# A spherical Bernstein theorem for minimal submanifolds of higher codimension

J. Jost, Y. L. Xin and Ling Yang  Max Planck Institute for Mathematics in the
Sciences, Inselstr. 22, 04103 Leipzig, Germany<EMAIL_ADDRESS>Institute of
Mathematics, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>Institute of Mathematics, Fudan University, Shanghai 200433, China.
<EMAIL_ADDRESS>
###### Abstract.

Combining the tools of geometric analysis with properties of Jordan angles and
angle space distributions, we derive a spherical and a Euclidean Bernstein
theorem for minimal submanifolds of arbitrary dimension and codimension, under
the condition that the Gauss image is contained in some geometrically defined
closed region of a Grassmannian manifold. The proof depends on the
subharmoncity of an auxiliary function, the Codazzi equations and geometric
measure theory.

###### 1991 Mathematics Subject Classification:

58E20,53A10.

The first author is supported by the ERC Advanced Grant FP7-267087. The second
named author and the third named author are partially supported by NSFC. They
are grateful to the Max Planck Institute for Mathematics in the Sciences in
Leipzig for its hospitality and continuous support.

##  1\. Introduction

This paper is a part of our systematic approach to the Bernstein problem in
higher codimension. The Bernstein problem has a spherical and a Euclidean
version, and the two are tightly related and essentially equivalent, as is
well known and as we shall explain in a moment in more detail.

The Euclidean version says that a complete $n$-dimensional minimal submanifold
$M$ of ${\tenmsb R}^{n+m}$, that is, of codimension $m$, has to be an affine
subspace if its Gauss image is contained in a sufficiently small subset of the
Grassmann manifold $\mathbb{G}_{m,n}$. Equivalently, it is affine when all of
its normal spaces $N$ satisfies $\langle N,Q_{0}\rangle>c_{0}$ for a fixed
reference space $Q_{0}$ and some positive constant $c_{0}$. In either
formulation, we are assuming that the tangent, or equivalently, the normal
spaces do not change their direction too much when we move across $M$. That
some such condition is necessary follows from an example of Lawson-
Osserman[14] with $\langle N,Q_{0}\rangle=1/9$. And that example tells us,
more precisely, that the condition has to be stricter for $m>1$ than in the
codimension 1 case, the setting of the classical Bernstein theorem [3] and its
extensions by Fleming [7], de Giorgi [5], Almgren [2], Simons [17], Moser
[15], and others.

The spherical Bernstein theorem concerns compact $(n-1)$-dimensional minimal
submanifolds of the sphere $S^{n+m-1}$, and analogously, the aim is to prove
that they are totally geodesic (i.e. equatorial) subspheres when their normal
planes do not change their directions too much.

As indicated, we are interested here in the case $m>1$, and we ask what the
optimal quantitative condition is. In previous work, we have shown

######  Theorem 1.1.

[11] Let $M$ be an $(n-1)$-dimensional compact minimal submanifold in
$S^{n+m-1}$. Suppose that there is a fixed oriented $m$-plane $Q_{0}$ and a
number $c_{0}>1/3$, such that $\langle N,Q_{0}\rangle\geq c_{0}$ holds for all
normal $m$-planes $N$ of $M$. Then $M$ is totally geodesic.

Our question here is whether this is optimal, that is, whether there exists a
counterexample for $c_{0}=1/3$, or whether one can move beyond. In this paper,
we show that the result continues to hold for $c_{0}=1/3$, that is,

######  Theorem 1.2.

Let $M$ be an $(n-1)$-dimensional compact minimal submanifold in $S^{n+m-1}$.
Suppose that there is a fixed oriented $m$-plane $Q_{0}$, such that $\langle
N,Q_{0}\rangle\geq 1/3$ holds for all normal $m$-planes $N$ of $M$. Then $M$
is totally geodesic.

