An interesting problem in submanifold geometry is to study the relationship between the geometric structure and topological properties of submanifolds in various ambient space. For example, Cao–Shen–Zhu [3] proved a complete non-compact oriented stable minimal hypersurface $ M^n(n\geq 3) $ in $ \mathbb{R}^{n+1} $ must have only one end. Their proof mainly used a Liouville theorem for harmonic maps due to Schoen–Yau [4]. Due to this connection with harmonic functions, Li–Wang [5] showed that a complete minimal hypersurface $ M^n $ in $ \mathbb{R}^{n+1} $ with finite index must have finite first $ L^2 $-Betti number, and $ M^n $ must have finitely many ends. So it is an interesting problem in geometry and topology to study vanishing theorems of harmonic forms on submanifolds in various ambient space.

Let $ (M, g) $ be a complete Riemannian manifold of dimension $ n $, $ d $ be the exterior differential operator on $ M $. Then the formal dual operator of $ d $ is defined by

$ \begin{equation*} \delta=(-1)^{n(l+1)+1}*d*, \end{equation*} $

where $ * $ is the Hodge star operator with respect to $ g $. Then the Hodge–Laplace–Beltrami operator $ \Delta $ acting on the space of smooth $ l $-form $ \Omega^l(M) $ is given by

$ \begin{equation*} \Delta=-(d\delta+\delta d). \end{equation*} $

A smooth $ l $-form $ \omega\in \Omega^l(M) $ is said to be harmonic if $ \Delta \omega=0 $. It is well-known that $ \omega $ is harmonic if and only if $ d\omega=0 $ and $ \delta \omega=0 $ when $ M $ is compact. Hence, for any $ p\geq 2 $, we say an $ l $-form $ \omega\in \Omega^l(M) $ is a $ p $-harmonic $ l $-form if it satisfies the following properties:

$ \begin{equation} \begin{cases} d\omega=0, \\ \delta(|\omega|^{p-2}\omega)=0. \end{cases} \end{equation} $

(1.1)

It is easy to see that when $ p=2 $ and $ M $ is compact, a $ p $-harmonic $ l $-form is exactly a harmonic $ l $-form. When $ l=0 $, it is a $ p $-harmonic function and the differential of a $ p $-harmonic function is a $ p $-harmonic $ 1 $-form.

There are various vanishing theorems for $ L^2 $ harmonic forms on complete submanifolds by assuming various geometric and analytic conditions. For example, Palmer [6] proved that there exist no nontrivial $ L^2 $ harmonic $ 1 $-forms on a complete stable minimal hypersurface in $ \mathbb{R}^{n+1} $. Cavalcante–Mirandola–Vitório [7] showed some finiteness and vanishing theorems for $ L^2 $ harmonic $ 1 $-forms on submanifolds in a nonpositive curved pinching manifold with some conditions about the first eigenvalues and the total curvature. Recently, by supposing that the submanifold is stable or has sufficiently small total curvature, Chao–Lv [8] established some vanishing theorems for $ L^2 $ harmonic $ 1 $-forms on complete submanifolds with weighted Poincaré inequality. Recall that we say that $ M $ satisfies a weighted Poincaré inequality with a nonnegative weight function $ \rho(x) $ if the inequality

$ \begin{equation} \int_{M}\rho(x)\varphi^{2}\mathrm{d}v\leq \int_{M}|\nabla\varphi|^{2}\mathrm{d}v \end{equation} $

(1.2)

holds true for all compactly supported smooth function $ \varphi\in C_0^{\infty}(M) $. We would like to point out that the weighted Poincaré inequalities have appeared in many important issues of analysis and mathematical physics. When the manifolds satisfy a weighted Poincaré inequality, many works have been conduced (see [9-13] and references therein).

Furthermore, when the ambient manifold is a Hadamard manifold, there are also many interesting results. Recall that a smooth manifold is called a Hadamard manifold if it is a complete, simply connected manifold with nonpositive sectional curvature. When $ M^{n} $ is a complete noncompact submanifold immersed in Hadamard manifold $ N^{n+m} $ with sectional curvature satisfying $ -k^{2}\leq K_{N}\leq 0 $, Cavalcante–Mirandola–Viótrio [7] obtained some vanishing theorems for $ L^{2} $ harmonic $ 1 $-forms if the traceless second fundamental form is small enough and $ \lambda_{1}(M)>\frac{(n-1)^{2}}{n}(k^{2}-\inf{|H|^{2}}) $. Subsequently, under the same assumptions except the lower bound of the first eigenvalue of Laplacian only depends on $ ||\phi||_{L^{n}}^2 $ instead of $ \inf_{M}|H|^2 $, Dung and Seo [2] obtained the similar vanishing theorems. On the other hand, assume that $ N^{n+m} $ has pure curvature tensor and the $ (l, n-l) $-curvature of $ N^{n+m} $ (see Definition 2.1) is not less than $ -k(k>0) $, Lin [14] proved a vanishing and finiteness theorem on $ M^n $ with flat normal bundle. Recently, under the same conditions except the $ (l, n-l) $-curvature of $ N^{n+m} $ to not less than $ -k\rho $ (where $ \rho $ is the weight function) and requiring $ M^{n} $ satisfies a weighted Poincaré inequality, Wang–Chao–Wu–Lv [1] obtained two vanishing theorems for harmonic $ l $-forms if the total curvature is small enough or $ M^{n} $ has at most Euclidean volume growth.

For general $ p $-harmonic forms, Zhang [15] proved that there is no nontrivial $ L^{q} (q>0) $ $ p $-harmonic $ 1 $-form on a complete manifold with nonnegative Ricci curvature. Inspired by Zhang's results, Chang–Guo–Sung [16] extended Zhang's results and obtained the compactness for any bounded set of $ p $-harmonic $ 1 $-forms. On complete noncompact submanifolds in a Hadamard manifold, if the first eigenvalue of the Laplacian has a suitable lower bound and the total curvature is sufficiently small, Han–Pan [17] obtained a vanishing theorem for $ L^{p} $ $ p $-harmonic $ 1 $-forms, which extends Cavalcante–Mirandola–Vitório's results. On complete noncompact submanifolds of $ N^{n+m} $ with pure curvature tensor, Dung–Tien [18] obtained some vanishing theorems for $ p $-harmonic $ l $-forms by assuming some appropriate conditions for the second fundamental form and the first eigenvalue of the Laplacian. Recently, Lin–Yang [19] proved some vanishing theorems of $ L^{q} $ $ p $-harmonic $ l $-forms on complete noncompact submanifolds in Hadamard manifold, which extends Lin's results in [14].

