Rigidity of Willmore submanifolds and extremal submanifolds in the unit sphere

Abstract

Let M be an n-dimensional $(n\ge 4)$ compact Willmore (or extremal) submanifold in the unit sphere $S^{n+p}$. Denote by ${\text {Ric}}$ and H the Ricci curvature and the mean curvature of M, respectively. It is proved that if $(\int _M ({\text {Ric}}_-^{\lambda })^\frac{n}{2})^\frac{2}{n}<A(n,\lambda ,H,\rho )~ (\text{ or }\ B(n,\lambda ,H,\rho ))$, then M is a totally umbilical sphere, where $A(n,\lambda ,H,\rho )$ and $B(n,\lambda ,H,\rho )$ are two explicit positive constants depending on n, $\lambda$, H, and $\rho$. This extends parts of the results from a pointwise Ricci curvature lower bound to an integral Ricci curvature lower bound.

1 Introduction

Let $x:M\rightarrow S^{n+p}$ be an n-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$. Choose a local orthonormal frame field $\{e_1,\ldots ,e_{n+p}\}$ in $S^{n+p}$ such that, restricted to M, the $\{e_1,\ldots ,e_{n}\}$ are tangent to M. We will make the following convention on the range of indices:

$$\begin{aligned} 1\le i,j,k,\ldots \le n; \qquad n+1\le \alpha ,\beta ,\gamma ,\ldots \le n+p. \end{aligned}$$

Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. Then we have

$$\begin{aligned} S=\sum \limits _{\alpha ,i,j}(h_{ij}^\alpha )^2,\quad \textbf{H}=\sum \limits _{\alpha }H^\alpha e_\alpha ,\quad H ^\alpha =\frac{1}{n}\sum \limits _k h^\alpha _{kk},\quad H=|\textbf{H}|, \end{aligned}$$

where $h_{ij}^\alpha$ are the components of the second fundamental form of M:

We define the following non-negative function on M:

$$\begin{aligned} \rho ^2=S-nH^2, \end{aligned}$$

(1.1)

which vanishes exactly at the umbilic points of M, the Willmore functional is

$$\begin{aligned} W(x)=\int \limits _M\rho ^ndv=\int \limits _M(S-nH^2)^\frac{n}{2}dv. \end{aligned}$$

(1.2)

It was shown in [4] that the Willmore functional is an invariant under the Möbius transformation of $S^{n+p}$. The Willmore submanifold was defined by [10].

Definition 1.1

$x:M\rightarrow S^{n+p}$ is called a Willmore submanifold if it is a critical point of the Willmore functional W(x).

In particular, when $n=2$, the functional essentially coincides with the well-known Willmore functional W(x) and its critical points are the Willmore surfaces. The Euler–Lagrange equation (i.e., Willmore equation) can be found in [10, (1.2)].

Li [10] also proved the following pointwise pinching theorem for compact Willmore submanifolds.

Theorem A

([10]). Let M be an n-dimensional $(n\ge 2)$ compact Willmore submanifold in $S^{n+p}$. If $\rho ^2\le \frac{n}{2-1/p},$ then either $\rho ^2\equiv 0$ and M is totally umbilical, or $\rho ^2\equiv \frac{n}{2-1/p}$. In the latter case, either $p=1$ and M is a Willmore torus $W_{m,n-m}=S^m(\sqrt{\frac{n-m}{n}})\times S^{n-m}(\sqrt{\frac{m}{n}})$; or $n=2$, $p=2$, and M is the Veronese surface.

Define

$$\begin{aligned} F(x)=\int \limits _M\rho ^2dv=\int \limits _M(S-nH^2)dv, \end{aligned}$$

(1.3)

which vanishes if and only if M is a totally umbilical submanifold. So the function F(x) measures the extent to which x(M) is a totally umbilical submanifold. Obviously, when $n=2$, F(x) reduces to the well-known Willmore functional W(x).

Definition 1.2

$x:M\rightarrow S^{n+p}$ is called an extremal submanifold if it is a critical point of the functional F(x).

Guo and Li [7] calculated the Euler–Lagrange equation of F(x) and proved the following rigidity theorem.

Theorem B

([7]). Let M be an n-dimensional $(n\ge 2)$ compact extremal submanifold in $S^{n+p}$. If $\rho ^2\le \frac{n}{2-1/p},$ then either $\rho ^2\equiv 0$ and M is totally umbilical, or $\rho ^2\equiv \frac{n}{2-1/p}$. In the latter case, either $p=1$, $n=2m$, and M is a Clifford torus $C_{m,m}=S^m(\sqrt{\frac{1}{2}})\times S^m(\sqrt{\frac{1}{2}})$; or $n=2$, $p=2$, and M is the Veronese surface.

Both of the above results are pointwise pinching theorems. It seems to be interesting to study the L$^q$-pinching condition. In [18] and [19], the authors obtain the following global pinching theorems for compact Willmore submanifolds and extremal submanifolds in a sphere.

Theorem C

([18]). Let M be a compact Willmore surface in the unit sphere $S^{2+p}$. There exists an explicit positive constant $C=\frac{(\sqrt{2}-1)\sqrt{\pi }}{12\sqrt{3}}B$ such that if $(\int _M\rho ^4)^{\frac{1}{2}}< C,$ then $\rho ^2=0$ and M is a totally umbilical sphere, where B is a constant.

Theorem D

([19]). Let M be an n-dimensional $(n\ge 3)$ compact extremal submanifold in $S^{n+p}$. There exists an explicit positive constant $A_n$ depending only on n such that if $(\int _M\rho ^n)^{\frac{2}{n}}<A_n,$ then M is a totally umbilical submanifold.