This might look like a small and insignificant step, but as in many examples
of geometric analysis, limiting cases often are much harder than those
involving strict inequalities. The reason is that one needs additional tools
to analyze possible limit configurations before one can deduce that they can’t
exist after all. Typically, in such cases, the analytical estimates need to be
supplemented by considerations of a more algebraic nature. That is also the
case here. We shall carefully utilize the information contained in the Codazzi
equations. Still, there exists a quantitative gap between the above positive
results and the counterexample of Lawson-Osserman, as the latter corresponds
to the value $c_{0}=1/9$ instead of the $1/3$ that we can currently achieve.
Nevertheless, we believe that since we can now rule out a counterexample for
$c_{0}=1/3$, we expect that one can go even beyond that value. Whether
$c_{0}=1/9$ is the largest value with a counterexample or whether a
counterexample different from the Lawson-Osserman one exists for some value
between $1/9$ and $1/3$, we currently do not know.

Let us now recall the relation between the spherical and the Euclidean
Bernstein problem and then state our results for the latter. Fleming’s idea
[7] was that by rescaling a nontrivial minimal graph in Euclidean space, one
obtains a nonflat minimal cone, and the intersection of that cone with the
unit sphere then is a compact minimal submanifold of the latter. Therefore,
conditions ruling out the latter can be translated into conditions ruling out
the former. Because of the noncompact nature of minimal graphs in Euclidean
space, as an important technical ingredient, one needs to invoke Allard’s
regularity theory.

We then obtain the following result concerning the Euclidean version of the
higher codimensional Bernstein problem (see Theorem 4.2), which is an
improvement of Theorem 1.1 in [11].

######  Theorem 1.3.

Let $f:=(f^{1},\cdots,f^{m})$ be a smooth ${\tenmsb R}^{m}$-valued function
defined everywhere on ${\tenmsb R}^{n}$. If its graph $M:=\text{graph
}f=\\{(x,f(x)):x\in{\tenmsb R}^{m}\\}$ is a minimal submanifold in ${\tenmsb
R}^{n+m}$, and

(1.1) $\Delta_{f}:=\Big{[}\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\Big{]}^{\frac{1}{2}}\leq 3,$

then $f^{1},\cdots,f^{m}$ have to be affine linear, that is, represent an
affine $n$-plane in ${\tenmsb R}^{n+m}$.

The 3 here corresponds to the $1/3$ in the preceding results as will become
clear below when we describe the geometry.

In geometric terms, we approach the Bernstein problem via the normal Gauss map
$\gamma$ of a minimal submanifold $M$ which takes values in the Grassmannian
manifold $\mathbb{G}_{m,n}$. The strategy which was first applied in [9] then
is to show that $\gamma$ is constant. This, of course, is equivalent to $M$
being affine. The main point here is that this Gauss map is harmonic.
Therefore, the strategy then leads to showing that under appropriate geometric
conditions, a harmonic map into a Grassmannian $\mathbb{G}_{m,n}$ has to be
constant. This strategy could be successfully applied in increasing
generality, in [9, 10, 11, 12]. Essentially, one tries to translate a
geometric restriction on the size of the Gauss image $\gamma(M)$ into the
subharmonicity of a composition function $\phi\circ\gamma$ where
$\phi:\gamma(M)\to{\tenmsb R}$ is a suitable scalar function, for instance a
strictly convex function. When we are in the spherical setting, the domain $M$
is compact, and so, such a subharmonic function $\phi\circ\gamma$ then has to
be constant by the maximum principle. One then utilizes this to deduce that
$\gamma$ is constant itself. In the Euclidean setting, $\phi\circ\gamma$ has
to be subjected to clever and subtle estimates to eventually reach the same
conclusion. As already mentioned, for the case $c_{0}=1/3$ which we are
treating in the present paper, this strategy by itself is not yet powerful
enough and needs to be supplemented by detailed algebraic considerations. In
more precise terms, we consider eigenspaces for Jordan angles between the
tangent or normal space of our minimal submanifold $M$ and some fixed
reference space. Invoking a theorem of Nomizu [16], we find that, generically,
they yield smooth subbundles of the tangent or normal bundle of $M$. This will
provide us with decompositions that can be algebraically exploited.

