# On biharmonic submanifolds in non-positively curved manifolds

Yong Luo

###### Abstract

In the biharmonic submanifolds theory there is a generalized Chen’s conjecture
which states that biharmonic submanifolds in a Riemannian manifold with non-
positive sectional curvature must be minimal. This conjecture turned out false
by a counter example of Y. L. Ou and L. Tang in [12]. However it remains
interesting to find out sufficient conditions which guarantee this conjecture
to be true. In this note we prove that:

1\. Any complete biharmonic submanifold (resp. hypersurface) $(M,g)$ in a
Riemannian manifold $(N,h)$ with non-positive sectional curvature (resp. Ricci
curvature) which satisfies an integral condition: for some $p\in(0,+\infty)$,
$\int_{M}|\vec{H}|^{p}du_{g}<+\infty,$ where $\vec{H}$ is the mean curvature
vector field of $M\hookrightarrow N$, must be minimal. This generalizes the
recent results due to N. Nakauchi and H. Urakawa in [14] and [15].

2\. Any complete biharmonic submanifold (resp. hypersurface) in a Reimannian
manifold of at most polynomial volume growth whose sectional curvature (resp.
Ricci curvature) is non-positive must be minimal.

3\. Any complete biharmonic submanifold (resp. hypersurface) in a non-
positively curved manifold whose sectional curvature (resp. Ricci curvature)
is smaller that $-\epsilon$ for some $\epsilon>0$ which satisfies that
$\int_{B_{\rho}(x_{0})}|\vec{H}|^{p+2}d\mu_{g}(p\geq 0)$ is of at most
polynomial growth of $\rho$, must be minimal.

We also consider $\varepsilon$-superbiharmonic submanifolds defined recently
in [18] by G. Wheeler and prove similar results for
$\varepsilon$-superbiharmonic submanifolds, which generalize the result in
[18].

##  1 Introduction

Let $f:(M^{m},g)\to(\mathbb{R}^{m+t},h)$ be an isometric immersion from a
Riemannian manifold $M$ of dimension $m$ into an $m+t$-dimensional Euclidean
space, where $t\geq 1$ and $\vec{H}$ is the mean curvature vector field of $M$
in $\mathbb{R}^{m+t}$. $M$ is said to have harmonic mean curvature vector
field if (cf. [7])

$\displaystyle\Delta\vec{H}=0,$ (1.1)

where $\Delta$ is the rough Laplacian on $(M,g)$.

We see that a submanifold with harmonic mean curvature vector field is a
natural generalization of minimal submanifolds.

Another natural generalization of minimal submanifolds is as follows. The
intrinsic bi-energy of $f$ is defined by

$\displaystyle E_{2}(f)=\frac{1}{2}\int_{M}|\tau(f)|^{2}d\mu_{g},$ (1.2)

where $\tau(f)$ is the tension field of $f$ and $d\mu_{g}$ is the volume
element on $(M,g)$. The critical points of the functional $E_{2}$ satisfy the
following E-L equation ( see (2.7))

$\displaystyle-\Delta\vec{H}=\sum_{i=1}^{m}R^{N}(e_{i},\vec{H})e_{i},$ (1.3)

where $R^{N}$ is the Riemann curvature tensor of the ambient manifold $N$ and
$\\{e_{i},i=1,...,m\\}$ is a local orthonormal frame of $M$.

A submanifold satisfying equation (1.3) is called a biharmonic submanifold and
we see that submanifolds with harmonic mean curvature vector fields in
Euclidean spaces are biharmonic submanifolds. A natural question concerning
biharmonic submanifolds is under what conditions they are minimal
submanifolds. We have the following generalized Chen’s conjecture (cf. [4] [5]
[6] [7] [11] [12] and [17] etc.).
Generalized Chen’s Conjecture. Any biharmonic submanifold in a non-positively
curved manifold is minimal.

Though it turned out that this conjecture is false by a counter example of Y.
L. Ou and L. Tang (cf. [12]), it remains interesting to find out sufficient
conditions which guarantee biharmonic submanifolds to be minimal. Recently N.
Nakauchi and H. Urakawa proved:

######  Theorem 1.1 ([14] [15]).

Assume that $(M,g)$ is a complete Riemannian manifold of dimension $m$ and
$(N,h)$ a Riemannian manifold of dimension $m+t(t\geq 1)$ whose sectional
curvature (Ricci curvature if t=1) is non-positive. If $f:(M,g)\to(N,h)$ is a
biharmonic isometric immersion with mean curvature vector field satisfying

$\int_{M}|\vec{H}|^{2}dM<+\infty,$

then $f$ is minimal.

In this paper, we generalize this result to the following:

######  Theorem 1.2.

Assume that $(M,g)$ is a complete Riemannian manifold of dimension $m$ and
$(N,h)$ a Riemannian manifold of dimension $m+t(t\geq 1)$ whose sectional
curvature is non-positive. If $f:(M,g)\to(N,h)$ is a biharmonic isometric
immersion with mean curvature vector field satisfying

$\int_{M}|\vec{H}|^{p}dM<+\infty,$

for some $p\in(0,+\infty)$, then $f$ is minimal. Furthermore, if $M$ is a
hypersurface, then we only assume that $N$ has non-positive Ricci curvature.

######  Remark 1.3.

In a recent preprint [13], S. Maeta proved the above theorem for a complete
biharmonic submanifold $M\hookrightarrow N$ with conditions:
1\. $N$ has non-positive Ricci curvature, $M$ is a hypersurface and satisfies
$\int_{M}|\vec{H}|^{\alpha}dM<+\infty$ for $1+\varepsilon<a<+\infty$, where
$\varepsilon$ is a positive constant, or
2\. $N$ has non-positive sectional curvature and $M$ has general codimensions
satisfying $\int_{M}|\vec{H}|^{\alpha}dM<+\infty$ for
$1+\varepsilon<a<3-\varepsilon$, where $\varepsilon$ is a positive constant.

Clearly our assumption is weaker.

We also find another sufficient condition to guarantee the Generalized Chen’s
conjecture to be true, to state our theorem, we first give a definition.

######  Definition 1.4.

Let $(M,g)$ be a complete Riemannian manifold and $x_{0}\in M$. Then we say
$(M,g)$ is of at most polynomial volume growth, if there exists a nonnegative
integer $s$ such that

$\displaystyle Vol_{g}(B_{\rho}(x_{0}))\leq C\rho^{s},$ (1.4)

where $B_{\rho}(x_{0})$ is the geodesic ball centered at $x_{0}$ with radius
$\rho$ and $C$ is a positive constat independent of $\rho$.

######  Remark 1.5.

We see that for any $\rho$ there is a constant $C_{\rho}$ which depends on
$\rho$ such that $Vol_{g}(B_{\rho}(x_{0}))\leq C_{\rho}\rho^{m}$, where $m$ is
the dimension of $M$. Hence the volume growth depends on the ”blow up” rate of
the metric at infinity.

We have

######  Theorem 1.6.

Assume that $(M,g)$ is a complete $m$-dimensional Riemannian manifold of at
most polynomial volume growth and $(N,h)$ a Riemannian manifold of dimension
$m+t(t\geq 1)$ whose sectional curvature is non-positive. Let
$f:(M,g)\to(N,h)$ be an isometric immersion, then it is biharmonic if and only
if it is minimal. Furthermore, if $M$ is a hypersurface, we only assume that
$(N,h)$ has non-positive Ricci curvature.