For each $x\in M$, let $R_m(x)$ be the smallest eigenvalue of the Ricci tensor at x, and ${\text {Ric}}_-^\lambda (x)=\max \{0,(n-1)\lambda -R_m(x)\}$ for $\lambda \in \textbf{R}$. Define

$$\begin{aligned} \Vert {\text {Ric}}_-^\lambda \Vert _q=\left( \,\int \limits _M({\text {Ric}}_-^\lambda )^q\right) ^{\frac{1}{q}}. \end{aligned}$$

It is obvious that $\Vert {\text {Ric}}_-^\lambda \Vert _q=0$ if and only if ${\text {Ric}}\ge (n-1)\lambda$.

Chen and Wei proved the following rigidity theorem.

Theorem E

([5]). Let M be an n-dimensional $(n\ge 4)$ closed submanifold in a space form $M^{n+p}_c$ with parallel mean curvature. Denote by H the norm of the parallel mean curvature of M. Assume $c+H^2>0$. Given $\lambda$ satisfying $(n-2)(c+H^2)<(n-1)\lambda \le (n-1)(c+H^2)$, if $\Vert {\text {Ric}}_-^\lambda \Vert _{n/2}<\epsilon (n,c,\lambda ,H)$, then M is a totally umbilical sphere $S^n(\frac{1}{\sqrt{c+H^2}})$. Here

$$\begin{aligned} \epsilon (n,c,\lambda ,H)=\frac{P_n}{1+\frac{c_++H^2}{(n-1)\lambda -(n-2)(c+H^2)}P_n}\frac{1}{C^2(n)}, \end{aligned}$$

where $P_n=\frac{(n+2)(n-2)^2}{4n^2(n-1)^2}$.

In [14], Shu studied the rigidity of Willmore submanifolds in terms of Ricci curvatures and obtained the following theorem.

Theorem F

([14]). Let M be an n-dimensional ($n\ge 5$) compact Willmore submanifold in the unit sphere $S^{n+p}$. If the Ricci curvature ${\text {Ric}}$, H, and $\rho$ of M satisfy

$$\begin{aligned} {\text {Ric}}\ge (n-2)+(n-2)H\rho +(n-1)H^2, \end{aligned}$$

then either M is totally umbilic, or M is a Willmore torus $W_{m,m}=S^m(\sqrt{\frac{1}{2}})\times S^m(\sqrt{\frac{1}{2}})$.

When $n=2$, all minimal surfaces are Willmore surfaces (see [10, (1.3)]). But there are many compact non-minimal Willmore surface (see [1, 2, 6, 11, 13]). When $n\ge 3$, minimal submanifolds are not Willmore submanifolds in general, for example, Clifford minimal tori $S^{m}(\sqrt{\frac{m}{n}})\times S^{n-m}(\sqrt{\frac{n-m}{n}})$ are not Willmore submanifolds when $n\ne 2m$. In [8], the authors proved that all n-dimensional minimal Einstein submanifolds in a sphere are Willmore submanifolds. Motivated by everything above, we shall prove the following global pinching theorems for compact Willmore and extremal submanifolds in $S^{n+p}$.

The PhD thesis [20] of the first author studied the rigidity of extremal submanifolds in terms of the Ricci curvature.

Theorem G

([20]). Let M be an n-dimensional ($n\ge 4$) compact extremal submanifold in the unit sphere $S^{n+p}$. If the Ricci curvature of M, H, and $\rho$ satisfy

$$\begin{aligned} {\text {Ric}}\ge (n-2)+(n-2)H\rho +(n-1)H^2, \end{aligned}$$

then M is either totally umbilic, a Clifford torus $S^m(\sqrt{\frac{1}{2}})\times S^m(\sqrt{\frac{1}{2}})$ in $S^{n+1}$, or $CP^2(\frac{3}{4})$ in $S^7$. Here $CP^2(\frac{3}{4})$ denotes the 2-dimensional complex projective space minimally immersed in $S^7$ with constant holomorphic sectional curvature.

Now we extend parts of Theorems F and G from a pointwise Ricci curvature lower bound to an integral Ricci curvature lower bound. Our main result in this paper is the following:

Theorem 1.3

Let M be an n-dimensional $(n\ge 4)$ Willmore submanifold in the unit sphere $S^{n+p}$. Given $\lambda$ satisfying

$$\begin{aligned} (n-2)+(n-2)H\rho +(n-1)H^2<(n-1)\lambda \le (n-1)(1+H^2_0)+(n-2)H_0\rho _0, \end{aligned}$$

(1.4)

where $H_0=\max _{x\in M}H$ and $\rho _0=\max _{x\in M}\rho$, if

$$\begin{aligned} \Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}<A(n,\lambda ,H,\rho ), \end{aligned}$$

then M is a totally umbilical sphere. Here

$$\begin{aligned} A(\lambda ,n,H,\rho )=\frac{1}{\frac{n^3(n-1)}{(n-2)^2}+\frac{(1+H^2_0)}{(n-1)(\lambda -H^2_0)-(n-2)(1+H_0\rho _0)}}\frac{1}{C^2(n)}. \end{aligned}$$

Corollary 1.4

Let M be a 4-dimensional compact Willmore submanifold in the unit sphere $S^{4+p}$. If

$$\begin{aligned} {\text {Ric}} > 2+2H\rho +3H^2, \end{aligned}$$

then M is totally umbilic.

Remark 1.5

It is easy to see that $\Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}=0$ if and only if ${\text {Ric}}\ge (n-1)\lambda$. From (1.4), we know this means ${\text {Ric}}>(n-2)+(n-2)H\rho +(n-1)H^2$. When $n\ge 5$, this generalizes Theorem F in the sense of strict inequality. Theorem 1.3 extends Theorem F to $n=4$.