The paper is organized as follows. In Section 2, the basic geometric concepts
on Grassmannian manifolds, those of Jordan angles, angle spaces,
multiplicities, anti-involutive automorphisms and angle space distributions,
are introduced, which will play an important role in our statement, and the
connection between the $w$-function on Grassmannian manifolds and the Jordan
angles is revealed. Let $Q_{0}$ be a fixed point on $\mathbb{G}_{m,n}$, then
the composition of $w(\cdot,Q_{0})$ and the normal Gauss map yields a smooth
function on $M$, an arbitrary submanifold in ${\tenmsb R}^{n+m}$, which is
called the $w$-function. Based on [6], [11] and the properties of Jordan
angles, one can prove $v:=w^{-1}$ is a subharmonic function whenever $M$ has
parallel mean curvature and $v\leq 3$. We also explore the second fundamental
form of $M$ provided that $v\leq 3$, $\Delta v\equiv 0$ and $|B|^{2}\neq 0$
everywhere, and discover that $M$ is a simple austere submanfold, a notion
introduced by Harvey-Lawson [8] and further studied by Bryant [4]. These are
the main points of Section 3. Finally in Section 4, combining the Codazzi
equations and basic properties of angle spaces, we derive a spherical
Bernstein theorem, and also the corresponding Euclidean Bernstein theorem.

##  2\. Jordan angles and angle space distributions

Let ${\tenmsb R}^{n+m}$ be an $(n+m)$-dimensional Euclidean space. The
oriented $m$-spaces in ${\tenmsb R}^{n+m}$ constitute the Grassmann manifold
$\mathbb{G}_{m,n}$, which is the Riemannian symmetric space of compact type
$SO(n+m)/SO(n)\times SO(m)$.

Let $P,Q_{0}$ be 2 points in $\mathbb{G}_{m,n}$. The Jordan angles between $P$
and $Q_{0}$ are the critical values of the angle $\theta$ between a nonzero
vector $u$ in $P$ and its orthogonal projection $u^{*}$ in $Q_{0}$ as $u$ runs
through $P$. If $\theta$ is a nonzero Jordan angle between $P$ and $Q_{0}$
determined by a unit vector $u$ in $P$ and its projection $u^{*}$ in $Q_{0}$,
then $u$ is called an angle direction of $P$ relative to $Q_{0}$, and the
$2$-plane spanned by $u$ and $u^{*}$ is called an angle 2-plane between $P$
and $Q_{0}$ (see [18]).

Now, we give a slightly different description for the Jordan angles, which
will be useful later.

Denote by $\mathcal{P}_{0}$ the orthogonal projection of ${\tenmsb R}^{n+m}$
onto $Q_{0}$ and by $\mathcal{P}$ the orthogonal projection of ${\tenmsb
R}^{n+m}$ onto $P$. Then for an arbitrary vector $u\in P$ and $\varepsilon\in
Q_{0}$,

(2.1) $\displaystyle\langle\mathcal{P}_{0}u,\varepsilon\rangle$
$\displaystyle=\langle\mathcal{P}_{0}u+(u-\mathcal{P}_{0}u),\varepsilon\rangle=\langle
u,\varepsilon\rangle$ $\displaystyle=\langle
u,\mathcal{P}\varepsilon+(\varepsilon-\mathcal{P}\varepsilon)\rangle=\langle
u,\mathcal{P}\varepsilon\rangle.$

This means that $\mathcal{P}$ is adjoint to $\mathcal{P}_{0}$ with respect to
the canonical Euclidean inner product. Moreover,

(2.2)
$\langle(\mathcal{P}\circ\mathcal{P}_{0})u,v\rangle=\langle\mathcal{P}_{0}u,\mathcal{P}_{0}v\rangle=\langle
u,(\mathcal{P}\circ\mathcal{P}_{0})v\rangle$

holds for any $u,v\in P$, which implies that $\mathcal{P}\circ\mathcal{P}_{0}$
is a nonnegative definite self-adjoint transformation on $P$.