About the volume growth of a complete Riemannian manifold, the most celebrated
result is the volume comparison theory due to Bishop and Gromov. By Bishop-
Gromov’s volume comparison theorem, we see that manifolds with non-negative
Ricci curvature have Euclidean volume growth (that is, $s=dimM$ in definition
1.4 ) and hence complete biharmonic submanifolds with nonnegative Ricci
curvature are minimal, by theorem 1.6. But the assumption of nonnegative Ricci
curvature seems too strong. For volume comparison results of complete
Riemannian manifold, we refer to a survey paper by G. Wei (see[19]) and
references therein.

We see all the above results give restrictions on the submanifolds. In the
following we weaken the assumption of integral of the mean curvature vector
field by giving a condition on the target manifolds which guarantees the
generalized Chen’s conjecture to be true. In the following we let
$B_{\rho}(x_{0})$ be a geodesic ball of $M$ centered at $x_{0}$ of radius
$\rho$.

######  Theorem 1.7.

Assume that $(M,g)$ is a complete biharmonic submanifold in a Riemannian
manifold $(N,h)$ whose sectional curvature is smaller that $-\epsilon$ for
some constant $\epsilon>0$ and
$\int_{B_{\rho}(x_{0})}|\vec{H}|^{p+2}d\mu_{g}(p\geq 0)$ is of at most
polynomial growth of $\rho$, then $M$ is a minimal submanifold. Furthermore,
if $M$ is a hypersurface, we only assume that the Ricci curvature of $(N,h)$
is smaller than $-\epsilon$.

We say a function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is of at most polynomial
growth, if $f(\rho)\leq C(1+\rho^{s})$ as $\rho\to\infty$, for some positive
constant $C$ independent of $\rho$ and $s$ a positive integer.

If we look carefully at Ou and Tang’s counter example (cf. [12]) for the
generalized Chen’s conjecture we can see that their example is complete and
the sectional curvature of the target manifold is negative and tends to zero
at infinity.

Question: Is any complete biharmonic submanifold in a Riemannian manifold
$(N,h)$ whose sectional curvature is smaller that $-\epsilon$ for some
constant $\epsilon>0$ minimal?

For other type sufficient conditions which guarantee the generalized Chen’s
conjecture to be true, we refer the readers to papers [1] [3] [16]. In [3],
Caddeo etc. proved that any biharmonic submanifold in hyperbolic 3-space
$\mathcal{H}^{3}(-1)$ is minimal, and any pseudo-umbilical biharmonic
submanifold $M^{m}\subset\mathcal{H}^{n}(-1)$ with $m\neq 4$ is minimal. It is
also shown in [1] that any biharmonic hypersurface of $\mathcal{H}^{n}(-1)$
with at most two distinct principal curvatures is minimal.

In a recent paper, G. Wheeler proposed a notion of
$\varepsilon$-superbiharmonic submanifolds which is a generalization of
submanifolds with harmonic mean curvature vector fields, as follows:

######  Definition 1.8 ([18]).

Let $M$ be a submanifold in $N$ with metric $\langle.,.\rangle$, then we call
$M$ an $\varepsilon$-superbiharmonic submanifold, if

$\langle\Delta\vec{H},\vec{H}\rangle\geq(\varepsilon-1)|\nabla\vec{H}|^{2},$

where $\varepsilon\in[0,1]$ is a constant.

######  Remark 1.9.

When $\varepsilon=0$, we see that an $\varepsilon$-superbiharmonic submanifold
satisfying $\Delta|\vec{H}|^{2}\geq 0$, i.e. the square length of the mean
curvature vector field of a 0-superbiharmonic submanifold is a superharmonic
function.

We have

######  Theorem 1.10 ([18]).

Let $N^{m+t}$ be a complete Riemannian manifold. Suppose $f:M^{m}\to N^{m+t}$
is a proper $\varepsilon$-superbiharmonic submanifold for $\varepsilon>0$.
Assume in addition that $f$ satisfies the curvature growth condition

$\displaystyle\lim_{\rho\to\infty}\frac{1}{\rho^{2}}\int_{f^{-1}(B_{\rho})}|\vec{H}|^{2}d\mu=0,$
(1.5)

where $B_{\rho}$ is a geodesic ball of $N$ with radius $\rho$. Then
$\vec{H}=0$ and $f$ is minimal.

We generalize this theorem to

######  Theorem 1.11.

Let $N^{m+t}$ be a complete Riemannian manifold. Suppose $f:M^{m}\to N^{m+t}$
is a proper $\varepsilon$-superbiharmonic submanifold for $\varepsilon>0$.
Assume in addition that $f$ satisfies the curvature growth condition

$\displaystyle\lim_{\rho\to\infty}\frac{1}{\rho^{2}}\int_{f^{-1}(B_{\rho})}|\vec{H}|^{2+a}d\mu=0,$
(1.6)

where $B_{\rho}$ is a geodesic ball of $N$ with radius $\rho$ and $a\geq 0$ is
a constant. Then $\vec{H}=0$ and $f$ is minimal.

######  Remark 1.12.

We say a map is proper, if the preimage of any compact subset of the ambient
manifold is compact. Note that properness implies completeness of the metric
of the domain manifold.

If we drop the assumption of properness of the immersion, we have

######  Theorem 1.13.

Suppose $f:M^{m}\to N^{m+t}$ is a complete $\varepsilon$-superbiharmonic
submanifold in $N^{m+t}$ for $\varepsilon>0$. Assume in addition that $f$
satisfies the condition

$\displaystyle\int_{M}|\vec{H}|^{2+a}d\mu<\infty,$ (1.7)

where $a\geq 0$ is a constant. Then $\vec{H}=0$ and $f$ is minimal.

Several words on the proof: The main argument in our proof is integral by
parts. But compared with some other results in this line, instead of using
integral by parts to the biharmonic submanifold equation, we integrate over
the tangential part of the biharmonic submanifolds equation, and when we
integrate by parts, we chose test functions more delicately and use the
Young’s inequality in a very subtle way.

Organization. In section 2 we give some preliminaries on submanifolds theory,
harmonic and biharmonic maps. Theorems 1.2, 1.6, 1.7, 1.11, 1.13 are proved in
section 3.

##  2 Preliminaries

###  2.1 Submanifolds

Assume that $f:M\to(N,h)$ is an immersion from a manifold of dimension $m$ to
a Riemannian manifold $(N,h)$ of dimension $m+t$. Then $M$ inherits a
Riemannian metric from $(N,h)$ by $g(X,Y):=h(df(X),df(Y))$ for any $X,Y\in TM$
and a volume form by $d\mu_{g}:=\sqrt{\det g}dx$. The second fundamental form
of $M\hookrightarrow N$, $B:TM\otimes TM\to NM$, is defined by

$B(X,Y):=D_{X}Y-\nabla_{X}Y,$

for any $X,Y\in TM$, where $D$ is the covariant derivative with respect to the
Levi-Civita connection on $N$, $\nabla$ is the Levi-Civita connection on $M$
with respect to the induced metric and $NM$ is the normal bundle of $M$. For
any normal vector field $\eta$ the Weingarten map associated with $\eta$,
$A_{\eta}:TM\to TM$ is defined by

$\displaystyle D_{X}\eta=-A_{\eta}X+\nabla^{\bot}_{X}\eta,$ (2.1)

where $\nabla^{\bot}$ is the normal connection and as is well known that $B$
and $A$ are related by

$\displaystyle\langle B(X,Y),\eta\rangle=\langle A_{\eta}X,Y\rangle.$ (2.2)

For any $x\in M$, let $\\{e_{1},e_{2},...,e_{m},e_{m+1},...,e_{m+t}\\}$ be a
local orthonormal basis of $N$ such that $\\{e_{1},...,e_{m}\\}$ is an
orthonormal basis of $T_{x}M$. Then $B$ is decomposed at $x$ as

$B(X,Y)=\sum_{\alpha=m+1}^{m+t}B_{\alpha}(X,Y)e_{\alpha}.$

The mean curvature vector field is defined as

$\displaystyle\vec{H}:=\frac{1}{m}\sum_{i=1}^{m}B(e_{i},e_{i})=\sum_{\alpha=m+1}^{m+t}H_{\alpha}e_{\alpha},$
(2.3)

where

$\displaystyle H_{\alpha}:=\frac{1}{m}\sum_{i=1}^{m}B_{\alpha}(e_{i},e_{i}).$
(2.4)

###  2.2 Harmonic and biharmonic maps

Let $f:(M,g)\to(N,h)$ be a map from a Riemannian manifold $(M,g)$ to a
Riemannian manifold $(N,h)$. The Dirichlet energy of $f$ is defined by

$E(f):=\frac{1}{2}\int_{M}|df|^{2}d\mu_{g}.$

The E-L equation of $E$ is

$\tau(f)=\rm tr\nabla df=0,$

where $\tau(f)$ is called the tension field of $f$. A map satisfying this E-L
equation is called a harmonic map.