Theorem 1.6

Let M be an n-dimensional $(n\ge 4)$ extremal submanifold in the unit sphere $S^{n+p}$. If

$$\begin{aligned} \Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}<B(\lambda ,n,H,\rho ), \end{aligned}$$

then M is a totally umbilical sphere, where

$$\begin{aligned} B(\lambda ,n,H,\rho )=\frac{1}{\frac{4n(n-1)^2}{(n-2)^2}+\frac{1}{(n-1)(\lambda -H^2_0)-(n-2)(1+H_0\rho _0)}}\frac{1}{C^2(n)}. \end{aligned}$$

Remark 1.7

It is easy to see that this generalizes Theorem G in the sense of strict inequality.

2 Preliminaries

In this section, we review some fundamental formulas for submanifolds. Let M be an n-dimensional compact submanifold in $S^{n+p}$. Thus the Gauss equations are as follows

$$\begin{aligned} R_{ijkl}=(\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk})+\sum _{\alpha }(h^{\alpha }_{ik}h^{\alpha }_{jl}-h^{\alpha }_{il}h^{\alpha }_{jk}). \end{aligned}$$

(2.1)

Thus we have

$$\begin{aligned} R_{ij}=(n-1)\delta _{ij}+n\sum _{\alpha }H^{\alpha }h^{\alpha }_{ij}-\sum _{\alpha ,k}h^{\alpha }_{ik}h^{\alpha }_{kj},\end{aligned}$$

(2.2)

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

(2.3)

The Codazzi and Ricci equations are given by

$$\begin{aligned} h^\alpha _{ijk}= & {} h^\alpha _{ikj}, \end{aligned}$$

(2.4)

$$\begin{aligned} R_{\alpha \beta ij}= & {} \sum _{k}h^\alpha _{ik}h^\beta _{kj}-\sum _{k}h^\beta _{ik}h^\alpha _{kj}. \end{aligned}$$

(2.5)

The Ricci identity shows that

$$\begin{aligned} h^\alpha _{ijkl}-h^\alpha _{ijlk}=\sum \limits _m h^\alpha _{mj}R_{mikl} +\sum \limits _m h^\alpha _{im}R_{mjkl}+\sum \limits _\beta h^\beta _{ij} R_{\beta \alpha kl}. \end{aligned}$$

(2.6)

Denote by $h_{ij}^\alpha$ the components of the second fundamental form of M. Define the following tensors:

$$\begin{aligned} \tilde{h}_{ij}^\alpha =h_{ij}^\alpha -H^\alpha \delta _{ij},~~~~~~\sigma _{\alpha \beta }=\sum _{i,j}h_{ij}^\alpha h_{ij}^\beta ,~~~~~~\tilde{\sigma }_{\alpha \beta }=\sum _{i,j}\tilde{h}_{ij}^\alpha \tilde{h}_{ij}^\beta . \end{aligned}$$

(2.7)

Then the $(p\times p)$-matrix $(\tilde{\sigma }_{\alpha \beta })$ is symmetric and can be assumed to be diagonalized for a suitable choice of $e_{n+1}, \ldots , e_{n+p}$. Set

$$\begin{aligned} \tilde{\sigma }_{\alpha \beta }=\tilde{\sigma }_\alpha \delta _{\alpha \beta }, \end{aligned}$$

(2.8)

we have by a direct calculation

$$\begin{aligned} \sum _k\tilde{h}^\alpha _{kk}= & {} 0,~~~~~\sigma _{\alpha \beta }=\tilde{\sigma }_{\alpha \beta }+nH^\alpha H^\beta ,~~~~~\rho ^2=\sum _\alpha \tilde{\sigma }_\alpha =S-nH^2, \end{aligned}$$

(2.9)

$$\begin{aligned} \sum _{i,j,k,\alpha }h^\beta _{kj}h^\alpha _{ij}h^\alpha _{ik}= & {} \sum _{i,j,k,\alpha }\tilde{h}^\beta _{kj}\tilde{h}^\alpha _{ij}\tilde{h}^\alpha _{ik}+2\sum _{i,j,\alpha }H^\alpha \tilde{h}^\alpha _{ij}\tilde{h}^\beta _{ij} +H^\beta \rho ^2+nH^2H^\beta , \end{aligned}$$

(2.10)

and

$$\begin{aligned} \sum _{\alpha ,\beta }\tilde{\sigma }_{\alpha \beta }^2=\sum _{\alpha }\tilde{\sigma }_{\alpha }^2 \le \left( \sum _{\alpha }\tilde{\sigma }_{\alpha }\right) ^2=\rho ^4. \end{aligned}$$

(2.11)

The above symbols and formulas are quoted from [10]. For the convenience of narration and the following proof, let us repeat it here.

From (2.2), we get

$$\begin{aligned} R_{ij}=(n-1)\delta _{ij}+(n-2)\sum _{\alpha }H^{\alpha }\tilde{h}^{\alpha }_{ij}+(n-1)H^2\delta _{ij}-\sum _{\alpha ,k}\tilde{h}^{\alpha }_{ik}\tilde{h}^{\alpha }_{kj}.\qquad \end{aligned}$$

(2.12)

Let $R_m$ be the smallest eigenvalue of the Ricci tensor. By using the Cauchy–Schwarz inequality $\sum _{\alpha }H^{\alpha }\tilde{h}^{\alpha }_{ij}\le H\rho$ and (2.12), we have

$$\begin{aligned} \frac{\rho ^2}{n}\le (n-1)(1+H^2)+(n-2)H\rho -R_m. \end{aligned}$$

(2.13)

Given $\lambda$ satisfying

$$\begin{aligned} (n-2)+(n-2)H\rho +(n-1)H^2<(n-1)\lambda \le (n-1)(1+H^2_0)+(n-2)H_0\rho _0, \end{aligned}$$