Inspired by these results, the main purpose of this article is to study $ L^{p} $ $ p $-harmonic forms on submanifolds in Hadamard manifolds. As usual, we define the space of the $ L^{p} $ $ p $-harmonic $ l $-forms on $ M $ by

$ \begin{equation*} H^{l, p}(L^{p}(M))=\left\{\omega\in\Omega^{l}(M)|d\omega=0, \delta(|\omega|^{p-2}\omega)=0, \int_{M}|\omega|^{p}\mathrm{d}v<\infty\right\}. \end{equation*} $

In this paper, we will prove the following theorems.

Theorem 1.1  Let $ M^{n} $ $ (n\geq 4) $ be an $ n $-dimensional complete submanifold immersed in a Hadamard manifold $ N^{n+m} $. Suppose that $ M $ satisfies the weighted Poincaré inequality and has flat normal bundle. Assume further that $ N $ has pure curvature tensor and the $ (l, n-l) $-curvature of $ N $ is not less than $ -k\rho (0\leq k\leq\frac{4}{p^{2}}) $ for $ 2\leq l\leq n-2 $. If the traceless second fundamental form $ \phi $ satisfies

$ \begin{equation} ||\phi||^{2}_{L^{n}(M)}<\frac{8[1+k_{p}(p-1)^{2}]-2kp^{2}}{np^{2}C(n)}, \end{equation} $

(1.3)

where $ C(n)>0 $ is the Sobolev constant depending only on the dimension $ n $. Then there is no nontrivial $ L^{p} $ $ p $-harmonic $ l $-form on $ M $.

We should remark that our Theorem 1.1 recovers Wang–Chao–Wu–Lv's results [1, Theorem 2] when $ p=2 $.

Theorem 1.2  Let $ M^{n} (n\geq 3) $ be an $ n $-dimensional complete noncompact submanifold in a complete simply connected Riemannian manifold $ N $ with sectional curvature $ -k^{2}\leq K_{N}\leq 0 $, where $ k $ is a constant. Assume that the traceless second fundamental from $ \phi $ satisfies

$ \begin{equation} ||\phi||_{L^{n}}<\frac{4}{p^{2}}\sqrt{\frac{1}{n(n-1)C(n)}}. \end{equation} $

(1.4)

In the case $ k\neq 0 $, assume further that the first eigenvalue of the Laplacian of $ M $ satisfies

$ \begin{equation} \lambda_{1}(M)>\frac{2n(n-1)p^{2}k^{2}}{8(p-1)n+8nk_{p}(p-1)^{2}-p^{2}(n-2)\sqrt{n(n-1)C(n)}||\phi||_{n}-2p^{2}(n-1)C(n)||\phi||^{2}_{n}}, \end{equation} $

(1.5)

where $ C(n) $ is the Sobolev constant. Then there is no nontrivial $ L^{p} $ $ p $-harmonic $ 1 $-form on $ M $.

We should remark that our Theorem 1.2 also generalizes Dung–Seo's results [2, Theorem 4.1] when $ p=2 $. And the upper bound of $ ||\phi||_{L^n(M)} $ and the lower bound of $ \lambda_1(M) $ depend only on the dimension of $ M $ and the curvature of the ambient space.

The rest of this article is arranged as follows. In Section 2, we recall some preliminary knowledge and lemmas. In Section 3, we will give a detailed proof of Theorem 1.1. In Section 4, we prove the Theorem 1.2.

In this section, we will recall some terminologies and notations about geometry of submanifolds and some useful lemmas which will be adopted in the proof of our theorems.

Let $ \iota: M\to N $ be an $ n $-dimensional submanifold isometrically immersed in an $ (n+m) $-dimensional Riemannian manifold $ (N, \bar{g}) $. Fix a point $ x \in M $ and a local orthonormal frame $ \{e_{1}, \cdots, e_{n+m}\} $ of $ N $ such that $ \{e_{1}, \cdots, e_{n}\} $ are tangent fields of $ M $ at $ x $ and $ \{e_{n+1}, \cdots, e_{n+m}\} $ is a local orthonormal frame of normal bundle $ NM $. In our paper, we also adopt the following ranges of indices: $ 1\leq i, j, \cdots\leq n $, $ n+1\leq \alpha, \beta, \cdots \leq n+m $.

For any $ X, Y\in \Gamma(TM) $, we have the following orthogonal decomposition

$ \begin{equation*} \bar{\nabla}_XY=\nabla_XY+A(X, Y), \end{equation*} $

where $ \bar{\nabla} $ is the Levi–Civita connection of the Riemannian manifold with respect to $ \bar{g} $ and $ A:\Gamma(TM)\times \Gamma(TM)\to \Gamma(NM) $ is the second fundamental form of the immersion. Then

$ \begin{equation*} A(X, Y)=\sum\limits_{\alpha}\langle\bar{\nabla}_{X}Y, e_{\alpha}\rangle e_{\alpha}. \end{equation*} $

We denote by $ h^{\alpha}_{ij}=\langle\bar{\nabla}_{e_{i}}e_{j}, e_{\alpha}\rangle $ the coefficients of the second fundamental form. Then the square norm $ |A|^2 $ of the second fundamental form and the mean curvature vector $ H $ are given by

$ \begin{equation*} |A|^2=\sum\limits_{\alpha}\sum\limits_{i, j}(h_{ij}^{\alpha})^2 \end{equation*} $

and

$ \begin{equation*} H=\frac{1}{n}H^{\alpha}e_{\alpha}=\frac{1}{n}\sum\limits_{\alpha}\sum\limits_{i}h_{ii}^{\alpha}e_{\alpha}. \end{equation*} $

A submanifold $ M $ is said to be minimal if $ H=0 $ identically. The traceless second fundamental form $ \phi $ is defined by

$ \begin{equation*} \phi(X, Y)=A(X, Y)-\langle X, Y\rangle H, \end{equation*} $

for any vector fields $ X, Y\in \Gamma(TM) $. It is easy to see check that

$ \begin{equation*} |\phi|^2=|A|^2-n|H|^2. \end{equation*} $

We say the immersion $ \iota: M^n\to N^{n+m} $ has finite total curvature if the $ L^n $-norm of the traceless second fundamental form is finite, i.e.,

$ \begin{equation*} ||\phi||_{L^n(M)}=\bigg(\int_{M}|\phi|^n \mathrm{d}v\bigg)^{\frac{1}{n}}<+\infty. \end{equation*} $

Let $ R $ and $ \bar{R} $ be the curvature tensors of $ M $ and $ N $, respectively. Recall that the definition of $ (l, n-l) $-curvature is given in [14], which appears naturally in the Weitzenbk formula.