For every nonzero $u\in P$,

(2.3) $\cos^{2}\angle(u,u^{*})=\frac{\langle u^{*},u^{*}\rangle}{\langle
u,u\rangle}=\frac{\langle\mathcal{P}_{0}u,\mathcal{P}_{0}u\rangle}{\langle
u,u\rangle}=\frac{\langle(\mathcal{P}\circ\mathcal{P}_{0})u,u\rangle}{\langle
u,u\rangle}.$

Hence $\theta$ is a Jordan angle between $P$ and $Q_{0}$ if and only if
$\mu:=\cos^{2}\theta$ is an eigenvalue of $\mathcal{P}\circ\mathcal{P}_{0}$,
and $u$ is an angle direction associated to $\theta$ if and only if it is an
eigenvector associated to the eigenvalue $\mu$, i.e.

(2.4) $(\mathcal{P}\circ\mathcal{P}_{0})u=\mu u=\cos^{2}\theta\ u.$

Therefore, all the angle directions associated to $\theta$ constitute a linear
subspace of $P$, which is called an angle space of $P$ relative to $Q_{0}$ and
we denote it by $P_{\theta}$. The dimension of $P_{\theta}$ is called the
multiplicity of $\theta$, which is denoted by $m_{\theta}$. If we denote by
$\text{Arg}(P,Q_{0})$ the set consisting of all the Jordan angles between $P$
and $Q_{0}$, then

(2.5) $P=\bigoplus_{\theta\in\text{Arg}(P,Q_{0})}P_{\theta}.$

and hence

(2.6) $m=\sum_{\theta\in\text{Arg}(P,Q_{0})}m_{\theta}.$

The angle spaces are mutually orthogonal to each other, and in particular

(2.7) $P_{0}=P\cap Q_{0},\qquad P_{\pi/2}=P\cap Q_{0}^{\bot}.$

Similarly, $\theta$ is a Jordan angle between $Q_{0}$ and $P$ if and only if
$\mu:=\cos^{2}\theta$ is an eigenvalue of $\mathcal{P}_{0}\circ\mathcal{P}$.
Denote by $(Q_{0})_{\theta}$ the angle space of $Q_{0}$ relative to $P$
associated to $\theta$, then $\varepsilon\in(Q_{0})_{\theta}$ if and only if
$(\mathcal{P}_{0}\circ\mathcal{P})\varepsilon=\cos^{2}\theta\ \varepsilon$,
and

(2.8) $Q_{0}=\bigoplus_{\theta\in\text{Arg}(Q_{0},P)}(Q_{0})_{\theta}$

with $\text{Arg}(Q_{0},P)$ denoting the set consisting of all the Jordan
angles between $Q_{0}$ and $P$.

Let $P^{\bot}$ and $Q_{0}^{\bot}$ be the orthogonal complements of $P$ and
$Q_{0}$, and denote by $\mathcal{P}_{0}^{\bot}$ and $\mathcal{P}^{\bot}$ the
orthogonal projections of ${\tenmsb R}^{n+m}$ onto $P^{\bot}$ and
$Q_{0}^{\bot}$, respectively. As above, the set consisting of all the Jordan
angles between $P^{\bot}$ and $Q_{0}^{\bot}$ is denoted by
$\text{Arg}(P^{\bot},Q_{0}^{\bot})$, $P_{\theta}^{\bot}$ denotes the angle
space associated to $\theta\in\text{Arg}(P^{\bot},Q_{0}^{\bot})$, and
$m_{\theta}^{\bot}:=\dim P_{\theta}^{\bot}$ denotes the multiplicity of
$\theta$. The following lemma reveals the close relationship between
$\text{Arg}(P,Q_{0})$, $\text{Arg}(Q_{0},P)$ and
$\text{Arg}(P^{\bot},Q_{0}^{\bot})$.