(2.14)

we can set

$$\begin{aligned} \Lambda :=(n-1)\lambda =(n-2)+(n-2)H_0\rho _0+(n-1)H^2_0+\delta \end{aligned}$$

(2.15)

for some $\delta >0$. Put ${\text {Ric}}_-^\lambda =\max \{0,(n-1)\lambda -R_m\}.$ By definition,

$$\begin{aligned} (n-2)+(n-2)H\rho +(n-1)H^2-R_m\le -\delta +\Lambda -R_m\le -\delta +{\text {Ric}}_-^\lambda . \end{aligned}$$

(2.16)

Lemma 1.8

([5, 17]). Let $M^n\, (n\ge 3)$ be a closed submanifold in $S^{n+p}$. Then for all $t>0$ and $f\in C^1(M)$, $f\ge 0$, we have

$$\begin{aligned} \int \limits _M|\nabla f|^2dv\ge k_1\left( \int \limits _{\,\,M} f^\frac{2n}{n-2}dv \right) ^{\frac{n-2}{n}}-k_2\int \limits _M(1+H^2)f^2dv, \end{aligned}$$

(2.17)

where

$$\begin{aligned} k_1=\frac{(n-2)^2}{4(n-1)^2(1+t)}\frac{1}{C^2(n)},~~k_2=\frac{(n-2)^2}{4(n-1)^2t},~~C(n)=2^n\frac{(n+1)^{1+1/n}}{(n-1)\omega _n^{1/n}}, \end{aligned}$$

and $\omega _n$ is the volume of the unit ball in $R^n$.

Lemma 1.9

([19]). Let M be an n-dimensional compact Riemannian submanifold in $S^{n+p}$. Then

$$\begin{aligned} |\nabla \tilde{h}|^2\ge |\nabla \rho _\varepsilon |^2, \end{aligned}$$

(2.18)

where $|\nabla \tilde{h}|^2=\sum _{\alpha ,i,j,k}(\tilde{h}^\alpha _{ijk})^2$, $\rho _\varepsilon =\sqrt{\sum _\alpha \sum _{i,j}(\tilde{h}_{ij}^\alpha +\varepsilon \delta _{ij})^2}>0$, $\varepsilon >0$.

From (2.17) and (2.18), we have

$$\begin{aligned} \int \limits _M|\nabla \tilde{h}|^2dv\ge \int \limits _M|\nabla \rho _\varepsilon |^2dv\ge k_1 \left( \int \limits _{\,\,M}\rho _\varepsilon ^\frac{2n}{n-2}dv\right) ^{\frac{n-2}{n}}-k_2\int \limits _M(1+H^2)\rho _\varepsilon ^2dv. \end{aligned}$$

(2.19)

Letting $\varepsilon \rightarrow 0$ in (2.19), we obtain

$$\begin{aligned} \int \limits _M|\nabla \tilde{h}|^2dv\ge k_1 \left( \int \limits _{\,\,M}\rho ^\frac{2n}{n- 2}dv\right) ^{\frac{n-2}{n}}-k_2\int \limits _M(1+H^2)\rho ^2dv. \end{aligned}$$

(2.20)

An argument similar to (2.20) shows that

$$\begin{aligned} \frac{n^2}{4}\int \limits _M\rho ^{n-2}|\nabla \tilde{h}|^2dv\ge k_1\left( \,\int \limits _M\rho ^\frac{n^2}{n-2}\right) ^\frac{n-2}{n}dv-k_2\int \limits _M(1+H^2)\rho ^ndv. \end{aligned}$$

(2.21)

3 Proof of Theorem 1.3

By use of (2.4), (2.6), and the definition of $\Delta$ and $\rho ^2$, we have

$$\begin{aligned} \frac{1}{2}\Delta \rho ^2= & {} \displaystyle \frac{1}{2}\Delta \left( \sum _{\alpha ,i,j} (h^\alpha _{ij})^2\right) -\frac{1}{2} \Delta (n H^2)\nonumber \\= & {} \displaystyle \sum _{\alpha ,i,j,k} (h^\alpha _{ijk})^2+\sum _{\alpha ,i,j,k} h^\alpha _{ij}h^\alpha _{kijk}- \frac{1}{2}\Delta (n H^2)\nonumber \\= & {} \displaystyle |\nabla h|^2-n^2|\nabla ^\perp \textbf{H}|^2+ \sum _{\alpha ,i,j,k} (h^\alpha _{ij}h^\alpha _{kki})_j +\sum _{\alpha ,i,j,k,m} h^\alpha _{ij} h^\alpha _{mk}R_{mijk}\nonumber \\{} & {} \displaystyle +\sum \limits _{\alpha ,i,j,m} h^\alpha _{ij}h^\alpha _{im}R_{mj} +\sum _{\alpha ,\beta ,i,j,k} h^\alpha _{ij} h^\beta _{ik} R_{\beta \alpha jk}-\frac{1}{2}\Delta (n H^2). \end{aligned}$$

(3.1)

From (2.1), (2.2), and (2.7), we have

$$\begin{aligned}{} & {} \displaystyle \sum _{\alpha ,i,j,k,m}h^\alpha _{ij} h^\alpha _{mk}R_{mijk}+\sum _{\alpha ,i,j,m} h^\alpha _{ij}h^\alpha _{im}R_{mj}\nonumber \\{} & {} \quad =\displaystyle \sum _{\alpha ,i,j,m}h^\alpha _{ij}\left( \sum _k h^\alpha _{mk}R_{mijk}+ h^\alpha _{im}R_{mj}\right) \nonumber \\{} & {} \quad =\displaystyle n\rho ^2-\sum _{\alpha ,\beta }\tilde{\sigma }^2_{\alpha \beta }+nH^2\rho ^2+ n\sum _{\alpha ,\beta ,i,j,k}H^\beta \tilde{h}^\beta _{kj}\tilde{h}^\alpha _{ij}\tilde{h}^\alpha _{ik}\nonumber \\{} & {} \qquad -\displaystyle \frac{1}{2}\sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }), \end{aligned}$$