To generalize the notion of harmonic maps, a natural way was proposed by J.
Eells and L. Lemaire ([9]) in 1983. They considered the bi-energy
$E_{2}(f):=\frac{1}{2}\int_{M}|\tau(f)|^{2}d\mu_{g}$. Critical points of the
bi-energy are called biharmonic maps and we see that all harmonic maps are
minimizers of the bi-energy functional $E_{2}$. In 1986, G. Y. Jiang ([10])
calculated the first and second variational formulas of the bi-energy
functional. The E-L equation of $E_{2}$ is

$\displaystyle\tau_{2}(f)=-\Delta\tau(f)-\sum_{i=1}^{m}R^{N}(\tau(f),df(e_{i}))df(e_{i})=0,$
(2.5)

where $\Delta$ is the rough Laplacian on $(M,g)$, $\\{e_{i},i=1,...,m\\}$ is a
local orthonormal frame on $M$ and $R^{N}$ is the Riemann curvature tensor of
$(N,h)$.

A map $f:(M,g)\to(N,h)$ satisfying equation (2.5) is called a biharmonic map.
Further if $f$ is an isometric immersion, it is called a biharmonic
submanifold.

Assume that $f$ is an isomeric immersion. we see $\\{df(e_{i})\\}$ is a local
orthonormal frame of $M$. In addition, for any $X,Y\in TM$,

$\displaystyle\nabla d(X,Y)=\nabla^{f}_{X}(df(Y))-df(\nabla_{X}Y),$

where $\nabla^{f}$ is the connection on the pullback bundle $f^{-1}TN$, whose
fibre at a point $x\in M$ is $T_{f(x)}N=TM\oplus NM$. Here $TM$ is the tangent
bundle and $NM$ is the normal bundle of $M$. Thus we see that
$\nabla^{f}_{X}(df(Y))-df(\nabla_{X}Y)=B(X,Y)$, and so

$\displaystyle\tau(f)=tr\nabla df=trB=m\vec{H}.$ (2.6)

Therefore a biharmonic submanifold satisfying the following equation:

$\displaystyle-\Delta\vec{H}-\sum_{i=1}^{m}R^{N}(e_{i},\vec{H})e_{i}=0,$ (2.7)

where $\Delta$ is the rough Laplacian on $(M,g)$, $\\{e_{i},i=1,...,m\\}$ is a
local orthonormal frame on $M$ and $R^{N}$ is the Riemann curvature tensor of
$(N,h)$.

By decomposing $-\Delta\vec{H}-\sum_{i=1}^{m}R^{N}(e_{i},\vec{H})e_{i}$ into
its tangential and normal parts, we see that a submanifold is biharmonic if
and only if it satisfies (cf. [2])

$\displaystyle\Delta^{\bot}\vec{H}-\sum_{i=1}^{m}B(A_{\vec{H}}e_{i},e_{i})+\sum_{i=1}^{m}(R^{N}(e_{i},\vec{H})e_{i})^{\bot}$
$\displaystyle=$ $\displaystyle 0,$ (2.8) $\displaystyle
m\nabla|\vec{H}|^{2}+4\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i}-\sum_{i=1}^{m}(R^{N}(e_{i},\vec{H})e_{i})^{T}$
$\displaystyle=$ $\displaystyle 0,$ (2.9)

where $(R^{N}(e_{i},\vec{H})e_{i})^{\bot},(R^{N}(e_{i},\vec{H})e_{i})^{T}$
denote the normal and tangential parts of $R^{N}(e_{i},\vec{H})e_{i}$
respectively. In particular, if $N$ is a space form of constant sectional
curvature $c$, then a submanifold is biharmonic if and only if (cf. [8, 17])

$\displaystyle\Delta^{\bot}\vec{H}-\sum_{i=1}^{m}B(A_{\vec{H}}e_{i},e_{i})+cm\vec{H}$
$\displaystyle=$ $\displaystyle 0,$ (2.10) $\displaystyle
m\nabla|\vec{H}|^{2}+4\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i}$
$\displaystyle=$ $\displaystyle 0.$ (2.11)

##  3 Proof of theorems

In this section we prove our theorems.

###  3.1 Proof of theorem 1.2.

###### Proof.

From equation $(\ref{equ4})$ we see that

$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\langle\vec{H},\Delta^{\bot}\vec{H}\rangle$
$\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
B(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle-2\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
A_{\vec{H}}e_{i},A_{\vec{H}}e_{i}\rangle-2\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle.$

Obviously if $N$ has non-positive sectional curvature, $-\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle\geq 0$. Furthermore, if $M$ is a
hypersurface, we see that $-\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle=-Ric^{N}(\vec{H},\vec{H})\geq 0.$
Therefore

$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
A_{\vec{H}}e_{i},A_{\vec{H}}e_{i}\rangle$ (3.1) $\displaystyle\geq$
$\displaystyle 2|\nabla^{\bot}\vec{H}|^{2}+2m|\vec{H}|^{4},$

where in the last inequality we used $\sum_{i=1}^{m}\langle
A_{\vec{H}}e_{i},A_{\vec{H}}e_{i}\rangle\geq m|\vec{H}|^{4}$, which can be
seem as follows: Let $x\in M$, when $\vec{H}(x)=0$, we are done. If
$\vec{H}(x)\neq 0$, set $e_{m+t}=\frac{\vec{H}}{|\vec{H}|}$, then
$\vec{H}(x)=H_{m+t}(x)e_{m+t}$ and $|\vec{H}|^{2}=H^{2}_{m+t}$. Hence we have
at $x$,

$\displaystyle\sum_{i=1}^{m}\langle A_{\vec{H}}e_{i},A_{\vec{H}}e_{i}\rangle$
$\displaystyle=$ $\displaystyle H^{2}_{m+t}\sum_{i=1}^{m}\langle
A_{e_{m+t}}e_{i},A_{e_{m+t}}e_{i}\rangle$ $\displaystyle=$ $\displaystyle
H^{2}_{m+t}|B_{m+t}|_{h}^{2}$ $\displaystyle\geq$ $\displaystyle mH^{4}_{m+t}$
$\displaystyle=$ $\displaystyle m|\vec{H}|^{4}.$

Let $\gamma:M\to\mathbb{R}^{+}$ be a cut off function such that

$\gamma=1\,\,\,\,on\,\,\,\,B_{\rho},\gamma=0\,\,\,\,on\,\,\,\,M\setminus
B_{2\rho},\,\,\,\,and\,\,\,\,|\nabla\gamma|\leq\frac{C}{\rho},$

for some constant $C$ independent of $\rho$. Here $B_{\rho}$ is a geodesic
ball of radius $\rho$ on $M$.