Definition 2.1  ([14, Definition 1.1]) For any point $ x\in N^{n+m} $, choose an orthonormal frame $ \{e_i\}_{i=1}^{n+m} $ of the tangent space $ T_{x}N $ and set

$ \begin{equation*} \bar{R}^{(l, n-l)}([e_{i_1}, e_{i_2}, \cdots, e_{i_{n}}])=\sum\limits_{r=1}^l\sum\limits_{s=l+1}^n\bar{R}_{i_ri_si_ri_s} \end{equation*} $

for $ 1\leq l\leq n-1 $, where the indices $ 1\leq i_1, i_2, \cdots, i_n\leq n+m $ are distinct with each other. We call $ \bar{R}^{(l, n-l)}([e_{i_1}, e_{i_2}, \cdots, e_{i_n}]) $ the $ (l, n-l) $-curvature of $ N^{n+m} $.

We should remark that the $ (1, n-1) $-curvature is nothing but the $ (n-1) $-th Ricci curvature, which is a curvature condition between Ricci curvature and sectional curvature (see [20]).

In order to get a good estimation in Weitzenböck formula, we also need to assume that $ M $ has flat normal bundle and $ N $ has pure curvature tensor. Recall that the submanifold $ M $ is said to have flat normal bundle if the curvature of normal bundle $ NM $ is zero, i.e.,

$ \begin{equation*} h^{\alpha}_{ij}h^{\alpha}_{jk}-h^{\alpha}_{kj}h^{\alpha}_{ji}=0. \end{equation*} $

In this case, there exists an orthonormal frame diagonalizing $ h_{ij}^{\alpha} $ simultaneously. One can easily see that any hypersurface has flat normal bundle.

Definition 2.2  ([21, Definition 4.5]) A Riemannian manifold $ N^{n+m} $ is said to have pure curvature tensor if for every $ x\in N $, there is an orthonormal basis $ \{e_1, \cdots, e_{n+m}\} $ of the tangent space $ T_xN $ such that

$ \begin{equation*} \langle \bar{R}(e_i, e_j)e_{k}, e_l\rangle=0, \end{equation*} $

when the set $ \{i, j, k, l\} $ contains more than two elements.

It is easy to see that all the $ 3 $-manifolds and conformally flat manifolds have pure curvature tensor (see [21]).

In the rest of this Section, we shall give some useful lemmas which play an important role in the proof of our theorems.

Lemma 2.3   ([18, 22, 23]) For $ p\geq 2, l\geq 1 $, let $ \omega $ be a $ p $-harmonic $ l $-form on an $ n $-dimensional complete Riemannian manifold $ M $. Then we have

$ \begin{equation} |\nabla(|\omega|^{p-2}\omega)|^2\geq (1+k_p)|\nabla|\omega|^{p-1}|^2, \end{equation} $

(2.1)

where $ k_p=\begin{cases} \frac{1}{\max\{l, n-l\}}, &\text{if}\; p=2\\ \frac{1}{(p-1)^2}\min\{1, \frac{(p-1)^2}{n-1}\}, &\text{if}\; p>2\; \text{and}\; l=1\\ 0, &\text{if}\quad p>2\; \text{and} \; 1<l\leq n-1 \end{cases} $.

Following the similar arguments as in [14], we shall give a Weitzenböck formula for any $ l $-form under the assumption that $ M $ has flat normal bundle and $ N $ has pure curvature tensor.

Lemma 2.4  Let $ M^n (n\geq 3) $ be a complete submanifold immersed in $ N^{n+m} $. Assume that $ M $ has flat noral bundle and $ N $ has pure curvature tensor. Then for any $ l $-form $ \omega\in \Omega^l(M) $ with $ 2\leq l\leq n-2 $, we have

$ \begin{equation} \begin{split} \frac{1}{2}\Delta|\omega|^2\geq &|\nabla \omega|^2+\langle \Delta\omega, \omega\rangle+\left(\inf\limits_{i_1, \cdots, i_n}\sum\limits_{r=1}^l\sum\limits_{s=l+1}^n\bar{K}_{i_ri_s}\right)|\omega|^2\\ &+\left(-\frac{n}{2}|\phi|^2+\frac{1}{2}\min\{l, n-l\}|A|^2\right)|\omega|^2, \end{split} \end{equation} $

(2.2)

where $ \bar{K}_{ij}=\bar{R}_{ijij}=\langle\bar{R}(e_i, e_j)e_j, e_i\rangle $.

Proof  For any $ l $-form $ \omega\in \Omega^l(M) $, we have the Bochner formula [24]

$ \begin{equation} \begin{split} \frac{1}{2}\Delta |\omega|^2=&|\nabla\omega|^2+\langle \Delta \omega, \omega\rangle+\langle \sum\limits_{j, k=1}^n \theta^k\wedge \iota(e_j)R(e_j, e_k)\omega, \omega\rangle\\ =&|\nabla\omega|^2+\langle \Delta \omega, \omega\rangle+lF_l(\omega), \end{split} \end{equation} $

(2.3)

where

$ \begin{equation*} F_l(\omega)=R_{ij}\omega^{ii_2\cdots i_l}\omega^j_{i_2\cdots i_l}-\frac{l-1}{2}R_{ijkm}\omega^{iji_3\cdots i_l}\omega^{km}_{i_3\cdots i_l}. \end{equation*} $

