} (n-1)\sum_{i=1}^n G^{ii}=\sum_{i=1}^nF^{ii}<\frac{2}{n}\sum_{i=1}^n G^{ii}+(n-2)\sum_{i=1}^n G^{ii}.\nonumber \end{align*} $$

In this case, we can conclude that

$$ \begin{align*} F^{ii}h^2_{ii}\geq \frac{\delta^2}{n}h^2_{11}\geq 2C_0 h_{11}.\nonumber \end{align*} $$

Step 3: We show that if $h_{11}\geq B$ , then for every $2\leq j\leq n,$

$$ \begin{align*} |h_{jj}|\leq A C_1,\nonumber \end{align*} $$

where $C_1 (>C_0)$ is a positive constant depending on $n,k,l,\delta ,\inf \rho ,|\rho |_{C^1}$ and $||\Psi ||_{C^2}$ .

Combining the results obtained in Steps 1 and 2, we conclude that

(4.6) $$ \begin{align} 0\geq &-\frac{2}{h_{11}}\sum_{j=2}^n G^{1j,j1}h^2_{11;j}-F^{ii}(\ln h_{11})^2_{;i}-\frac{C_0}{h_{11}}-C_0-2AC_0\nonumber\\ &+\frac{1}{2} F^{ii}h^2_{ii}+(1+A\phi') \sum_{i=1}^nF^{ii}. \end{align} $$

From (4.1) and the concavity of G, it follows that

$$ \begin{align*} 0&\geq -4A^2F^{ii}\Phi^2_{;i}-4AC_0+\frac{1}{2}F^{ii}h^2_{ii}+(1+A\phi') \sum_{i=1}^nF^{ii}\nonumber\\ &\geq\frac{1}{2}F^{ii}h^2_{ii}-4AC_0-\big[4A^2C_0-(1+A\phi')\big]\sum_{i=1}^n F^{ii}\nonumber\\ &\geq\bigg\{\frac{1}{2n(n-1)}\sum_{j=2}^n h^2_{jj}-4AC_0-\big[4A^2C_0-(1+A\phi')\big]\bigg\}\sum_{i=1}^n F^{ii}\nonumber \end{align*} $$

where we used again the fact that for $2\leq j\leq n,$

$$ \begin{align*} F^{jj}\geq \frac{1}{n}\sum_{i=1}^n G^{ii}=\frac{1}{n(n-1)}\sum_{i=1}^n F^{ii}.\nonumber \end{align*} $$

The desired estimate follows directly.

Step 4: We show that there exists a constant $C_2$ depending on $n,k,l,\delta ,\inf \rho ,|\rho |_{C^1}$ and $||\Psi ||_{C^2}$ such that

$$ \begin{align*} h_{11}\leq C_2.\nonumber \end{align*} $$

Without loss of generality, we may assume that

(4.7) $$ \begin{align} h_{11}\geq \max\bigg\{ B,\frac{AC_1}{\alpha}\bigg\}, \end{align} $$

where $0<\alpha <1$ will be determined later.

Comparing with the result derived in Step 3, assumption (4.7) implies

$$ \begin{align*} |h_{jj}|\leq \alpha h_{11}, ~\;~ \text{for} ~\;~j\geq 2,\nonumber \end{align*} $$

and then

$$ \begin{align*} \frac{1}{h_{11}}\leq \frac{1+\alpha}{h_{11}-h_{jj}}, ~\;~\forall ~\;~ j\geq 2.\nonumber \end{align*} $$