######  Lemma 2.1.

Let $P,Q_{0}\in\mathbb{G}_{m,n}$, then
$\text{Arg}(P,Q_{0})=\text{Arg}(Q_{0},P)$ and the multiplicities of each
corresponding Jordan angles are equivalent. If we denote

(2.9) $R_{\theta}:=P_{\theta}+(Q_{0})_{\theta}$

for each $\theta\in\text{Arg}(P,Q_{0})$, then $R_{\theta}\bot R_{\sigma}$
whenever $\theta\neq\sigma$, and

(2.10) $P+Q_{0}=\bigoplus_{\theta\in\text{Arg}(P,Q_{0})}R_{\theta}.$

For any $\theta\in(0,\pi/2]$, $\theta\in\text{Arg}(P^{\bot},Q_{0}^{\bot})$ if
and only if $\theta\in\text{Arg}(P,Q)$, and $m_{\theta}^{\bot}=m_{\theta}$,
$R_{\theta}=P_{\theta}\oplus P_{\theta}^{\bot}$. Moreover, for every
$\theta\in\text{Arg}(P,Q_{0})\cap(0,\pi/2)$, there exists
$\Phi_{\theta}:R_{\theta}\rightarrow R_{\theta}$, satisfying

(i) $|\Phi_{\theta}(\xi)|=|\xi|$ for every $\xi\in R_{\theta}$;

(ii) $\Phi_{\theta}^{2}=-\mathbf{Id}$;

(iii) $\Phi_{\theta}(P_{\theta})=P_{\theta}^{\bot}$,
$\Phi_{\theta}(P_{\theta}^{\bot})=P_{\theta}$;

(iv) For any nonzero vector $u\in P_{\theta}$ ($v\in P_{\theta}^{\bot}$),
$\Phi_{\theta}(u)$ ($\Phi_{\theta}(v)$) lies in the angle 2-plane generated by
$u$ ($v$); more precisely,

(2.11) $\displaystyle\sec\theta\ \mathcal{P}_{0}u$ $\displaystyle=\cos\theta\
u-\sin\theta\ \Phi_{\theta}(u),$ $\displaystyle\sec\theta\
\mathcal{P}_{0}^{\bot}v$ $\displaystyle=\cos\theta\ v-\sin\theta\
\Phi_{\theta}(v).$

$\Phi_{\theta}$ is called the anti-involutive automorphism associated to
$\theta$.

###### Proof.

Given any $\theta\in\text{Arg}(P,Q_{0})\cap[0,\pi/2)$ and $u\in P_{\theta}$,
letting $\varepsilon:=\mathcal{P}_{0}u$ gives

(2.12)
$(\mathcal{P}_{0}\circ\mathcal{P})\varepsilon=(\mathcal{P}_{0}\circ\mathcal{P}\circ\mathcal{P}_{0})u=\cos^{2}\theta\
\mathcal{P}_{0}u=\cos^{2}\theta\ \varepsilon.$

Hence $\theta\in\text{Arg}(Q_{0},P)$ and $\mathcal{P}_{0}$ is an injective
linear mapping from $P_{\theta}$ into $(Q_{0})_{\theta}$. On the other hand,
assuming $\theta\in\text{Arg}(Q_{0},P)\cap[0,\pi/2)$, one can deduce
$\theta\in\text{Arg}(P,Q_{0})$ and $\mathcal{P}$ is an injective linear
mapping from $(Q_{0})_{\theta}$ into $P_{\theta}$. Thus both $\mathcal{P}_{0}$
and $\mathcal{P}$ are linear isomorphisms whenever $\theta\in[0,\pi/2)$, and
then $\dim(Q_{0})_{\theta}=\dim P_{\theta}$. In conjunction with (2.5) and
(2.8),