From (2.8) and (2.9) in [14], we get

$ \begin{equation} F_l\geq \frac{1}{l}\left(\inf\limits_{i_1, \cdots, i_n}\sum\limits_{r=1}^l\sum\limits_{s=l+1}^n\bar{K}_{i_ri_s}\right)|\omega|^2+\frac{1}{l}\inf\limits_{i_1, \cdots, i_n}\{\sum\limits_{\alpha}(h_{i_1i_1}^{\alpha}+\cdots+h_{i_li_l}^{\alpha})(h_{i_{l+1}i_{l+1}}^{\alpha}+\cdots+h_{i_ni_n}^{\alpha})\}|\omega|^2. \end{equation} $

(2.4)

By direct calculation, we obtain

$ \begin{equation} \begin{split} &\sum\limits_{\alpha}(h_{i_1i_1}^{\alpha}+\cdots+h_{i_li_l}^{\alpha})(h_{i_{l+1}i_{l+1}}^{\alpha}+\cdots+h_{i_ni_n}^{\alpha})\\ =&\frac{1}{2}\sum\limits_{\alpha}[(h^{\alpha}_{i_{1}i_{1}}+...+h^{\alpha}_{i_{n}i_{n}})^{2}-(h^{\alpha}_{i_{1}i_{1}}+...+h^{\alpha}_{i_{l}i_{l}})^{2}-(h^{\alpha}_{i_{l+1}i_{l+1}}+...+h^{\alpha}_{i_{n}i_{n}})^{2}]\\ \geq &\frac{1}{2}\left[n^{2}|H|^{2}-l\sum\limits_{\alpha}\sum^{l}_{j=1}(h^{\alpha}_{i_{j}i_{j}})^{2}-(n-l)\sum\limits_{\alpha}\sum^{l}_{j=1}(h^{\alpha}_{i_{j}i_{j}})^{2}\right]\\ \geq &\frac{1}{2}(n^2|H|^2-\max\{l, n-l\}|A|^2)\\ =&-\frac{n}{2}(|A|^2-n|H|^2)+\frac{1}{2}(n-\max\{l, n-l\})|A|^2\\ =&-\frac{n}{2}|\phi|^2+\frac{1}{2}\min\{l, n-l\}|A|^2. \end{split} \end{equation} $

(2.5)

Combining (2.3), (2.4) and (2.5), we have (2.2).

As is well known, the Sobolev inequality holds on a complete submanifold in a complete simply connected manifold with nonpositive sectional curvature.

Lemma 2.5  ([1, 25]) Let $ M^n (n\geq 3) $ be an $ n $-dimensional complete noncompact submanifold in a complete sumply connected manifold with nonpositive sectional curvature. Then for any $ \varphi\in W_{0}^{1, 2}(M) $, we have

$ \begin{equation} \left(\int_{M}|\varphi|^{\frac{2n}{n-2}}\mathrm{d}v\right)^{\frac{n-2}{n}}\leq C(n)\int_{M}|\nabla \varphi|^2+|H|^2\varphi^2 \mathrm{d}v, \end{equation} $

(2.6)

where $ C(n) $ is the Sobolev constant which depends only on the dimension $ n $.

Besides, we also need the following estimate for the differential of forms.

Lemma 2.6  ([26, Lemma 2.5]) For any closed $ l $-form $ \omega\in \Omega^l(M) $ and $ f\in C^{\infty}(M) $, we have

$ \begin{equation} |d(f\omega)|=|df\wedge \omega|\leq |df|\cdot |\omega|. \end{equation} $

(2.7)

Finally, we also need the estimate of Ricci curvature for submanifolds, which will be used in the proof of Theorem 1.2.

Lemma 2.7  ([27, 28]) Let $ M^n $ be an $ n $-dimensional submanifold in a Riemannian manifold $ N $ with sectional curvature $ -k^2\leq K_N $ for some constant $ k $. Then the Ricci curvature $ Ric_{M} $ of $ M $ satisfies

$ \begin{equation*} Ric_{M}\geq (n-1)(|H|^2-k^2)-\frac{(n-2)\sqrt{n(n-1)}}{n}|H||\phi|-\frac{n-1}{n}|\phi|^2. \end{equation*} $

Proof  Let $ \omega $ be any $ p $-harmonic $ l $-form on $ M $ with $ 2\leq l\leq n-2 $, then we have

$ \begin{equation*} \begin{cases} d\omega=0, \\ \delta(|\omega|^{p-2}\omega)=0. \end{cases} \end{equation*} $

Applying Lemma 2.4 to the form $ |\omega|^{p-2}\omega $, we obtain

$ \begin{equation} \begin{split} \frac{1}{2}\Delta|\omega|^{2(p-1)}\geq &|\nabla (|\omega|^{p-2}\omega)|^2-\langle (d\delta+\delta d)(|\omega|^{p-2}\omega), |\omega|^{p-2}\omega\rangle\\ &+\left(\inf\limits_{i_1, \cdots, i_n}\sum\limits_{r=1}^l\sum\limits_{s=l+1}^n\bar{K}_{i_ri_s}\right)|\omega|^{2(p-1)}\\ &-\frac{n}{2}|\phi|^2|\omega|^{2(p-1)}+\frac{1}{2}\min\{l, n-l\}|A|^2|\omega|^{2(p-1)}. \end{split} \end{equation} $

(3.1)

By direct calculation, we have

$ \begin{equation} \frac{1}{2}\Delta|\omega|^{2(p-1)}=|\omega|^{p-1}\Delta|\omega|^{p-1}+|\nabla|\omega|^{p-1}|^{2}. \end{equation} $

(3.2)

Since the $ (l, n-l) $-curvature of $ N $ is not less than $ -k\rho $, we get

$ \begin{equation} \inf\limits_{i_1, \cdots, i_n}\sum\limits_{r=1}^l\sum\limits_{s=l+1}^n\bar{K}_{i_ri_s}\geq -k\rho. \end{equation} $

(3.3)

Combining (3.1)-(3.3) and $ \delta(|\omega|^{p-2}\omega) $, it follows that

$ \begin{equation*} \begin{split} |\omega|^{p-1}\Delta|\omega|^{p-1}\geq &(|\nabla(|\omega|^{p-2}\omega)|^{2}-|\nabla|\omega|^{p-1}|^{2})-\langle\delta d(|\omega|^{p-2}\omega), |\omega|^{p-2}\omega\rangle-k\rho|\omega|^{2(p-1)}\\ &-\frac{n}{2}|\phi|^{2}|\omega|^{2(p-1)}+\frac{1}{2}\min \{l, n-l\}|A|^{2}|\omega|^{2(p-1)}. \end{split} \end{equation*} $