(3.2)

where $N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })=\textrm{tr}[(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })^2],$ $\tilde{A}_\alpha =(\tilde{h}^\alpha _{ij})=(h^\alpha _{ij}-H^\alpha \delta _{ij}).$ By use of (2.5) and (2.7), we get

$$\begin{aligned} \sum _{\alpha ,i,j,k,}h^\alpha _{ij} h^\beta _{ki}R_{\beta \alpha jk} =-\frac{1}{2}\sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }). \end{aligned}$$

(3.3)

Putting (3.2) and (3.3) into (3.1), we obtain

$$\begin{aligned} \frac{1}{2}\Delta \rho ^2= & {} \displaystyle |\nabla h|^2-n^2|\nabla ^\perp \textbf{H}|^2+ \sum _{\alpha ,i,j,k} (h^\alpha _{ij}h^\alpha _{kki})_j\nonumber \\{} & {} \displaystyle +n\rho ^2-\sum _{\alpha ,\beta }\tilde{\sigma }^2_{\alpha \beta }+nH^2\rho ^2+ n\sum _{\alpha ,\beta ,i,j,k}H^\beta \tilde{h}^\beta _{kj}\tilde{h}^\alpha _{ij}\tilde{h}^\alpha _{ik}\nonumber \\{} & {} -\displaystyle \sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })-\frac{1}{2}\Delta (n H^2). \end{aligned}$$

(3.4)

Hence

$$\begin{aligned} \frac{1}{2}\rho ^{n-2}\Delta \rho ^2= & {} \displaystyle \rho ^{n-2}(|\nabla h|^2-n|\nabla ^\perp \textbf{H}|^2)+ \rho ^{n-2}\sum _{\alpha ,i,j,k} (h^\alpha _{ij}h^\alpha _{kki})_j-\frac{1}{2}\rho ^{n-2}\Delta (n H^2)\nonumber \\{} & {} \displaystyle -n(n-1)\rho ^{n-2}|\nabla ^\perp \textbf{H}|^2 +n\rho ^{n-2}\sum _{\alpha ,\beta ,i,j,k}H^\beta \tilde{h}^\beta _{kj}\tilde{h}^\alpha _{ij}\tilde{h}^\alpha _{ik}\nonumber \\{} & {} \displaystyle +\rho ^{n-2}\left[ n\rho ^2+nH^2\rho ^2 -\sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }- \tilde{A}_{\beta }\tilde{A}_{\alpha })-\sum _{\alpha ,\beta }\tilde{\sigma }^2_{\alpha \beta }\right] . \end{aligned}$$

(3.5)

On the other hand, Li [10] has given a characterization of Willmore submanifolds in the following Lemma 3.1.

Lemma 1.10

([10, Lemma 4.3]). Let M be an n-dimensional submanifold in the unit sphere $S^{n+p}$. Then M is a Willmore submanifold if and only if for $n+1\le \alpha \le n+p,$

$$\begin{aligned}{} & {} (n-1)\rho ^{n-2}\Delta ^\perp H^\alpha \nonumber \\{} & {} \quad = -2(n-1)\sum \limits _i (\rho ^{n-2})_i H^\alpha _{,i}-(n-1)H^\alpha \Delta (\rho ^{n-2})\nonumber \\{} & {} \qquad -\rho ^{n-2} \left( \sum \limits _\beta H^\beta {\tilde{\sigma }}_{\alpha \beta } +\sum \limits _{\beta ,i,j,k} {\tilde{h}}^\alpha _{ij}{\tilde{h}}^\beta _{ik}{\tilde{h}}^\beta _{kj}\right) +\sum \limits _{i,j}(\rho ^{n-2})_{i,j} (nH^\alpha \delta _{ij}-h^\alpha _{ij}), \end{aligned}$$

where $\Delta (\rho ^{n-2})=\sum _i (\rho ^{n-2})_{i,i}$, $\Delta ^\perp H^\alpha =\sum _i H^\alpha _{,ii}$, and $(\rho ^{n-2})_{i,j}$ is the Hessian of $\rho ^{n-2}$ with respect to the induced met ${\text {Ric}}$, $H^\alpha _{,i}$ and $H^\alpha _{,ij}$ are the components of the first and second covariant derivative of the mean curvature vector field $\textbf{H}$.

Using Stokes’ formula and Lemma 3.1, we have (see [10])

$$\begin{aligned}{} & {} \displaystyle -n(n-1)\int \limits _M\rho ^{n-2}|\nabla ^\perp \textbf{H}|^2dv +n\int \limits _M \rho ^{n-2} \left( \sum _{\alpha ,\beta ,i,j,m}H^\beta {\tilde{h}}^\beta _{mj}{\tilde{h}}^\alpha _{ji}{\tilde{h}}^\alpha _{im}\right) dv\nonumber \\{} & {} \quad =\displaystyle -\frac{1}{2}n(n+1)\int \limits _M\sum _i (\rho ^{n-2})_i(H^2)_idv -n\int \limits _M \rho ^{n-2}\sum _{\alpha ,\beta } H^\alpha H^\beta {\tilde{\sigma }}_{\alpha \beta }dv\nonumber \\{} & {} \qquad \displaystyle -n\int \limits _M \sum _{\alpha ,i,j} H^\alpha h^\alpha _{ij} (\rho ^{n-2})_{i,j}dv. \end{aligned}$$

(3.6)