(2.13) $\displaystyle\dim(Q_{0})_{\pi/2}$
$\displaystyle=m-\sum_{\theta\in\text{Arg}(Q_{0},P)\cap[0,\pi/2)}\dim(Q_{0})_{\theta}$
$\displaystyle=m-\sum_{\theta\in\text{Arg}(P,Q_{0})\cap[0,\pi/2)}\dim
P_{\theta}$ $\displaystyle=\dim P_{\pi/2}.$

This mean $\pi/2\in\text{Arg}(Q_{0},P)$ if and only if
$\pi/2\in\text{Arg}(P,Q_{0})$ and the multiplicities are equivalent.
Therefore, $\text{Arg}(Q_{0},P)=\text{Arg}(P,Q_{0})$ and
$\dim(Q_{0})_{\theta}=\dim P_{\theta}=m_{\theta}$ for every
$\theta\in\text{Arg}(P,Q_{0})$.

Let $\theta,\sigma$ be distinct Jordan angles between $P$ and $Q_{0}$, it has
been shown above that $P_{\theta}\bot P_{\sigma}$ and
$(Q_{0})_{\theta}\bot(Q_{0})_{\sigma}$. For any $u\in P_{\theta}$ and
$\varepsilon\in(Q_{0})_{\sigma}$,

$\langle u,\varepsilon\rangle=\langle
u,\mathcal{P}(\varepsilon)\rangle\in\langle P_{\theta},P_{\sigma}\rangle=(0),$

which implies $P_{\theta}\bot(Q_{0})_{\sigma}$ and similarly
$(Q_{0})_{\theta}\bot P_{\sigma}$. Hence $R_{\theta}\bot R_{\sigma}$ and then
(2.10) immediately follows from (2.5) and (2.8).

If $\theta\in(0,\pi/2)$ is a Jordan angle between $P$ and $Q_{0}$, then for
any nonzero vector $u\in P_{\theta}$, we have
$|\mathcal{P}_{0}u|=\cos\theta|u|$ and
$(\mathcal{P}\circ\mathcal{P}_{0})u=\cos^{2}\theta\ u$. Denote

(2.14) $\varepsilon:=\sec\theta\ \mathcal{P}_{0}u,$

then $|\varepsilon|=|u|$. Now we put

(2.15) $v:=-\sec\theta\csc\theta(\mathcal{P}^{\bot}\circ\mathcal{P}_{0})u\in
P^{\bot},$

then

(2.16) $\displaystyle\varepsilon$ $\displaystyle=\sec\theta\
\mathcal{P}_{0}u=\sec\theta\big{[}(\mathcal{P}\circ\mathcal{P}_{0})u+(\mathcal{P}^{\bot}\circ\mathcal{P}_{0})u\big{]}$
$\displaystyle=\cos\theta\ u-\sin\theta\ v$

which implies $v\in\text{span}\\{u,\mathcal{P}_{0}u\\}\subset R_{\theta}$ and
$|v|=|u|$. Since

$\displaystyle(\mathcal{P}^{\bot}\circ\mathcal{P}_{0}^{\bot})v$
$\displaystyle=-\sec\theta\csc\theta(\mathcal{P}^{\bot}\circ\mathcal{P}_{0}^{\bot}\circ(\mathbf{Id}-\mathcal{P})\circ\mathcal{P}_{0})u$
$\displaystyle=\sec\theta\csc\theta(\mathcal{P}^{\bot}\circ(\mathbf{Id}-\mathcal{P}_{0})\circ\mathcal{P}\circ\mathcal{P}_{0})u$
$\displaystyle=\cot\theta(\mathcal{P}^{\bot}\circ(\mathbf{Id}-\mathcal{P}_{0}))u$
$\displaystyle=-\cot\theta(\mathcal{P}^{\bot}\circ\mathcal{P}_{0})u=\cos^{2}\theta\
v,$