As a consequence,

$$ \begin{align*} \sum_{j=2}^nF^{jj}(\ln h_{11})^2_{; j}&=\sum_{j=2}^n\frac{F^{jj}-F^{11}}{h^2_{11}}h^2_{11;j}+\sum_{j=2}^n\frac{F^{11}h^2_{11;j}}{h^2_{11}}\nonumber\\ &\leq\frac{1+\alpha}{h_{11}}\sum_{j=2}^n\frac{F^{jj}-F^{11}}{h_{11}-h_{jj}}h^2_{11;j}+\sum_{j=2}^n\frac{F^{11}h^2_{11;j}}{h^2_{11}}\nonumber\\ &=\frac{1+\alpha}{h_{11}}\sum_{j=2}^n\frac{G^{11}-G^{jj}}{\eta_{jj}-\eta_{11}}h^2_{11;j}+\sum_{j=2}^n\frac{F^{11}h^2_{11;j}}{h^2_{11}}\nonumber\\ &=-\frac{1+\alpha}{h_{11}}\sum_{j=2}^n G^{1j,j1}h^2_{11;j}+\sum_{j=2}^n\frac{F^{11}h^2_{11;j}}{h^2_{11}}\nonumber \end{align*} $$

where we used the basic fact that for every $2\leq j\leq n,$

$$ \begin{align*} -G^{1j,j1}=\frac{G^{11}-G^{jj}}{\eta_{jj}-\eta_{11}}.\nonumber \end{align*} $$

Hence, substituting this estimate back into (4.6), we can conclude that

$$ \begin{align*} 0&\geq -F^{11}|\nabla\ln h_{11}|^2-\frac{C_0}{h_{11}}-C_0-2AC_0+C_0 h_{11}\nonumber\\ &=-4A^2 F^{11}|\nabla\Phi|^2-\frac{C_0}{h_{11}}-C_0-2AC_0+C_0 h_{11},\nonumber \end{align*} $$

contradiction occurs once $h_{11}$ is permitted to approach to $+\infty $ , and simultaneously $F^{11}$ approaches to $0$ .

5 Proof of Theorem 1.2

In this section, we use the degree theory for nonlinear elliptic equation developed by Y. Y. Li in [Reference Li17] to prove Theorem 1.2.

We define

$$ \begin{align*} C^{4,\alpha}_{\text{ad}}(\mathbb{S}^n)=\big\{\rho\in C^{4,\alpha}(\mathbb{S}^n):\text{the graph of} \rho \text{is spacelike and} (\eta, k)-\text{convex}\big\},\nonumber \end{align*} $$

and consider the family of functionals

$$ \begin{align*} \mathcal{F}: C^{4,\alpha}_{\text{ad}}(\mathbb{S}^n)\times [0,1]\rightarrow C^{2,\alpha}(\mathbb{S}^n)\nonumber \end{align*} $$

defined by

$$ \begin{align*} \mathcal{F}(\rho,t)=G\big(\lambda(\rho)\big)-\Xi\,(\rho,\zeta, u, t),\nonumber \end{align*} $$

where

$$ \begin{align*} \Xi\,(\rho,\zeta, u, t)=t\Psi(\rho,\zeta, u)+(1-t)\Theta(\rho,\zeta,u)\nonumber \end{align*} $$

and

$$ \begin{align*} \Theta(\rho,\zeta,u)=(n-1)u^p \,\rho \tanh(\rho) , ~\;~ p>1.\nonumber \end{align*} $$

Claim 1 There exists an unique $\hat \rho \in C^{4,\alpha }_{\text {ad}}(\mathbb {S}^n)$ such that

$$ \begin{align*} G(\lambda(\hat\rho))=\Theta(\hat\rho,\zeta,u(\hat\rho)).\nonumber \end{align*} $$

In fact, when we choose $\hat \rho :\mathbb {S}^n\rightarrow \mathbb {R}$ to be of constant value, then the graphical hypersurface determined by $\hat \rho $ is spacelike and umbilical, with principal curvatures $\tanh (\hat \rho )$ and support function $\cosh (\hat \rho )$ . Hence, $G(\lambda (\hat \rho ))=(n-1)\tanh (\hat \rho )$ , and once there holds further $\hat \rho \cosh ^p(\hat \rho )=1$ , then $\hat \rho $ is a solution. Clearly, such a $\hat \rho $ does exist, since the continuous function $f(x)=x\cosh ^p(x)$ satisfies $f(0)=0$ and $f(1)>1$ .