By integral by parts and (3.1) one gets

$\displaystyle-\int_{M}\nabla\gamma^{b}|\vec{H}|^{a}\nabla|\vec{H}|^{2}d\mu_{g}$
(3.2) $\displaystyle=$
$\displaystyle\int_{M}\gamma^{b}|\vec{H}|^{a}\Delta|\vec{H}|^{2}d\mu_{g}$
$\displaystyle\geq$ $\displaystyle
2\int_{M}\gamma^{b}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+2m\int_{M}\gamma^{b}|\vec{H}|^{a+4}d\mu_{g},$

where $a,b$ are positive constants to be determined later.

On the other hand one has

$\displaystyle-\int_{M}\nabla\gamma^{b}|\vec{H}|^{a}\nabla|\vec{H}|^{2}d\mu_{g}$
(3.3) $\displaystyle=$
$\displaystyle-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}-2a\int_{M}\gamma^{b}|\vec{H}|^{a-2}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}.$

From (3.2)-(3.3) we obtain,

$\displaystyle
2\int_{M}\gamma^{b}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+2m\int_{M}\gamma^{b}|\vec{H}|^{a+4}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}.$

Now let $b=a+4$ we get

$\displaystyle
2\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+2m\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}$
(3.4) $\displaystyle\leq$
$\displaystyle-2\int_{M}(a+4)\gamma^{a+3}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}.$ $\displaystyle\leq$
$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(a+4)^{2}\int_{M}\gamma^{a+2}|\vec{H}|^{a+2}|\nabla\gamma|^{2}d\mu_{g}$
$\displaystyle=$
$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(a+4)^{2}\int_{M}\gamma^{q}|\vec{H}|^{q}\gamma^{a+2-q}|\vec{H}|^{a+2-q}|\nabla\gamma|^{2}d\mu_{g},$

where $q$ is a constant belonging to $(0,a+2)$.

By Young’s inequality,

$\displaystyle(a+4)^{2}\int_{M}\gamma^{q}|\vec{H}|^{q}\gamma^{a+2-q}|\vec{H}|^{a+2-q}|\nabla\gamma|^{2}d\mu_{g}$
(3.5) $\displaystyle\leq$
$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}$ $\displaystyle+$
$\displaystyle
C(a,q)\int_{M}\gamma^{(a+2-q)\frac{a+4}{a+4-q}}|\vec{H}|^{(a+2-q)\frac{a+4}{a+4-q}}|\nabla\gamma|^{\frac{2(a+4)}{a+4-q}}d\mu_{g}$
$\displaystyle=$
$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}+C(a,q)\int_{M}\gamma^{a+4-\frac{2(a+4)}{a+4-q}}|\vec{H}|^{a+4-\frac{2(a+4)}{a+4-q}}|\nabla\gamma|^{\frac{2(a+4)}{a+4-q}}d\mu_{g}$

where $C(a,q)$ is a constant depending on $a,q$.

From (3.4)-(3.5) we obtain

$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}$
$\displaystyle\leq$ $\displaystyle
C(a,q)\frac{1}{\rho^{\frac{2(a+4)}{a+4-q}}}\int_{M}\gamma^{a+4-\frac{2(a+4)}{a+4-q}}|\vec{H}|^{a+4-\frac{2(a+4)}{a+4-q}}d\mu_{g}$
$\displaystyle\leq$ $\displaystyle
C(a,q)\frac{1}{\rho^{\frac{2(a+4)}{a+4-q}}}\int_{M}|\vec{H}|^{a+4-\frac{2(a+4)}{a+4-q}}d\mu_{g}$

Note that when $q$ varies from $0$ to $a+2$, $a+4-\frac{2(a+4)}{a+4-q}$ varies
from $a+2$ to $0$. Now by assumption we assume that for some constant
$p\in(0,+\infty)$

$\int_{M}|\vec{H}|^{p}d\mu_{g}<+\infty,$

then we can choose $q$ and $a$ such that $a+4-\frac{2(a+4)}{a+4-q}=p,$ and so

$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}$
$\displaystyle\leq$ $\displaystyle
C(a,q)\frac{1}{\rho^{\frac{2(a+4)}{a+4-q}}}\int_{M}|\vec{H}|^{p}d\mu_{g}.$

Let $\rho\to\infty$ we have

$\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}=0,$

which implies that $\vec{H}=0$.

This completes the proof of theorem 1.2. $\hfill\Box$

###  3.2 Proof of theorem 1.6.

The proof of theorem 1.6 is quite similar to the proof of theorem 1.2. But we
still give all the details.

###### Proof.

From the beginning of the proof of theorem 1.2, we see that if $N$ has non-
positive sectional curvature or $M$ is a hypersurface and $N$ has non-positive
Ricci curvature, then $-\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle\geq 0$.

From equation $(\ref{equ4})$ we see that

$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\langle\vec{H},\Delta^{\bot}\vec{H}\rangle$ (3.6)
$\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
B(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle-2\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle$ $\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2m|\vec{H}|^{4}.$

As before, let $\gamma:M\to\mathbb{R}^{+}$ be a cut off function such that

$\gamma=1\,\,\,\,on\,\,\,\,B_{\rho},\gamma=0\,\,\,\,on\,\,\,\,M\setminus
B_{2\rho},\,\,\,\,and\,\,\,\,|\nabla\gamma|\leq\frac{C}{\rho},$

for some constant $C$ independent of $\rho$. Here $B_{\rho}$ is a geodesic
ball of radius $\rho$ on $M$.

By integral by parts and (3.6) one gets

$\displaystyle-\int_{M}\nabla\gamma^{b}|\vec{H}|^{a}\nabla|\vec{H}|^{2}d\mu_{g}$
(3.7) $\displaystyle=$
$\displaystyle\int_{M}\gamma^{b}|\vec{H}|^{a}\Delta|\vec{H}|^{2}d\mu_{g}$
$\displaystyle\geq$ $\displaystyle
2\int_{M}\gamma^{b}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+2m\int_{M}\gamma^{b}|\vec{H}|^{a+4}d\mu_{g},$

where $a,b$ are positive constants to be determined later.

On the other hand one has

$\displaystyle-\int_{M}\nabla\gamma^{b}|\vec{H}|^{a}\nabla|\vec{H}|^{2}d\mu_{g}$
(3.8) $\displaystyle=$
$\displaystyle-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}-2a\int_{M}\gamma^{b}|\vec{H}|^{a-2}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}.$

From (3.7)-(3.8) we obtain,

$\displaystyle
2\int_{M}\gamma^{b}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+2m\int_{M}\gamma^{b}|\vec{H}|^{a+4}dM\leq-2\int_{M}b\gamma^{b-1}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}.$ (3.9)

Now let $\frac{a+4}{a+2}=\frac{b}{b-2}$, i.e. $b=a+4$. Using Young’s
inequality, we have

$\displaystyle-2\int_{M}(a+4)\gamma^{a+3}\nabla\gamma|\vec{H}|^{a}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle
d\mu_{g}$ (3.10) $\displaystyle\leq$
$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}+C(a)\int_{M}|\nabla\gamma|^{\frac{a+4}{2}}d\mu_{g},$

where $C(a)$ is a constant depending on $a$.

From (3.9)-(3.10) we obtain

$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}$
(3.11) $\displaystyle\leq$ $\displaystyle
C(a)\int_{M}|\nabla\gamma|^{\frac{a+4}{2}}d\mu_{g}$ $\displaystyle\leq$
$\displaystyle C(a)\frac{1}{\rho^{\frac{a+4}{2}}}Vol_{g}(B_{2\rho}).$

By assumption, there exists an integer $s\geq 0$ such that

$Vol_{g}(B_{2\rho})\leq C\rho^{s},$

where $C$ is a constant independent of $\rho$.