$\theta\in\text{Arg}(P^{\bot},Q_{0}^{\bot})$, $v\in P_{\theta}^{\bot}$ and

(2.17)
$\Phi_{1,\theta}:u\mapsto-\sec\theta\csc\theta(\mathcal{P}^{\bot}\circ\mathcal{P}_{0})u$

is injective from $P_{\theta}$ into $P_{\theta}^{\bot}$, which implies
$m_{\theta}^{\bot}\geq m_{\theta}$. (2.16) then becomes

(2.18) $\sec\theta\ \mathcal{P}_{0}u=\cos\theta\ u-\sin\theta\
\Phi_{1,\theta}(u).$

Similarly, if $\theta\in(0,\pi/2)$ is a Jordan angle between $P^{\bot}$ and
$Q_{0}^{\bot}$, then $\theta\in\text{Arg}(P,Q_{0})$ and

(2.19)
$\Phi_{2,\theta}:v\mapsto-\sec\theta\csc\theta(\mathcal{P}\circ\mathcal{P}_{0}^{\bot})v$

is also injective from $P_{\theta}^{\bot}$ into $P_{\theta}$, which implies

(2.20) $\sec\theta\ \mathcal{P}_{0}^{\bot}v=\cos\theta\ v-\sin\theta\
\Phi_{2,\theta}(v)$

and $m_{\theta}\geq m_{\theta}^{\bot}$. Therefore
$m_{\theta}=m_{\theta}^{\bot}$, and both $\Phi_{1,\theta}$ and
$\Phi_{2,\theta}$ are isomorphisms.

For any $\theta\in\text{Arg}(P,Q_{0})\cap(0,\pi/2)$,
$P_{\theta}^{\bot}=\Phi_{1,\theta}(P_{\theta})\subset R_{\theta}$ gives
$P_{\theta}\oplus P_{\theta}^{\bot}\subset R_{\theta}$. On the other hand,
$P_{\theta}\cap(Q_{0})_{\theta}\subset(P\cap Q_{0})\cap P_{\theta}=P_{0}\cap
P_{\theta}=(0)$ implies $\dim R_{\theta}=\dim
P_{\theta}+\dim(Q_{0})_{\theta}=2m_{\theta}=\dim P_{\theta}+\dim
P_{\theta}^{\bot}$, then $R_{\theta}=P_{\theta}\oplus P_{\theta}^{\bot}$. Now
we define $\Phi_{\theta}:R_{\theta}\rightarrow R_{\theta}$

(2.21) $u+v\mapsto\Phi_{1,\theta}(u)+\Phi_{2,\theta}(v)\qquad\forall u\in
P_{\theta},v\in P_{\theta}^{\bot}.$

Then $\Phi_{\theta}$ is an isometric automorphism,
$\Phi_{\theta}(P_{\theta})=P_{\theta}^{\bot}$,
$\Phi_{\theta}(P_{\theta}^{\bot})=P_{\theta}$ and (2.11) immediately follows
from (2.18) and (2.20). For any $u\in P_{\theta}$, (2.17) and (2.19) implies

$\displaystyle\Phi_{\theta}^{2}(u)$
$\displaystyle=\sec^{2}\theta\csc^{2}\theta(\mathcal{P}\circ\mathcal{P}_{0}^{\bot}\circ\mathcal{P}^{\bot}\circ\mathcal{P}_{0})u$
$\displaystyle=\sec^{2}\theta\csc^{2}\theta(\mathcal{P}\circ\mathcal{P}_{0}^{\bot}\circ(\mathbf{Id}-\mathcal{P})\circ\mathcal{P}_{0})u$
$\displaystyle=-\sec^{2}\theta\csc^{2}\theta(\mathcal{P}\circ\mathcal{P}_{0}^{\bot}\circ\mathcal{P}\circ\mathcal{P}_{0})u$
$\displaystyle=-\csc^{2}\theta(\mathcal{P}\circ\mathcal{P}_{0}^{\bot})u=\csc^{2}\theta(\mathcal{P}\circ(\mathbf{Id}-\mathcal{P}_{0})u)$
$\displaystyle=-u$

and similarly $\Phi_{\theta}^{2}(v)=-v$ for each $v\in P_{\theta}^{\bot}$.
Hence $\Phi_{\theta}^{2}=-\mathbf{Id}$.