Suppose that there exists another $\rho \in C^{4,\alpha }_{\text {ad}}(\mathbb {S}^n)$ such that $\mathcal {F}(\rho , 0)=0$ and furthermore that $\max \rho =\rho (\zeta _0)>\hat \rho $ . As in obtaining the $C^0$ estimates, we have

$$ \begin{align*} \Theta(\rho(\zeta_0),\zeta_0, u(\zeta_0))&=(n-1)\cosh^p(\rho(\zeta_0))\rho(\zeta_0)\tanh(\rho(\zeta_0))\nonumber\\ &=G(\lambda(\rho(\zeta_0)))\leq (n-1)\tanh(\rho(\zeta_0)),\nonumber \end{align*} $$

which is a contradiction, as $f(x)=x\cosh ^p(x)$ is a monotone increasing function. Therefore, $\max \rho \leq \hat \rho $ . The same argument implies $\min \rho \geq \hat \rho $ , and finally $\rho \equiv \hat \rho $ .

Since for the spacelike hypersurface determined by function $\rho $ ,

$$ \begin{align*} \eta^i_j(\rho)&=\frac{n\phi'}{\phi\upsilon^2}\delta^i_j-\frac{\phi'}{\phi^3\upsilon^4}\big(\phi^2\delta^i_j-\rho^i\rho_j)\nonumber\\ &\quad +\frac{1}{\phi^2\upsilon}\bigg[\delta^i_j\bigg(\sigma^{mq}+\frac{\rho^m\rho^q}{\phi^2\upsilon^2}\bigg)\rho_{,mq}-\bigg(\sigma^{im}+\frac{\rho^i\rho^m}{\phi^2\upsilon^2}\bigg)\rho_{,mj}\bigg],\nonumber \end{align*} $$

then

(5.1) $$ \begin{align} \delta_\rho \mathcal{F}(\vartheta, t):=\frac{d}{ds}\big[G(\rho+s\vartheta)-\Xi(\rho+s\vartheta , t)\big]\bigg|_{s=0}=a^{ij}\vartheta_{,ij}+b^i\vartheta_i+c\vartheta, \end{align} $$

where the matrix

$$ \begin{align*} a^{ij}=G_m^q\frac{\partial \eta^m_q}{\partial\rho_{, ij}}=\frac{1}{\upsilon} \bigg[\bigg(\sum _{m=1}^n G^m_m\bigg)g^{ij}-G^i_m g^{mj}\bigg]\nonumber \end{align*} $$

is positive definite and

$$ \begin{align*} c=G_i^j\frac{\partial \eta^i_j(\rho)}{\partial\rho}-\frac{\partial~ \Xi(\rho,\zeta, t)}{\partial\rho}.\nonumber \end{align*} $$

We are interested in (5.1) when $t=0$ , that is, when $\rho \equiv \hat \rho $ . In this case, we have

$$ \begin{align*} u\equiv \cosh(\hat\rho), ~\,~\eta^i_j\equiv(n-1)\tanh(\hat\rho)\delta^i_j,~\,~ G\equiv(n-1)\tanh(\hat\rho),~\,~ G_i^j\equiv\frac{1}{n}\delta^i_j,\nonumber \end{align*} $$

and consequently when $t=0$

$$ \begin{align*} c=-p(n-1)\hat\rho\cosh^{p-2}(\hat\rho)\sinh^2(\hat\rho)-(n-1)\cosh^{p-1}(\hat\rho)\sinh(\hat\rho)<0,\nonumber \end{align*} $$

where we used the condition that $\hat \rho \cosh ^p(\hat \rho )=1$ .

From the strong maximum principle, it follows that the unique solution to $\delta _\rho \mathcal {F}(\cdot ,0)(\vartheta )=0\nonumber $ is $\vartheta \equiv 0$ , or equivalently that $\ker (\delta _\rho \mathcal {F})=\{0\}$ at $t=0$ . And the standard theory of second-order elliptic equations implies further that $\delta _\rho \mathcal {F}$ is invertible at $t=0$ , as required.

Claim 3 For all $t\in [0,1]$ , the admissible solution $\rho ^t$ of $\mathcal {F}(\rho ,t)=0$ , if any, satisfies

$$ \begin{align*} \underline\varrho\leq\rho^t\leq\overline\varrho,\nonumber \end{align*} $$

where $0<\underline \varrho <\overline \varrho $ are constants independent of t.