By a direct computation, we get (see [10])

$$\begin{aligned}{} & {} \displaystyle \int \limits _M \rho ^{n-2}\sum \limits _{\alpha ,i,j,k}(h^\alpha _{ij}h^\alpha _{kki})_jdv\nonumber \\{} & {} \quad =\displaystyle n\int \limits _M \left( \sum \limits _{\alpha ,i,j} H^\alpha h^\alpha _{ij}(\rho ^{n-2})_{i,j}\right) dv +\frac{n^2}{2}\int \limits _M \sum \limits _i (\rho ^{n-2})_i(H^2)_i dv, \end{aligned}$$

(3.7)

and

$$\begin{aligned} -\frac{1}{2}\int \limits _M \rho ^{n-2}\Delta (nH^2)dv =\frac{n}{2}\int \limits _M\sum \limits _i (\rho ^{n-2})_i(H^2)_idv. \end{aligned}$$

(3.8)

By use of (3.6), (3.7), and (3.8), integrating (3.5) over M, we have

$$\begin{aligned} \displaystyle \frac{1}{2}\int \limits _M\rho ^{n-2}\Delta \rho ^2dv= & {} \displaystyle \int \limits _M\rho ^{n-2}(|\nabla h|^2-n|\nabla ^\perp \textbf{H}|^2)dv\nonumber \\{} & {} +\displaystyle n\int \limits _M\rho ^{n-2} \left( H^2\rho ^2-\sum _{\alpha ,\beta }H^{\alpha }H^{\beta }\tilde{\sigma }_{\alpha \beta }\right) dv+n\int \limits _M\rho ^ndv\nonumber \\{} & {} -\displaystyle \int \limits _M\rho ^{n-2}\sum _{\alpha ,\beta }(N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }) +\tilde{\sigma }_{\alpha \beta }^2)dv. \end{aligned}$$

(3.9)

We have by a direct calculation

$$\begin{aligned} H^2\rho ^2-\sum _{\alpha ,\beta }H^{\alpha }H^{\beta }\tilde{\sigma }_{\alpha \beta }\ge 0, \end{aligned}$$

(3.10)

$$\begin{aligned} |\nabla h|^2-n|\nabla ^\perp \mathbf{{H}}|^2=|\nabla \tilde{h}|^2, \end{aligned}$$

(3.11)

where $|\nabla ^\perp \mathbf{{H}}|^2=\sum \nolimits _{\alpha ,i}(H^\alpha _{,i})^2$, $|\nabla \tilde{h}|^2=\sum \nolimits _{\alpha ,i,j,k}(\tilde{h}^\alpha _{ijk})^2=\sum \nolimits _{\alpha ,i,j,k}(h^\alpha _{ijk}-H^\alpha _{,k}\delta _{ij})^2$. Thus

$$\begin{aligned} \displaystyle \frac{1}{2}\int \limits _M\rho ^{n-2}\Delta \rho ^2dv\ge & {} \displaystyle \int \rho ^{n-2}|\nabla \tilde{h}|^2dv\nonumber \\{} & {} + \displaystyle \int \limits _M\rho ^{n-2}\sum _{\alpha ,\beta }[n\rho ^2-N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })-\tilde{\sigma }_{\alpha \beta }^2]dv. \end{aligned}$$

(3.12)

It is easy to see that

$$\begin{aligned} \frac{1}{2}\int \limits _M\rho ^{n-2}\Delta \rho ^2dv=-\frac{4(n-2)}{n^2}\int \limits _M|\nabla \rho ^{\frac{n}{2}}|^2dv. \end{aligned}$$

(3.13)

Putting (3.13) into (3.12) yields

$$\begin{aligned} \displaystyle 0\ge & {} \displaystyle \frac{4(n-2)}{n^2}\int \limits _M|\nabla \rho ^{\frac{n}{2}}|^2dv+\int \rho ^{n-2}|\nabla \tilde{h}|^2dv\nonumber \\{} & {} + \displaystyle \int \limits _M\rho ^{n-2}\sum _{\alpha ,\beta }[n\rho ^2-N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })-\tilde{\sigma }_{\alpha \beta }^2]dv. \end{aligned}$$

(3.14)

Set $Q=n\rho ^2-\sum _{\alpha ,\beta }(N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }) +\tilde{\sigma }_{\alpha \beta }^2).$ Now we estimate the lower bound of Q based on $\rho ^2$ and ${\text {Ric}}_-^{\lambda }$. For a fixed $\alpha$, we choose $\{e_i\}$ such that $A_\alpha$ is diagonalized, $A_\alpha ={\text {diag}}\{\lambda ^\alpha _1,\ldots ,\lambda ^\alpha _n\}$, then (2.12) gives

$$\begin{aligned} \sum _j\sum _{\beta \ne \alpha }(\tilde{h}^\beta _{ij})^2=(n-1)(1+H^2)+(n-2)\sum _{\gamma }H^{\gamma }\tilde{h}^\gamma _{ii}-(\tilde{\lambda }^\alpha _i)^2-R_{ii},\qquad \end{aligned}$$

(3.15)

and

$$\begin{aligned}{} & {} \displaystyle \sum _{\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }) =\sum _{\beta \ne \alpha }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })\nonumber \\{} & {} \quad =\displaystyle \sum _{\beta \ne \alpha }\sum _{ij}(\tilde{h}^\beta _{ij})^2(\tilde{\lambda }^\alpha _i-\tilde{\lambda }^\alpha _j)^2 \le 4\sum _{\beta \ne \alpha }\sum _{ij}(\tilde{h}^\beta _{ij})^2(\tilde{\lambda }^\alpha _i)^2\nonumber \\{} & {} \quad \le \displaystyle 4\sum _i[(n-1)(1+H^2)+(n-2)\sum _{\gamma }H^{\gamma }\tilde{h}^\gamma _{ii}-(\tilde{\lambda }^\alpha _i)^2-R_{m}](\tilde{\lambda }^\alpha _i)^2\nonumber \\{} & {} \quad \le \displaystyle 4[(n-1)(1+H^2)+(n-2)H\rho -R_m]N(\tilde{A}_{\alpha })-4N(\tilde{A}^2_{\alpha }). \end{aligned}$$