Therefore we have

$\displaystyle\int_{M}\gamma^{a+4}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}\gamma^{a+4}|\vec{H}|^{a+4}d\mu_{g}\leq
C(a)\rho^{s-\frac{a+4}{2}}.$

Chosing $a$ such that $a>\max\\{0,2s-4\\}$ and letting $\rho\to+\infty$ in the
above inequality, we get

$\int_{M}|\vec{H}|^{a}|\nabla^{\bot}\vec{H}|^{2}d\mu_{g}+(2m-1)\int_{M}|\vec{H}|^{a+4}d\mu_{g}=0,$

which implies $\vec{H}=0$.

This completes the proof of theorem 1.6. $\hfill\Box$

###  3.3 Proof of theorem 1.7.

From equation $(\ref{equ4})$ and the assumption that the sectional curvature
of $(N,h)$ is small that $-\epsilon$ we see that

$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\langle\vec{H},\Delta^{\bot}\vec{H}\rangle$
$\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
B(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle-2\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle$ $\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2m|\vec{H}|^{4}+2m\epsilon|\vec{H}|^{2}$
$\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2m\epsilon|\vec{H}|^{2}.$

Let $\gamma:M\to\mathbb{R}^{+}$ be a cut off function such that

$\gamma=1\,\,\,\,on\,\,\,\,B_{\rho},\gamma=0\,\,\,\,on\,\,\,\,M\setminus
B_{2\rho},\,\,\,\,and\,\,\,\,|\nabla\gamma|\leq\frac{C}{\rho},$

for some constant $C$ independent of $\rho$. Here $B_{\rho}$ is a geodesic
ball of radius $\rho$ on $M$. Then

$\displaystyle-\int_{M}\nabla|\vec{H}|^{2}\nabla|\vec{H}|^{a}\gamma^{2}d\mu_{g}$
(3.12) $\displaystyle=$
$\displaystyle\int_{M}\Delta|\vec{H}|^{2}|\vec{H}|^{a}\gamma^{2}d\mu_{g}$
$\displaystyle\geq$ $\displaystyle
2\int_{M}|\nabla^{\bot}\vec{H}|^{2}|\vec{H}|^{a}\gamma^{2}d\mu_{g}+2m\epsilon\int_{M}|\vec{H}|^{a+2}\gamma^{2}d\mu_{g},$

where $a$ is a nonnegative constant.

On the other hand we have

$\displaystyle-\int_{M}\nabla|\vec{H}|^{2}\nabla|\vec{H}|^{a}\gamma^{2}d\mu_{g}$
(3.13) $\displaystyle=$
$\displaystyle-\int_{M}2\langle\nabla^{\bot}\vec{H},\vec{H}\rangle(a|\vec{H}|^{a-2}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle+2|\vec{H}|^{a}\gamma\nabla\gamma)d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-4\int_{M}\langle\nabla^{\bot}\vec{H},\vec{H}\rangle|\vec{H}|^{a}\gamma\nabla\gamma
d\mu_{g}$ $\displaystyle\leq$ $\displaystyle
2\int_{M}|\nabla^{\bot}\vec{H}|^{2}|\vec{H}|^{a}\gamma^{2}d\mu_{g}+\frac{C}{\rho^{2}}\int_{B_{2\rho}\setminus
B_{\rho}}|\vec{H}|^{a+2}d\mu_{g}.$

From (3.12)-(3.13), we get

$\displaystyle
2m\epsilon\int_{B_{\rho}}|\vec{H}|^{a+2}d\mu_{g}\leq\frac{C}{\rho^{2}}\int_{B_{2\rho}\setminus
B_{\rho}}|\vec{H}|^{a+2}d\mu_{g}.$

Set $f(\rho)=\int_{B_{\rho}}|\vec{H}|^{a+2}d\mu_{g}$, from the above
inequality we see that when $\rho$ is big enough, we have

$f(\rho)\leq\frac{C}{\rho^{2}}f(2\rho).$

This implies that $f(\rho)\leq\frac{C}{\rho^{2n}}f(2^{n}\rho)$, where $C$ is a
constant independent of $\rho$. By assumption we have $f(2^{n}\rho)\leq
C(1+2^{ns}\rho^{s})$ for some positive integer $s$, as $\rho$ is big enough,
hence $f(\rho)\leq\frac{C(1+2^{ns}\rho^{s})}{\rho^{2n}}$. Let $2n>s$ one gets
$\lim_{\rho\to\infty}f(\rho)=0,$ i.e. $\int_{M}|\vec{H}|^{a+2}d\mu_{g}=0$.
Therefore, $\vec{H}=0$.

Furthermore, if $M$ is a hypersurface and the Ricci curvature of $(N,h)$ is
smaller that $-\epsilon$, then

$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\langle\vec{H},\Delta^{\bot}\vec{H}\rangle$
$\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
B(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle-2\sum_{i=1}^{m}\langle
R^{N}(e_{i},\vec{H})e_{i},\vec{H}\rangle$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\sum_{i=1}^{m}\langle
B(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle-2Ric(\vec{H},\vec{H})$
$\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2m|\vec{H}|^{4}+2\epsilon|\vec{H}|^{2}$
$\displaystyle\geq$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\epsilon|\vec{H}|^{2}.$

Then by the same argument as before we can prove that $\vec{H}=0$.

This finishes the proof of theorem 1.7. $\hfill\Box$

###  3.4 Proof of theorem 1.11.

###### Proof.

We prove theorem 1.11 in two steps.

Step 1. At any point $x\in M$, $\vec{H}=0$ or $\nabla\vec{H}=0$.

Let $\tilde{\gamma}:N\to\mathbb{R}^{+}$ be a cut off function defined as

$\tilde{\gamma}=1\,\,\,\,on\,\,\,\,B_{\rho},\tilde{\gamma}=0\,\,\,\,on\,\,\,\,N\setminus
B_{2\rho},\,\,\,\,|D\tilde{\gamma}|\leq\frac{C}{\rho}$

and define $\gamma:M\to\mathbb{R}^{+}$ by

$\gamma:=\tilde{\gamma}\circ f.$

By integral by parts we have

$\displaystyle\int_{M}|\vec{H}|^{a}\langle\vec{H},\Delta\vec{H}\rangle\gamma^{2}d\mu_{g}=$
$\displaystyle-$
$\displaystyle\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g}$ (3.14) $\displaystyle-$ $\displaystyle
a\int_{M}|\vec{H}|^{a-2}\langle\nabla\vec{H},\vec{H}\rangle^{2}\gamma^{2}d\mu_{g}.$

On the other hand, by assumption we have

$\displaystyle\int_{M}|\vec{H}|^{a}\langle\vec{H},\Delta\vec{H}\rangle\gamma^{2}d\mu_{g}\geq(\varepsilon-1)\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}.$
(3.15)

From (3.14)-(3.15), we have

$\displaystyle\varepsilon\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}$
$\displaystyle=$
$\displaystyle-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g}-a\int_{M}|\vec{H}|^{a-2}\langle\nabla\vec{H},\vec{H}\rangle^{2}\gamma^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g},$

which implies by using Young’s inequality that

$\displaystyle\varepsilon\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}\leq\frac{\varepsilon}{2}\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}+\frac{2}{\varepsilon}\int_{M}|\nabla\gamma|^{2}|\vec{H}|^{a+2}d\mu_{g}.$
(3.16)

Therefore

$\displaystyle\int_{f^{-1}(B_{\rho})}|\vec{H}|^{a}|\nabla\vec{H}|^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle\frac{4}{\varepsilon^{2}}\frac{C}{\rho^{2}}\int_{f^{-1}(B_{2\rho})}|\vec{H}|^{a+2}d\mu_{g}.$

Let $\rho\to\infty$ in this inequality we get

$\displaystyle\int_{M}|\vec{H}|^{a}|\nabla\vec{H}|^{2}d\mu_{g}\leq 0,$ (3.17)

and hence $\vec{H}=0$ or $\nabla\vec{H}=0$.