By using the refined Kato inequality (2.1) and dividing both sides of the above inequality by $ |\omega|^{p-2} $, we obtain

$ \begin{equation} \begin{split} |\omega|\Delta|\omega|^{p-1}\geq &(p-1)^{2}k_{p}|\omega|^{p-2}|\nabla|\omega||^{2}-\langle\delta d(|\omega|^{p-2}\omega), \omega\rangle-k\rho|\omega|^{p}-\frac{n}{2}|\phi|^{2}|\omega|^{p}\\ &+\frac{1}{2}\min\{l, n-l\}|A|^{2}|\omega|^{p}. \end{split} \end{equation} $

(3.4)

Let $ r(x) $ be the geodesic distance on $ M $ from a fixed point $ x_0\in M $ to $ x $. Then we take a cutoff function $ \varphi\in C_0^{\infty}(M) $ satisfying

$ \begin{equation} \varphi(x)=\begin{cases} 1, & \text{on}\; B_r(x_0), \\ \in [0, 1]\;\text{and}\; |\nabla \varphi|\leq \frac{2}{r}, &\text{on}\;B_{2r}(x_0)\setminus B_{r}(x_0), \\ 0, &\text{on} \; M\setminus B_{2r}(x_0), \end{cases} \end{equation} $

(3.5)

where $ B_r(x_0) $ is the open geodesic ball of radius $ r $ and center at $ x_0 $ of $ M $.

Multiplying both sides of the above inequality (3.4) and integrating over $ M $, it holds

$ \begin{equation} \begin{split} &\int_{M}\varphi^2|\omega|\Delta|\omega|^{p-1}\mathrm{d}v\\ \geq &(p-1)^2k_p\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v-\int_{M}\langle d(|\omega|^{p-2}\omega), d(\varphi^2\omega)\rangle \mathrm{d}v\\ &-k\int_{M}\rho\varphi^2|\omega|^{p}\mathrm{d}v-\frac{n}{2}\int_{M}\varphi^2|\phi|^2|\omega|^{p}\mathrm{d}v+\frac{1}{2}\min\{l, n-l\}\int_{M}\varphi^2|A|^2|\omega|^p\mathrm{d}v. \end{split} \end{equation} $

(3.6)

For the term in the left hand side, using integration by parts yields

$ \begin{equation} \begin{split} &\int_{M}\varphi^2|\omega|\Delta |\omega|^{p-1}\mathrm{d}v\\ =&-\int_{M}\langle \nabla(\varphi^2|\omega|), \nabla|\omega|^{p-1}\rangle \mathrm{d}v\\ =&-2(p-1)\int_{M}\varphi|\omega|^{p-1}\langle\nabla\varphi, \nabla|\omega|\rangle \mathrm{d}v-(p-1)\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\\ \leq& 2(p-1)\int_{M}\varphi |\omega|^{p-1}|\nabla\varphi||\nabla|\omega||\mathrm{d}v-(p-1)\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v. \end{split} \end{equation} $

(3.7)

By using Lemma 2.6 and $ d\omega=0 $ for the second term in the right hand side, we get

$ \begin{equation} \begin{split} \int_{M}\langle d(|\omega|^{p-2}\omega), d(\varphi^2\omega)\rangle \mathrm{d}v \leq &\int_{M}\left|d(|\omega|^{p-2}\omega)\right|\cdot |d(\varphi^2\omega)|\mathrm{d}v\\ =&\int_{M}\left|d|\omega|^{p-2}\wedge \omega\right|\cdot | d\varphi^2 \wedge \omega|\mathrm{d}v\\ =&2(p-2)\int_{M}\varphi|\omega|^{p-1}\left|\nabla|\omega|\right|\cdot |\nabla \varphi|\mathrm{d}v. \end{split} \end{equation} $

(3.8)

Combining (3.6), (3.7) and (3.8) implies that

$ \begin{equation} \begin{split} 0 \geq & -2(2p-3)\int_{M}\varphi|\omega|^{p-1}|\nabla \varphi|\left|\nabla |\omega|\right|\mathrm{d}v+[(p-1)+(p-1)^2k_p]\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\\ &-k\int_{M}\rho\varphi^2|\omega|^{p}\mathrm{d}v-\frac{n}{2}\int_{M}\varphi^2|\phi|^2|\omega|^{p}\mathrm{d}v+\frac{1}{2}\min\{l, n-l\}\int_{M}\varphi^2|A|^2|\omega|^p\mathrm{d}v. \end{split} \end{equation} $

(3.9)

Next, we shall give some estimates of (3.9). Firstly, by Schwarz's inequality, we have

$ \begin{equation} \begin{split} &2\int_{M}\varphi|\omega|^{p-1}|\nabla \varphi|\left|\nabla |\omega|\right|\mathrm{d}v \\ \leq &\varepsilon_1\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+\frac{1}{\varepsilon_1}\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v, \end{split} \end{equation} $

(3.10)

where $ \varepsilon_1>0 $ is a constant. Secondly, by using the weighted Poincaré inequality and the Schwarz's inequality, it is easy to deduce that

$ \begin{equation} \begin{split} &\int_{M}\rho\varphi^2|\omega|^p \mathrm{d}v\\ =&\int_{M}\rho\left(\varphi|\omega|^{\frac{p}{2}}\right)^2\mathrm{d}v \leq \int_{M}|\nabla(\varphi|\omega|^{\frac{p}{2}})|^2\mathrm{d}v\\ \leq &(1+\varepsilon_2)\left(\frac{p}{2}\right)^2\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+\bigg(1+\frac{1}{\varepsilon_2}\bigg)\int_{M}|\nabla\varphi|^2|\omega|^{p}\mathrm{d}v, \end{split} \end{equation} $

(3.11)

where $ \varepsilon_2>0 $ is a constant. Finally, we shall give an estimate to the forth term in the right hand side. Denote by