(3.16)

Summing over $\alpha$, we get

$$\begin{aligned}{} & {} \displaystyle \sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })\nonumber \\{} & {} \quad \le \displaystyle 4[(n-1)(1+H^2)+(n-2)H\rho -R_m]\rho ^2-4\sum _{\alpha }N(\tilde{A}^2_{\alpha }). \end{aligned}$$

(3.17)

Obviously

$$\begin{aligned} \sum _{\alpha }N(\tilde{A}^2_{\alpha })\ge \frac{1}{n}\sum _{\alpha }(N(\tilde{A}_{\alpha }))^2=\frac{1}{n}\sum _{\alpha }\tilde{\sigma }_{\alpha }^2. \end{aligned}$$

(3.18)

Substituting (3.18) into (3.17), we have

$$\begin{aligned} \displaystyle \sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha }) \le \displaystyle 4[(n-1)(1+H^2)+(n-2)H\rho -R_m]\rho ^2-\frac{4}{n}\sum _{\alpha }\tilde{\sigma }_{\alpha }^2. \end{aligned}$$

(3.19)

Combining (2.11) with (3.19), we obtain

$$\begin{aligned} Q\ge & {} \displaystyle n\rho ^2-4[(n-1)(1+H^2)+(n-2)H\rho -R_m]\rho ^2+\frac{4-n}{n}\rho ^4\nonumber \\= & {} \displaystyle (4-n)(\frac{\rho ^2}{n}-1)\rho ^2-4[(n-2)+(n-1)H^2+(n-2)H\rho -R_m]\rho ^2.\nonumber \\ \end{aligned}$$

(3.20)

From (2.13), (2.16), and (3.20), by using $n\ge 4$, we get

$$\begin{aligned} Q\ge & {} \displaystyle -n[(n-2)+(n-1)H^2+(n-2)H\rho -R_m]\rho ^2\nonumber \\\ge & {} \displaystyle -n[-\delta +\Lambda -R_m]\rho ^2, \end{aligned}$$

(3.21)

i.e.,

$$\begin{aligned} Q\ge & {} \displaystyle -n(-\delta +{\text {Ric}}_-^{\lambda })\rho ^2. \end{aligned}$$

(3.22)

By Lemma 2.1, (2.21), (3.22), and (3.14), we get

$$\begin{aligned} 0\ge & {} \displaystyle \frac{4(n-1)}{n^2} \left[ k_1\left( \,\int \limits _M\rho ^\frac{n^2}{n-2}dv\right) ^\frac{n-2}{n}-k_2\int \limits _M(1+H^2)\rho ^ndv\right] \nonumber \\{} & {} +\displaystyle n\delta \int \limits _M\rho ^ndv-n\int \limits _M({\text {Ric}}_-^{\lambda })\rho ^ndv. \end{aligned}$$

(3.23)

Applying the Hölder inequality, we obtain

$$\begin{aligned} 0\ge & {} \displaystyle \frac{4(n-1)}{n^2}[k_1\Vert \rho ^n\Vert _\frac{n}{n-2}-k_2(1+H_0^2)\int \limits _M\rho ^ndv]\nonumber \\{} & {} +\displaystyle n\delta \int \limits _M\rho ^ndv-n\Vert {\text {Ric}}^{\lambda }_{-}\Vert _\frac{n}{2}\vert \vert \rho ^n\Vert _\frac{n}{n-2}, \end{aligned}$$

(3.24)

i.e.,

$$\begin{aligned} 0\ge & {} \displaystyle \frac{4(n-1)}{n^2}k_1\Vert \rho ^n\Vert _\frac{n}{n-2}-n\Vert {\text {Ric}}_-^{\lambda }\Vert _\frac{n}{2}\Vert \rho ^n\Vert _\frac{n}{n-2}\nonumber \\{} & {} +\displaystyle \left[ n\delta -\frac{4(n-1)}{n^2}k_2(1+H_0^2)\right] \int \limits _M\rho ^ndv. \end{aligned}$$

(3.25)

By taking

$$\begin{aligned} t=\frac{(n-2)^2(1+H_0^2)}{\delta n^3(n-1)} \end{aligned}$$

such that $[n\delta -\frac{4(n-1)}{n^2}k_2(1+H_0^2)]=0$, we have

$$\begin{aligned} 0\ge \left\{ \frac{4(n-1)}{n^2}k_1-n\Vert {\text {Ric}}_-^{\lambda }\Vert _\frac{n}{2}\right\} \Vert \rho ^n\Vert _\frac{n}{n-2}. \end{aligned}$$

Therefore, under the assumption $\Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}<\frac{4(n-1)}{n^3}k_1$, it is easy to see from the above that $\rho ^2=0$ and M is totally umbilical.

4 Proof of Theorem 1.6

First of all, Guo and Li calculated the Euler–Lagrange equation of F(x) given by (1.3).