Step 2. $\nabla\vec{H}=0$ implies $\vec{H}=0$.

Now let $x\in M$ such that $\nabla\vec{H}(x)=0$. We choose an orthonormal
basis $\\{e_{i},i=1,...,m\\}$ of $T_{x}M$ and an orthonormal basis
$\\{\nu^{\alpha},\alpha=1,...,t\\}$ of $(T_{x}M)^{\bot}$.

Computing directly one gets

$\displaystyle
0=\langle\nabla_{e_{i}}\vec{H},e_{j}\rangle=\sum_{\alpha}H_{\alpha}B_{\alpha}(e_{i},e_{j}),$
(3.18)

for any $1\leq i,j\leq m$. Taking trace over this equality we get

$\sum_{\alpha}(H_{\alpha})^{2}=|\vec{H}|^{2}=0.$

Therefore $\vec{H}=0$.

Combing step 1 and 2, we see that if $f:(M,g)\to(N^{m+t},h)$ is a biharmonic
submanifold satisfying the assumptions of theorem 1.11, then $\vec{H}=0$.
$\hfill\Box$

###  3.5 Proof of theorem 1.13.

The proof is quite similar as before. For the convenience of readers we give
the details.

###### Proof.

Let $\gamma:M\to\mathbb{R}^{+}$ be a cut off function such that

$\gamma=1\,\,\,\,on\,\,\,\,B_{\rho},\gamma=0\,\,\,\,on\,\,\,\,M\setminus
B_{2\rho},\,\,\,\,and\,\,\,\,|\nabla\gamma|\leq\frac{C}{\rho},$

for some constant $C$ independent of $\rho$. Here $B_{\rho}$ is a geodesic
ball of radius $\rho$ on $M$.

By integral by parts we get

$\displaystyle\int_{M}|\vec{H}|^{a}\langle\vec{H},\Delta\vec{H}\rangle\gamma^{2}d\mu_{g}=$
$\displaystyle-$
$\displaystyle\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g}$ (3.19) $\displaystyle-$ $\displaystyle
a\int_{M}|\vec{H}|^{a-2}\langle\nabla\vec{H},\vec{H}\rangle^{2}\gamma^{2}d\mu_{g}.$

On the other hand, by assumption we have

$\displaystyle\int_{M}|\vec{H}|^{a}\langle\vec{H},\Delta\vec{H}\rangle\gamma^{2}d\mu_{g}\geq(\varepsilon-1)\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}.$
(3.20)

Hence we have, by combining the above two inequalities, that

$\displaystyle\varepsilon\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}$
$\displaystyle=$
$\displaystyle-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g}-a\int_{M}|\vec{H}|^{a-2}\langle\nabla\vec{H},\vec{H}\rangle^{2}\gamma^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle-2\int_{M}\vec{H}|^{a}\gamma\nabla\gamma\langle\nabla\vec{H},\vec{H}\rangle
d\mu_{g},$

which implies by using Young’s inequality that

$\displaystyle\varepsilon\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}\leq\frac{\varepsilon}{2}\int_{M}\vec{H}|^{a}|\nabla\vec{H}|^{2}\gamma^{2}d\mu_{g}+\frac{2}{\varepsilon}\int_{M}|\nabla\gamma|^{2}|\vec{H}|^{a+2}d\mu_{g}.$
(3.21)

Noth that

$\int_{B_{2\rho}}|\vec{H}|^{a+2}d\mu_{g}\leq\int_{M}|\vec{H}|^{a+2}d\mu_{g},$

therefore one gets

$\displaystyle\int_{B_{\rho}}|\vec{H}|^{a}|\nabla\vec{H}|^{2}d\mu_{g}\leq\frac{4}{\varepsilon^{2}}\frac{C}{\rho^{2}}\int_{M}|\vec{H}|^{a+2}d\mu_{g}.$
(3.22)

Let $\rho\to\infty$ in this inequality we get

$\displaystyle\int_{M}|\vec{H}|^{a}|\nabla\vec{H}|^{2}d\mu_{g}\leq 0,$ (3.23)

and hence $\vec{H}=0$ or $\nabla\vec{H}=0$.

Now using the same procedure as step 2 of the proof of theorem 1.11, we can
prove that $\vec{H}=0$. $\hfill\Box$

Acknowledgement. The author would like to express his appreciation to
Professor Guofang Wang for his suggestions on this paper. He is also very
appreciated with Dr. Shun Maeta for pointing out a mistake in the manuscript.

## References

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Yong Luo

School of mathematics and statistics,

Wuhan university, Hubei 430072, China

<EMAIL_ADDRESS>


# CodeLL: A Lifelong Learning Dataset to Support the Co-Evolution of Data and
Language Models of Code

Martin Weyssow∗, Claudio Di Sipio†, Davide Di Ruscio†, Houari Sahraoui∗ ∗DIRO,
Université de Montréal<EMAIL_ADDRESS><EMAIL_ADDRESS>†University of l’Aquila<EMAIL_ADDRESS>
(2018)

###### Abstract.

Motivated by recent work on lifelong learning applications for language models
(LMs) of code, we introduce CodeLL, a lifelong learning dataset focused on
code changes. Our contribution addresses a notable research gap marked by the
absence of a long-term temporal dimension in existing code change datasets,
limiting their suitability in lifelong learning scenarios. In contrast, our
dataset aims to comprehensively capture code changes across the entire release
history of open-source software repositories. In this work, we introduce an
initial version of CodeLL, comprising 71 machine-learning-based projects mined
from Software Heritage. This dataset enables the extraction and in-depth
analysis of code changes spanning 2,483 releases at both the method and API
levels. CodeLL enables researchers studying the behaviour of LMs in lifelong
fine-tuning settings for learning code changes. Additionally, the dataset can
help studying data distribution shifts within software repositories and the
evolution of API usages over time.

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: 21st International Conference on Mining Software
Repositories; April 2024; Lisbon, Portugal

##  1\. Introduction

The abundance of open-source software (OSS) data has played a pivotal role in
advancing the field of automated software engineering. Lately, large-scale
pre-training datasets, exemplified by The Pile (Gao et al., 2020) and The
Stack (Kocetkov et al., 2022) have enabled the emergence of large language
models (LLMs) tailored for programming languages understanding and generation
(Chen et al., 2021; Roziere et al., 2023; Nijkamp et al., 2023). From another
perspective, datasets and benchmarks such as CodeXGlue (Lu et al., 2021),
CodeSearchNet (Husain et al., 2019), XLCoST (Zhu et al., 2022) or CodeNet
(Puri et al., 2021) have been proven to be essential for fine-tuning language
models (LMs) of code111We use “language models” to refer to both “large
language models” and “pre-trained language models”. across diverse code-
related tasks.