$ \begin{equation*} E(\varphi)=C(n)\left(\int_{{\rm supp}\;\varphi }|\phi|^n\mathrm{d}v\right)^{\frac{2}{n}}, \end{equation*} $

where $ {{\rm supp}\; \varphi} $ is the compact support set of $ \varphi $. Then using the Hölder inequality, Sobolev inequality and the Schwarz's inequality, we obtain

$ \begin{equation} \begin{split} \int_{M}|\phi|^2\varphi^2|\omega|^p\mathrm{d}v\leq& \left(\int_{{\rm supp}\; \varphi}|\phi|^n \mathrm{d}v\right)^{\frac{2}{n}}\cdot \left(\int_{M}(|\varphi||\omega|^{\frac{p}{2}})^{\frac{2n}{n-2}}\mathrm{d}v\right)^{\frac{n-2}{n}}\\ \leq & E(\varphi)\left\{\int_{M}|\nabla(\varphi|\omega|^{\frac{p}{2}})|^2\mathrm{d}v+\int_{M}|H|^2\varphi^2|\omega|^{p}\mathrm{d}v\right\}\\ \leq &E(\varphi)\left\{(1+\varepsilon_3)\left(\frac{p}{2}\right)^2\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\right.\\ &+\left.\left(1+\frac{1}{\varepsilon_3}\right)\int_{M}|\nabla\varphi|^2|\omega|^p \mathrm{d}v+\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v\right\}, \end{split} \end{equation} $

(3.12)

where $ \varepsilon_3>0 $ is a constant. Therefore, combining (3.9)-(3.12), it follows that

$ \begin{equation} \begin{split} &\left\{(p-1)+(p-1)^2k_p-(2p-3)\varepsilon_1-\frac{p^2}{4}(1+\varepsilon_2)k\right.\\ &\left.-\frac{np^2}{8}(1+\varepsilon_3)E(\varphi)\right\}\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\\ &-\frac{n}{2}E(\varphi)\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v+\frac{1}{2}\min\{l, n-l\}\int_{M}\varphi^2|A|^2|\omega|^p\mathrm{d}v\\ \leq &\left[\frac{2p-3}{\varepsilon_1}+\bigg(1+\frac{1}{\varepsilon_2}\bigg)+\frac{n}{2}\left(1+\frac{1}{\varepsilon_3}\right)E(\varphi)\right]\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v. \end{split} \end{equation} $

(3.13)

Since $ ||\phi||_{L^n(M)}^2<\frac{8[p-1+(p-1)^2k_p]-2kp^2}{np^2C(n)} $, it implies that

$ \begin{equation*} p-1+(p-1)^2k_p-\frac{kp^2}{4}-\frac{np^2}{8}E(\varphi)>0. \end{equation*} $

Then we can choose sufficiently small $ \varepsilon_1 $, $ \varepsilon_2 $ and $ \varepsilon_3 $ such that

$ \begin{equation*} p-1+(p-1)^2k_p-(2p-3)\varepsilon_1-\frac{p^2}{4}(1+\varepsilon_2)k-\frac{np^2}{8}(1+\varepsilon_3)E(\varphi)>0. \end{equation*} $

Moreover, due to

$ \begin{equation*} E(\varphi)<\frac{8[p-1+(p-1)^2k_p]-2kp^2}{np^2}<\frac{8[p-1+(p-1)^2k_p]}{np^2}<\min\{l, n-l\}, \end{equation*} $

we have

$ \begin{equation*} \min\{l, n-l\}-E(\varphi)>0. \end{equation*} $

Note that $ |A|^2-n|H|^2=|\phi|^2\geq 0 $, then it follows from (3.13) that

$ \begin{equation} \begin{split} &B\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+\frac{1}{2}E(\varphi)\int_{M}(|A|^2-n|H|^2)\varphi^2|\omega|^{p}\mathrm{d}v+C\int_{M}\varphi^2|A|^2|\omega|^{p}\mathrm{d}v\\ \leq &D\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v, \end{split} \end{equation} $

(3.14)

where

$ \begin{equation*} \begin{split} &B=p-1+(p-1)^2k_p-(2p-3)\varepsilon_1-\frac{p^2}{4}(1+\varepsilon_2)k-\frac{np^2}{8}(1+\varepsilon_3)E(\varphi)>0, \\ &C=\frac{1}{2}\left(\min\{l, n-l\}-E(\varphi)\right)>0, \\ &D=\frac{2p-3}{\varepsilon_1}+\bigg(1+\frac{1}{\varepsilon_2}\bigg)+\frac{n}{2}\left(1+\frac{1}{\varepsilon_3}\right)E(\varphi). \end{split} \end{equation*} $

Since $ \varphi $ is a compactly supported nonnegative smooth function on $ M $ and $ |\nabla \varphi|^2\leq\frac{4}{r^2} $, we have

$ \begin{eqnarray} &&B\int_{B_{r}(x_0)}|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+\frac{1}{2}E(\varphi)\int_{B_r(x_0)}(|A|^2-n|H|^2)|\omega|^{p}\mathrm{d}v +C\int_{B_{r}(x_0)}|A|^2|\omega|^{p}\mathrm{d}v \\ &\leq &D\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v\leq \frac{4D}{r^2}\int_{B_{2r}(x_0)}|\omega|^p\mathrm{d}v. \end{eqnarray} $

(3.15)

Since $ \omega\in L^{p}(M) $, letting $ r\to +\infty $ in (3.15), it follows that

$ \begin{equation*} |\omega|^{p-2}|\nabla|\omega||=0, (|A|^2-n|H|^2)|\omega|^p=0, |A|^2|\omega|^p=0. \end{equation*} $

If $ |\omega| $ is not identically zero, then $ |\nabla|\omega||=0, |A|=0 $, and $ |H|=0 $. It implies that $ |\omega| $ is a constant and $ M $ is a minimal submanifold. Since $ |H|=0 $, the Sobolev inequality (2.6) becomes

$ \begin{equation*} \left(\int_{M}|\varphi|^{\frac{2n}{n-2}}\mathrm{d}v\right)^{\frac{n-2}{n}}\leq C(n)\int_{M}|\nabla \varphi|^2 \mathrm{d}v, \end{equation*} $

for any $ \varphi\in C_0^{\infty}(M) $. Therefore, $ M^n $ has infinite volume according to [29, Proposition 2.4]. This contradicts the fact that $ \int_{M}|\omega|^p\mathrm{d}v<\infty $. Therefore, $ \omega\equiv 0 $ on $ M^n $. This completes the proof.