In fact, there exist positive constants $R_1$ and $R_2$ such that

$$ \begin{align*} \Theta(r, \zeta, \cosh(r))<(n-1)\tanh(r), ~\:~ r<R_1,\nonumber\\ \Theta(r, \zeta, \cosh(r))>(n-1)\tanh(r), ~\;~ r>R_2.\nonumber \end{align*} $$

Setting $\underline \varrho =\min \{r_1, R_1\}$ and $\overline \varrho =\max \{r_2, R_2\}$ , then for all $t\in [0,1]$ ,

$$ \begin{align*} \Xi (r, \zeta, \cosh(r), t)&<(n-1)\tanh(r),~\;~ r<\underline\varrho\, ,\nonumber\\ \Xi (r, \zeta, \cosh(r), t)&>(n-1)\tanh(r),~\;~ r>\overline\varrho.\nonumber \end{align*} $$

The claim follows then from the comparison principle. Furthermore, it is trivial to check that $\Theta (\rho ,\zeta ,u)$ satisfies as well the second item in Assumption 1.1. Following the same arguments as in Sections 3 and 4, we conclude that for $t\in [0,1]$ , $\rho ^t$ is spacelike and equation $\mathcal {F}(\rho , t)=0$ is uniformly elliptic. According to the regularity theory developed in [Reference Evans10] and [Reference Krylov16], there is constant C depending only on $k,l,n, \underline \varrho , \overline \varrho $ and $||\Psi ||_{C^{2,\alpha }(\mathbb {S}^n)}$ such that for all $t\in [0,1]$

$$ \begin{align*} u(\rho^t)<C~\;~ \text{and} ~\;~ |\rho^t|_{C^{4,\alpha}(\mathbb{S}^n)}< C.\nonumber \end{align*} $$

Moreover, there are positive constants $\mu $ and $C_{\mathcal {W}}$ such that for all $t\in [0,1]$ ,

$$ \begin{align*} \Xi(\rho^t(\zeta),\zeta, u(\rho^t), t)>\mu, ~\;~ \zeta\in\mathbb{S}^n,\nonumber \end{align*} $$

and

$$ \begin{align*} |\kappa_i(\rho^t)|<C_{\mathcal{W}}, ~\;~ i\in\{1,2,\dots,n\}.\nonumber \end{align*} $$

Let

$$ \begin{align*} \Gamma=\bigg\{\lambda\in \Gamma_k~\big|~G(\lambda)>\frac{1}{2}\mu,~|\lambda|<2\sqrt{n}(n-1)C_{\mathcal{W}}\bigg\}\nonumber \end{align*} $$

and

$$ \begin{align*} \mathcal{B}_R=\bigg\{\rho\in C^{4,\alpha}(\mathbb{S}^n) ~\big|~ \frac{1}{2}\,\underline\varrho<\rho<2\,\overline\varrho,~\frac{1}{2}<u(\rho)<R, ~|\rho|_{C^{4,\alpha}(\mathbb{S}^n)}< R,~ \lambda(\rho)\in \Gamma \bigg\}.\nonumber \end{align*} $$

From Claim 3, it yields that there is some sufficiently large constant R such that for all $t\in [0,1]$ , the admissible solution $\rho ^t$ of $\mathcal {F} (\cdot ,t)=0$ , if any, belongs to $\mathcal {B}_R$ , while $ \mathcal {F}(\cdot , t)=0$ has no solution on $\partial \mathcal {B}_R$ . Therefore, the degree $\deg (\mathcal {F}(\cdot , t), \mathcal {B}_R, 0)$ is well defined for $0\leq t\leq 1$ . Using Claims 2 and 3 and the homotopic invariance of the degree, we have

$$ \begin{align*} \deg\big(\mathcal{F}(\cdot, 1), \mathcal{B}_R, 0\big)=\deg\big(\mathcal{F}(\cdot, 0),\mathcal{B}_R, 0\big)=\pm 1.\nonumber \end{align*} $$

So we obtain a solution at $t=1$ . This completes the proof of Theorem 1.2.