In contrast to most previous research in code intelligence that often views
these models as static entities, our stance aligns with prior research
(Weyssow et al., 2023b; Gao et al., 2023; Weyssow et al., 2023a),
conceptualizing LMs of code as dynamic and adaptable. In essence, we contend
that viewing models of code as dynamically evolving over time opens up
exciting opportunities to study their behaviour in lifelong learning
scenarios. This paradigm shift underscores the necessity for mining software
repositories and constructing datasets that incorporate concepts related to
the long-term evolution of code and repositories. Furthermore, it sheds light
on a potential research gap in the current landscape. First, the datasets and
benchmarks mentioned previously lack a temporal dimension and are commonly
utilized for pre-training or fine-tuning models at a specific point in time.
The absence of temporality compromises their ability to capture the inherent
dynamic evolution in software development and repositories. Alternatively,
code change datasets such as CodeChangeNet (Lin et al., 2023) and MegaDiff
(Monperrus et al., 2021), only encompass code changes over two specific points
in time, rendering them unsuitable for pre-training or fine-tuning LMs of code
in a lifelong learning setting. Furthermore, while prior research effectively
studied LMs of code in lifelong learning settings (Weyssow et al., 2023b; Gao
et al., 2023), there remains an absence of datasets incorporating notions of
code evolution over time. Consequently, in the context of lifelong learning,
where LMs of code evolve as dynamic entities accumulating knowledge over time,
the nature of the existing datasets poses significant limitations.

To address this gap, we present an initial version of Code Lifelong Learning
(CodeLL), a dataset conceived for supporting lifelong co-evolution of software
repositories and LMs of code. To build CodeLL, we leverage Software Heritage
(SWH) (Cosmo and Zacchiroli, 2017) platform that stores, collects, and
maintains software repositories from multiple sources, including GitHub, PyPi,
and GitLab. The main rationale behind this choice is that SWH enables the
fingerprinting of large and reproducible datasets through queries and
timestamps (Lefeuvre et al., 2023). Consequently, it renders the extraction of
our raw datasets reproducible and facilitates its extension in the future. In
the scope of this paper, we collect 71 Python repositories related to machine
learning that often use fast-evolving third-party libraries (Dilhara et al.,
2021), which constitutes an interesting scope of analysis for studying
lifelong code changes. For each repository, we extract all its associated
releases and Python files, resulting in a total of 2,483 releases and 970,277
files. Next, we design mapping heuristics to map files, methods, and API calls
between two consecutive releases to analyze how methods content and API calls
evolve over the entire lifespan of the repositories.

By conducting a code change analysis of one of the mined repositories (see
Section 3), we highlight potential research opportunities spanning from the
fine-tuning of LLMs of code for learning lifelong code changes and API
evolution to analyzing potential distribution shifts of the data over time.
Furthermore, the descriptive statistics of our data (see Section 3)
underscores the potential for building a more massive version of CodeLL, that
could be used for continual pre-training of LLMs, by mining more repositories
as incorporating 71 repositories already produces a relatively large-scale
dataset.

##  2\. Data Collection methodology

Our data collection methodology is depicted in Figure 1. We describe the data
mining and code change analysis phases in Section 2.2 and Section 2.3,
respectively.

Figure 1. Data collection methodology.

###  2.1. CodeLL Data Model

Before delving into the description of the data collection methodology, we
present the CodeLL data schema modeled as a class diagram222Interested readers
can refer to (Lefeuvre et al., 2023) for a comprehensive presentation of the
SWH general data model. in Figure 2. The diagram illustrates an Archive as a
general software repository stored on SWH, with a unique originURL and
multiple releases. Each Release is identified by a branch name and a date,
i.e., the timestamp of that specific release. A release contains a list of
SourceCodeFile entities, each linked to its previous release through the
previous_release association. A SourceCodeFile is characterized by imports and
composed of methods; a Method contains a list of function_calls. It is worth
noting that imports are stored as a list of strings since analysing them in
depth is not in the scope of this paper. These entities are specialized forms
of a MappedEntity, characterized by a unique hash ID, signifying the mapping
between SourceCodeFile, Method, and FunctionCall originating from two distinct
releases. The mapping_type attribute of MappedEntity denotes whether an entity
is effectively mapped, added, or removed from one release to another.

###  2.2. Data Mining

#### SWH dump

To collect the initial set of artifacts, we rely on the
2021-03-23-popular-3k-python dataset available on SWH. The dump contains a
total number of 2,197 Python artifacts originating from four different data
sources, i.e., 580 GitHub and 135 GitLab repositories, and 827 PyPI and 655
Debian packages. We leverage this dump as it encompasses only popular software
repositories and packages from diverse data sources.333See
https://tinyurl.com/swhdataset for more details.

#### Filtering

From the initial dump, we exclude Debian packages due to the absence of .py
files. Next, we filter out repositories and packages unrelated to machine
learning by relying on GitHub topics444https://github.com/topics and PyPI
classifiers.555https://pypi.org/classifiers/ As mentioned in the introduction,
machine learning projects tend to use APIs that evolve at a fast pace, e.g.,
PyTorch, which is interesting for studying the lifelong evolution of the
projects. Specifically, we focus on artifacts labeled with terms like “machine
learning” and “neural network”, as illustrated in Figure 1. In cases where
artifacts do not align with any search term, we automatically evaluate whether
they use any third-party libraries related to machine learning. Inclusion is
then determined based on this assessment. This initial filtering process
results in 140 archives, comprising 44 from PyPI, 96 from GitHub, and none
from GitLab. Given our primary objective of releasing a dataset tailored for
lifelong learning studies, we focus the data mining on archive releases, i.e.,
a version of the archive at a certain timestamp. We discard archives with less
than two releases as it does not enable any code evolution analysis, and end
up with 71 archives composed of 26 PyPI packages and 45 GitHub repositories.
From those archives, we extracted 2,483 releases. Another reason to limit our
mining efforts is motivated by the potential for a single archive to contain a
substantial number of releases, and the SWH API imposes restrictions on the
number of requests per user. Nonetheless, we plan on expanding the dataset by
incorporating all the artifacts contained in the 2021-03-23-popular-3k-python
SWH dump in the future.

#### Code extraction

For each release, we statically analyze the .py files they contain. Within
each file, we retrieve imports, standalone functions, and class methods.
Furthermore, we extract the function and API calls within each function and
method, and store their code positions, i.e., start and end offsets.
Importantly, we establish implicit links between API calls and imports; e.g.,
“torch.mean” is connected to “import torch”. This information storage
facilitates the application of API-related downstream tasks using CodeLL.

Figure 2. The CodeLL data model.

###  2.3. Code Change Analysis

Given two releases $r_{m}$ and $r_{n}$ of a software archive, the code change
analysis involves mapping files, methods, and function/API calls from both
releases. Note that $r_{m}$ and $r_{n}$ do not have to be consecutive
releases, i.e., the code change analysis can be performed between any archive
releases.

#### Files mapping

We start by mapping files with identical relative paths. To handle cases where
a file in $r_{m}$ changes path in $r_{n}$, as in the case of code refactoring,
we rely on the following heuristics. We ensure a high Levenshtein similarity
ratio between both file names ($\geq$ 0.75) and verify the presence of
identical functions/methods in their content.

#### Methods mapping

We map functions and methods belonging to two mapped files $f_{r_{m}}$ and
$f_{r_{n}}$ if they belong to the same class (None if the function is
standalone), have the same names and parameters. If a method from $f_{r_{m}}$
remains unmapped, we map it with a method from $f_{r_{n}}$ that belongs to the
same class and has the same name.

#### Function calls mapping

We map function and API calls belonging to two mapped methods $m_{f_{r_{m}}}$
and $m_{f_{r_{n}}}$ if they have the same code positions. If a function/API
call in $m_{f_{r_{m}}}$ remains unmapped, we map it with an identical
function/API call in $m_{f_{r_{n}}}$ with the closest position.

Finally, unmapped files, methods and function/API calls $\in r_{m}$ are tagged
as “removed”, while unmapped elements $\in r_{n}$ are tagged as “added”.