Proof  Let $ \omega $ be a $ p $-harmonic $ 1 $-form on $ M $. Then the Bochner formula (2.3) becomes

$ \begin{equation} \frac{1}{2}\Delta |\omega|^2=|\nabla \omega|^2+\langle \Delta \omega, \omega\rangle+ Ric_{M}(\omega^{\sharp}, \omega^{\sharp}). \end{equation} $

(4.1)

This together with Lemma 2.7 gives

$ \begin{equation} \begin{split} \frac{1}{2}\Delta |\omega|^2\geq &|\nabla \omega|^2+\langle \Delta \omega, \omega\rangle\\ &+\left\{(n-1)(|H|^2-k^2)-\frac{(n-2)\sqrt{n(n-1)}}{n}|H||\phi|-\frac{n-1}{n}|\phi|^2\right\}|\omega|^2. \end{split} \end{equation} $

(4.2)

Applying the Bochner formula (4.2) to the form $ |\omega|^{p-2}\omega $, we obtain

$ \begin{equation} \begin{split} \frac{1}{2}\Delta|\omega|^{2(p-1)}&\geq|\nabla(|\omega|^{p-2}\omega)|^{2}-\langle(\delta d+d\delta)(|\omega|^{p-2}\omega), |\omega|^{p-2}\omega\rangle\\ &+\left\{(n-1)(|H|^2-k^2)-\frac{(n-2)\sqrt{n(n-1)}}{n}|H||\phi|-\frac{n-1}{n}|\phi|^2\right\}|\omega|^{2(p-1)}. \end{split} \end{equation} $

(4.3)

Combining (3.2) and $ \delta(|\omega|^{p-2}\omega)=0 $, it follows that

$ \begin{equation} \begin{split} |\omega|^{p-1}\Delta|\omega|^{p-1}&\geq (|\nabla(|\omega|^{p-2}\omega)|^{2}-|\nabla|\omega|^{p-1}|^{2})-\langle\delta d(|\omega|^{p-2}\omega), |\omega|^{p-2}\omega\rangle\\ &+\left\{(n-1)(|H|^2-k^2)-\frac{(n-2)\sqrt{n(n-1)}}{n}|H||\phi|-\frac{n-1}{n}|\phi|^2\right\}|\omega|^{2(p-1)}. \end{split} \end{equation} $

(4.4)

By using the refined Kato inequality (2.1) and dividing both sides of the above inequality by $ |\omega|^{p-2} $, we get

$ \begin{equation} \begin{split} |\omega|\Delta|\omega|^{p-1} &\geq k_{p}(p-1)^{2}|\omega|^{p-2}|\nabla|\omega||^{2}-\langle\delta d(|\omega|^{p-2}\omega), \omega\rangle\\ &+\left\{(n-1)(|H|^2-k^2)-\frac{(n-2)\sqrt{n(n-1)}}{n}|H||\phi|-\frac{n-1}{n}|\phi|^2\right\}|\omega|^{p}. \end{split} \end{equation} $

(4.5)

Similarly, multiplying both sides of the inequality (4.5) by $ \varphi^2 $ and integrating over $ M $ gives

$ \begin{equation} \begin{split} &\int_{M}\varphi^2|\omega|\Delta |\omega|^{p-1}\mathrm{d}v\\ \geq & (p-1)^2k_p\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v-\int_{M}\langle d(|\omega|^{p-2}\omega), d(\varphi^2\omega)\rangle \mathrm{d}v\\ &+(n-1)\int_{M}\varphi^2|H|^2|\omega|^{p}\mathrm{d}v-(n-1)k^2\int_{M}\varphi^2|\omega|^{p}\mathrm{d}v\\ &-\frac{(n-2)\sqrt{n(n-1)}}{n}\int_{M}\varphi^2|\phi||H||\omega|^p \mathrm{d}v-\frac{n-1}{n}\int_{M}\varphi^2|\phi|^2|\omega|^{p}\mathrm{d}v, \end{split} \end{equation} $

(4.6)

where $ \varphi $ is the cutoff function defined in (3.5). Applying the estimates (3.7) and (3.8), it follows that

$ \begin{equation} \begin{split} 0\leq& 2(2p-3)\int_{M}\varphi|\omega|^{p-1}|\nabla \varphi||\nabla |\omega||\mathrm{d}v-[p-1+(p-1)^2k_p]\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\\ &-(n-1)\int_{M}\varphi^2|H|^2|\omega|^{p}\mathrm{d}v+(n-1)k^2\int_{M}\varphi^2|\omega|^{p}\mathrm{d}v\\ &+\frac{(n-2)\sqrt{n(n-1)}}{n}\int_{M}\varphi^2|\phi||H||\omega|^p \mathrm{d}v+\frac{n-1}{n}\int_{M}\varphi^2|\phi|^2|\omega|^{p}\mathrm{d}v. \end{split} \end{equation} $

(4.7)

Then using the Young's inequality yields

$ \begin{equation} \begin{split} 2\int_{M}\varphi^2|\phi||H||\omega|^{p}\mathrm{d}v=&2\int_{M}\varphi |H||\omega|^{\frac{p}{2}}\cdot \varphi |\phi||\omega|^{\frac{p}{2}} \mathrm{d}v\\ \leq &\varepsilon_{4}\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v +\frac{1}{\varepsilon}_4\int_{M}\varphi^2|\phi|^2|\omega|^p \mathrm{d}v, \end{split} \end{equation} $

(4.8)

where $ \varepsilon_4>0 $ is a constant.