Theorem 1.11

([7]). Let M be an n-dimensional submanifold in the unit sphere $S^{n+p}$. Then M is an extremal submanifold if and only if it satisfies for $n+1\le \alpha \le n+p$,

$$\begin{aligned} \sum _{\beta ,i,j,k}\tilde{h}^\alpha _{ij}\tilde{h}^\beta _{ik}\tilde{h}^\beta _{kj}=-(n-1)\Delta ^\perp H^\alpha -\sum _\beta H^\beta \tilde{\sigma }_{\alpha \beta } -H^\alpha \rho ^2+\frac{n}{2}H^\alpha \rho ^2. \end{aligned}$$

(4.1)

From (4.1), we have

$$\begin{aligned} n\sum _{\alpha ,\beta ,i,j,k}H^\beta \tilde{h}^\beta _{kj}\tilde{h}^\alpha _{ij}\tilde{h}^\alpha _{ik}= -n(n-1)\sum _{\beta }H^\beta \Delta ^\perp H^\beta -n\sum _{\alpha ,\beta } H^\beta H^\alpha \tilde{\sigma }_{\alpha \beta } +\frac{n(n-2)}{2}H^2\rho ^2.\nonumber \\ \end{aligned}$$

(4.2)

Putting (4.2) into (3.4) yields

$$\begin{aligned} \frac{1}{2}\Delta \rho ^2= & {} \displaystyle |\nabla h|^2-n^2|\nabla ^\perp \textbf{H}|^2+ \sum _{\alpha ,i,j,k} (h^\alpha _{ij}h^\alpha _{kki})_j+n\rho ^2-\sum _{\alpha ,\beta }\tilde{\sigma }^2_{\alpha \beta }+nH^2\rho ^2\nonumber \\{} & {} \displaystyle -n(n-1)\sum _{\beta }H^\beta \Delta ^\perp H^\beta -n\sum _{\alpha ,\beta } H^\beta H^\alpha \tilde{\sigma }_{\alpha \beta } +\frac{n(n-2)}{2}H^2\rho ^2\nonumber \\{} & {} -\displaystyle \sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })-\frac{1}{2}\Delta (n H^2). \end{aligned}$$

(4.3)

Integrating (4.3) over M and using Stokes’ formula, we have

$$\begin{aligned} \displaystyle \frac{1}{2}\int \limits _M\Delta \rho ^2dv= & {} \displaystyle \int \limits _M(|\nabla h|^2-n|\nabla ^\perp \textbf{H}|^2)dv+\int \limits _M\left( n\rho ^2-\sum _{\alpha ,\beta }\tilde{\sigma }^2_{\alpha \beta }\right) dv\nonumber \\{} & {} \displaystyle +\int \limits _Mn \left( H^2\rho ^2-\sum _{\alpha ,\beta } H^\beta H^\alpha \tilde{\sigma }_{\alpha \beta }\right) dv+\int \limits _M\frac{n(n-2)}{2}H^2\rho ^2dv\nonumber \\{} & {} -\displaystyle \int \limits _M\sum _{\alpha ,\beta }N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })dv. \end{aligned}$$

(4.4)

By (3.10) and (3.11),

$$\begin{aligned} 0\ge & {} \displaystyle \int \limits _M|\nabla \tilde{h}|^2dv+\frac{n(n-2)}{2}\int \limits _MH^2\rho ^2dv\nonumber \\{} & {} \displaystyle +n\int \limits _M\rho ^2dv-\int \limits _M\sum _{\alpha ,\beta }(N(\tilde{A}_{\alpha }\tilde{A}_{\beta }-\tilde{A}_{\beta }\tilde{A}_{\alpha })+\tilde{\sigma }_{\alpha \beta }^2)dv. \end{aligned}$$

(4.5)

Substituting (3.22) into (4.5), by the definition of Q, we have

$$\begin{aligned} 0\ge & {} \displaystyle \int \limits _M|\nabla \tilde{h}|^2dv+\frac{n(n-2)}{2}\int \limits _MH^2\rho ^2dv\nonumber \\{} & {} \displaystyle -\int \limits _M n(-\delta +{\text {Ric}}_-^{\lambda })\rho ^2dv. \end{aligned}$$

(4.6)

From (2.20), we get

$$\begin{aligned} 0\ge & {} \displaystyle k_1 \left( \,\int \limits _M\rho ^\frac{2n}{n-2}dv\right) ^\frac{n-2}{n}-k_2\int \limits _M(1+H^2)\rho ^2dv,\nonumber \\{} & {} +\displaystyle \frac{n(n-2)}{2}\int \limits _MH^2\rho ^2dv-n\int \limits _M (-\delta +{\text {Ric}}_-^{\lambda })\rho ^2dv, \end{aligned}$$

(4.7)

i.e.,

$$\begin{aligned} \begin{array}{lcl} 0&{}\ge &{}\displaystyle k_1\left( \,\int \limits _M\rho ^\frac{2n}{n-2}dv\right) ^\frac{n-2}{n}-n\int \limits _M({\text {Ric}}_-^\lambda )\rho ^2dv\\ &{}&{}-\displaystyle \int \limits _M(k_2-n\delta )\rho ^2dv+\int \limits _M\left[ \frac{n(n-2)}{2}-k_2\right] H^2\rho ^2dv. \end{array} \end{aligned}$$

(4.8)

We choose $t=\frac{(n-2)^2}{4\delta n(n-1)^2}$ such that $k_2=n\delta$. By using Hölder’s inequality, from the above, we obtain

$$\begin{aligned} 0\ge & {} \displaystyle k_1\Vert \rho ^2\Vert _{\frac{n}{n-2}}-n\Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}\Vert \rho ^2\Vert _{\frac{n}{n-2}}. \end{aligned}$$

(4.9)

Therefore, under the assumption

$$\begin{aligned} \Vert {\text {Ric}}_-^\lambda \Vert _\frac{n}{2}<\frac{k_1}{n}=\frac{(n-2)^2\delta }{4n(n-1)^2\delta +(n-2)^2}\frac{1}{C^2(n)}, \end{aligned}$$

it is easy to see from (4.9) that $\rho ^2=0$ and M is totally umbilical.