##  3\. Data overview

In this section, we present an overview of CodeLL by first discussing
descriptive statistics of the dataset. Subsequently, we conduct a code change
analysis of one selected repository from our dataset. Finally, we discuss the
storage and reusability of CodeLL.

#### Descriptive statistics

Figure 3. Number of releases per repository. Figure 4. Number of Python files
per release.

This initial version of CodeLL comprises 71 software repositories, 2,483
releases, and 970,277 Python files. Figure 3 illustrates the distribution of
releases per software repository, revealing that the majority feature between
0 and 40 releases, with certain repositories exhibiting a notably higher
release count. This diversity facilitates examining software repository
evolution across a spectrum of lifespans. Furthermore, Figure 4 depicts the
distribution of files per release, underscoring the capacity for extensive
evolution analysis across a substantial number of files.

#### Lifelong code change analysis

Figure 5. Gensim project file changes. Figure 6. Correlation between changed
files and methods in each Gensim release, with Levenshtein distance computed
and averaged across changed methods.

For illustrative reasons, we analyse the evolution of the Gensim project,
which comprises 78 releases. In Figure 5, we analyse how files change
throughout the releases. This analysis helps pinpointing releases where either
substantial refactoring occur, e.g., at release #15, or instances where a
significant amount of files changed from one release to another, e.g., between
releases #52-70. Furthermore, Figure 6 shows a linear correlation between the
number of files and methods that changed for each release of Gensim. The
regression plot also depicts the average Levenshtein distance computed for
each altered method, facilitating the identification of releases with
substantial changes at the code level. Overall, these analyses are pivotal in
identifying distinctive patterns in the evolution of a software repository,
streamlining the application of subsequent data analyses or downstream tasks
to specific releases.

#### Storage and reusability

⬇

{

”repository”: str, ”branch” : str, ”prev_branch” : str,

”date”: str, ”id”: integer, ”imports”: List[str],

”methods”: List[{

”id”: integer, ”class”: str, ”name”: str,

”params”: List[str], ”content”: str,

”function_calls”: List[{

”id”: integer, ”expression”: str,

”start_offset”: integer, ”end_offset”: integer,

”mapping”: integer | ”added” | ”removed”

},

”mapping”: integer | ”added” | ”removed”

}],

”mapping”: integer | ”added” | ”removed”

}

Figure 7. Description of one line of the CodeLL dataset.

Our dataset and code are available online.666 https://github.com/martin-
wey/CodeLL-Dataset We provide one .jsonl file per mined repository, and
Listing 7 illustrates the structure of one line of the .jsonl files. We also
provide our code to mine more data from SWH, and generate the .jsonl files, as
well as a Python notebook to compute statistics.

##  4\. Research opportunities

Lifelong learning for code and LLMs. The primary objective of CodeLL is to
enable the study of (L)LMs of code in lifelong learning settings. As
demonstrated in previous work (Weyssow et al., 2023b; Gao et al., 2023), pre-
trained language models of code suffer from catastrophic forgetting, i.e.,
they forget about past data, in scenarios where the model continuously learns
on new data over time. However, these prior studies do not consider learning
from the lifelong evolution of software repositories and leverage rather
synthetic datasets not encompassing any temporal dimension. In this context,
our dataset enables researchers to study how LMs of code can (1) continuously
acquire knowledge over the whole lifespans of software repositories during
fine-tuning, or (2) adapt to a specific software release at inference using
prompting techniques. In its current state, CodeLL is not suitable for pre-
training LLMs, but subsequent large-scale versions of CodeLL might enable
exploring continual pre-training of LLMs.

Data distribution shift. Analyzing CodeLL in terms of the evolution of the
data distribution over time presents a compelling research opportunity. As
discussed in Section 3, conducting a lifelong analysis of code changes within
an archive enables the identification of software releases marked by
significant code changes. However, it is crucial to note that releases
featuring code changes may not necessarily pose challenges for LMs of code,
particularly if the distribution of these changes does not deviate
significantly from the training data distribution. Consequently, investigating
the dynamic shifts in release distribution allows for pinpointing instances
where LMs may exhibit weak generalization. These challenges, particularly
those associated with out-of-distribution generalization and long-tailed
distributions, have demonstrated a substantial impact on the effectiveness of
the model (Zhou et al., 2023; Weyssow et al., 2023b).

API evolution. The problem of API evolution has been studied from different
perspectives, including investigating community trends (Alrubaye et al., [n.
d.]; Kula et al., 2018), and performing automated migration (Nguyen et al.,
2022; Xu et al., 2019). Despite these efforts, our current understanding is
that there is a gap in the literature concerning the adaptation of (L)LMs of
code to either changes in API usage or in the APIs themselves over extended
periods. While prior work enhanced LLMs of code with API knowledge (Patil et
al., 2023; Qin et al., 2023), it remains unclear how to continuously adapt
LLMs to new APIs over time in an effective manner.

##  5\. Limitations

This first version of CodeLL presents some limitations related to the
constrained scope of mined software repositories (see Section 2.2) and the
code change analysis (see Section 2.3). First, we limited the mining to
machine learning-related archives, resulting in the mining of 2,483 releases,
a time-consuming process as SWH API restricts the number of requests allowed
per hour. We believe this initial release of CodeLL already enables a broad
range of applications as described in Section 4. We intend to broaden the
dataset with more repositories in subsequent versions of CodeLL. Additionally,
the current method employed for mapping files, methods, and function calls
between two releases relies on heuristics. We believe the applied heuristics
enable mapping releases elements in an accurate way. However, we acknowledge
that refining these heuristics may further improves the mappings for more
specific code changes, especially at the line or API call level.

##  6\. Related work

We identified code change datasets as the most related to our contribution. To
the best of our knowledge, there is no existing dataset such as CodeLL
facilitating lifelong learning, data distribution shift, and code change/API
evolution applications over the entire lifespan of software repositories.
Existing datasets, including MegaDiff (Monperrus et al., 2021), CodRep (Chen
and Monperrus, 2018), or ManySStuBs4J (Karampatsis and Sutton, 2020), focus on
Java code changes without incorporating timestamps or considering the temporal
evolution of code changes. Other prior work, exemplified by APIDiff (Brito et
al., 2018) and PyMIgBench (Islam et al., 2023), enable detecting API changes.
Our dataset not only encompasses the analysis of API changes but also enables
studying how API usages within a repository evolve throughout its lifespan.
Nonetheless, we recognize that incorporating the taxonomy of code changes from
PyMIgBench (Islam et al., 2023) in CodeLL would enhance our dataset. Finally,
CodeChangeNet (Lin et al., 2023) comprises 2M data samples with code changes
ranging between 3 to 100 tokens across six languages. Similar to other
datasets, CodeChangeNet does not encompass any notion of code changes over
time, rendering it unsuitable for lifelong learning applications.
Nevertheless, our plans include expanding our tool and dataset to support
multiple programming languages.

##  7\. Conclusion and future work

In this paper, we introduce CodeLL, a lifelong learning dataset to support the
co-evolution of source code data and LMs of code. The initial version of
CodeLL encompasses 71 Python machine learning software repositories,
comprising 2,483 releases and nearly 1 million files. CodeLL records code
changes at both the method and function/API call levels throughout the entire
release history of code repositories, facilitating the exploration of lifelong
learning applications for pre-trained language models and LLMs of code. Our
dataset opens up a wide range of future work opportunities related to lifelong
learning for code, identifying distribution shifts in software repository
releases, and learning code change and API evolution. As future work, we aim
to expand CodeLL with additional software repositories spanning diverse
programming languages. Finally, we plan to delve into downstream tasks and
applications using CodeLL.

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