Combining the estimates (3.10), (4.7) and (4.8), we deduce that

$ \begin{equation} \begin{split} 0\leq &\left\{ (2p-3)\varepsilon_1-[p-1+(p-1)^2k_p]\right\}\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v\\ &+\frac{2p-3}{\varepsilon_1}\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v+(n-1)k^2\int_{M}\varphi^2|\omega|^p\mathrm{d}v\\ &+\left[\frac{n-1}{n}+\frac{(n-2)\sqrt{n(n-1))}}{2n\varepsilon_4}\right]\int_{M}\varphi^2|\phi|^2|\omega|^p\mathrm{d}v\\ &+\left[\frac{(n-2)\sqrt{n(n-1)}\varepsilon_4}{2n}-(n-1)\right]\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v. \end{split} \end{equation} $

(4.9)

When $ k\neq 0 $, we need to estimate the second term of the second line in (4.9). More precisely, by using the monotonicity of $ \lambda_1(B_r(x_0)) $, it holds that

$ \begin{equation} \lambda_1(M)\leq \lambda_1(B_r(x_0))\leq \frac{\int_{B_{r(x_0)}}|\nabla \varphi|^2\mathrm{d}v}{\int_{B_r(x_0)}\varphi^2\mathrm{d}v}, \end{equation} $

(4.10)

for any $ \varphi\in C_0^{\infty}(M) $. Then substituting $ \varphi|\omega|^{\frac{p}{2}} $ into the inequality (4.10) and using the Young's inequality again gives

$ \begin{equation} \begin{split} &\lambda_1(M)\int_{M}\varphi^2|\omega|^p\mathrm{d}v \leq \int_{M}|\nabla(\varphi|\omega|^{\frac{p}{2}})|^2\mathrm{d}v\\ \leq & \frac{p^2(1+\varepsilon_5)}{4}\int_{M}\varphi^2|\omega|^{p-2}|\nabla |\omega||^2\mathrm{d}v+\left(1+\frac{1}{\varepsilon_5}\right)\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v, \end{split} \end{equation} $

(4.11)

where $ \varepsilon_5>0 $ is a constant.

Finally, combining these inequalities (3.12), (4.9) and (4.11), we obtain that

$ \begin{equation} B\int_{M}\varphi^2|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+C\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v\leq D\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v, \end{equation} $

(4.12)

where $ B, C, D $ are constants given by

$ \begin{equation} \begin{split} B:=&[p-1+(p-1)^2k_p]-(2p-3)\varepsilon_1-\frac{(n-1)k^2p^2}{4\lambda_1(M)}(1+\varepsilon_5)\\ &-\left[\frac{n-1}{n}+\frac{(n-2)\sqrt{n(n-1)}}{2n\varepsilon_4}\right]E(\varphi)\cdot \frac{p^2(1+\varepsilon_3)}{4}, \\ C:=&(n-1)-\frac{(n-2)\sqrt{n(n-1)}}{2n}\varepsilon_4-\left[\frac{n-1}{n}+\frac{(n-2)\sqrt{n(n-1)}}{2n\varepsilon_4}\right]E(\varphi), \\ D:=&\frac{2p-3}{\varepsilon_1}+\frac{(n-1)k^2}{\lambda_1(M)}\left(1+\frac{1}{\varepsilon_5}\right)+\left[\frac{n-1}{n}+\frac{(n-2)\sqrt{n(n-1)}}{2n\varepsilon_4}\right]E(\varphi)\left(1+\frac{1}{\varepsilon_3}\right). \end{split} \end{equation} $

(4.13)

Since the traceless second fundamental form $ \phi $ satisfies $ ||\phi||_{L^n(M)}<\frac{4}{p^2}\sqrt{\frac{1}{n(n-1)C(n)}} $, we have

$ \begin{equation*} E(\varphi)<\frac{16}{n(n-1)p^4}\leq \frac{1}{n(n-1)}. \end{equation*} $

Then we can choose $ \varepsilon_4=\sqrt{E(\varphi)}<\frac{1}{\sqrt{n(n-1)}} $ such that

$ \begin{equation*} \begin{split} C=&(n-1)-\frac{(n-2)\sqrt{n(n-1)}}{2n}\cdot 2\sqrt{E(\varphi)}-\frac{n-1}{n}E(\varphi)\\ \geq & (n-1)-\frac{n-2}{n}-\frac{1}{n^2}=(n-1)\left(1-\frac{n-1}{n^2}\right)>0. \end{split} \end{equation*} $

By the assumption on the first eigenvalue $ \lambda_1(M) $ and $ ||\phi||_{L^n(M)} $, we can choose $ \varepsilon_1, \varepsilon_3, \varepsilon_5 $ small enough such that $ B>0 $. In addition, it is obvious that $ D>0 $.

On the other hand, since $ \varphi $ is a compactly supported nonnegative smooth function on $ M $ and $ |\nabla \varphi|^2\leq \frac{4}{r^2} $, by using (4.12), we obtain

$ \begin{equation} \begin{split} &B\int_{B_r(x_0)}|\omega|^{p-2}|\nabla|\omega||^2\mathrm{d}v+C\int_{B_{r}(x_0)}|H|^2|\omega|^p\mathrm{d}v\\ \leq &B\int_{M}\varphi^2|\omega|^p|\nabla|\omega||^2\mathrm{d}v+C\int_{M}\varphi^2|H|^2|\omega|^p\mathrm{d}v\\ \leq &D\int_{M}|\nabla \varphi|^2|\omega|^p\mathrm{d}v\\ \leq& \frac{4D}{r^2}\int_{B_{2r}(x_0)}|\omega|^p \mathrm{d}v. \end{split} \end{equation} $

(4.14)

Since $ \int_{M}|\omega|^p\mathrm{d}v<+\infty $, taking $ r\to +\infty $ in the inequality (4.14) implies that

$ \begin{equation*} |\omega|^{p-2}|\nabla|\omega||^2\equiv 0, |H|^2|\omega|^p\equiv 0. \end{equation*} $

If $ |\omega| $ is not identically zero, then $ |\nabla|\omega||=0 $ and $ |H|=0 $. Hence, $ |\omega| $ is a constant and $ M $ is a minimal submanifold. However, due to the infinite volume of complete minimal submanifold in a Riemannian manifold of nonpositive sectional curvature, we have $ \int_{M}|\omega|^p\mathrm{d}v=+\infty $. This contradicts the fact that $ \int_{M}|\omega|^p\mathrm{d}v<+\infty $. Therefore, $ \omega\equiv 0 $ on $ M $. The proof is completed.

The authors would like to thank Prof. Xi Zhang for his useful discussions and helpful comments. The research was supported by the National Key R and D Program of China 2020YFA0713100 and the Natural Science Foundation of Jiangsu Province (Grants No BK20230900). Both authors are partially supported by NSF in China No.12141104.