It remains to prove $m_{\pi/2}=m_{\pi/2}^{\bot}$. By (2.13),

(2.22) $\displaystyle m_{\pi/2}^{\bot}$ $\displaystyle=\dim
P_{\pi/2}^{\bot}=\dim(P^{\bot}\cap Q_{0})$
$\displaystyle=\dim(Q_{0})_{\pi/2}=\dim P_{\pi/2}$ $\displaystyle=m_{\pi/2}.$

∎

Denote

(2.23)
$r:=\sum_{\theta\in\text{Arg}(P,Q_{0})\cap(0,\pi/2]}m_{\theta}=\sum_{\theta\in\text{Arg}(P^{\bot},Q_{0}^{\bot})\cap(0,\pi/2]}m_{\theta}^{\bot}$

then $0\in\text{Arg}(P,Q_{0})$ if and only if $r<m$, and $m_{0}=m-r$.
Similarly $0\in\text{Arg}(P^{\bot},Q_{0}^{\bot})$ if and only if $r<n$, and
$m_{0}^{\bot}=n-r$.

It is well-known that $\mathbb{G}_{m,n}$ can be viewed as a submanifold of
some Euclidean space via the Plücker embedding. The restriction of the
Euclidean inner product on $\mathbb{G}_{m,n}$ is denoted by
$w:\mathbb{G}_{m,n}\times\mathbb{G}_{m,n}\rightarrow{\tenmsb R}$. Let
$\\{\varepsilon_{1},\cdots,\varepsilon_{m}\\}$ and $\\{u_{1},\cdots,u_{m}\\}$
denote oriented orthonormal bases of $Q_{0}$ and $P$, respectively, then

(2.24) $\displaystyle w(P,Q_{0})$ $\displaystyle=\langle
u_{1}\wedge\cdots\wedge
u_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=\left|\begin{array}[]{ccc}\langle
u_{1},\varepsilon_{1}\rangle&\cdots&\langle u_{1},\varepsilon_{m}\rangle\\\
&\cdots&\\\ \langle u_{m},\varepsilon_{1}\rangle&\cdots&\langle
u_{m},\varepsilon_{m}\rangle\end{array}\right|$
$\displaystyle=\left|\begin{array}[]{ccc}\langle\mathcal{P}_{0}u_{1},\varepsilon_{1}\rangle&\cdots&\langle\mathcal{P}_{0}u_{1},\varepsilon_{m}\rangle\\\
&\cdots&\\\
\langle\mathcal{P}_{0}u_{m},\varepsilon_{1}\rangle&\cdots&\langle\mathcal{P}_{0}u_{m},\varepsilon_{m}\rangle\end{array}\right|$
$\displaystyle=\langle\mathcal{P}_{0}u_{1}\wedge\cdots\wedge\mathcal{P}_{0}u_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle.$

Note that the definition of $w(P,Q_{0})$ does not depend on the choices of
oriented orthonormal bases of $P$ and $Q_{0}$.

As shown above, one can choose an orthonormal basis $\\{u_{1},\cdots,u_{m}\\}$
of $P$, such that $P=u_{1}\wedge\cdots\wedge u_{m}$, and each $u_{\alpha}$ is
an angle direction of $P$ relative to $Q_{0}$. More precisely,

(2.25)
$(\mathcal{P}\circ\mathcal{P}_{0})u_{\alpha}=\cos^{2}\theta_{\alpha}u_{\alpha},$

where $\theta_{\alpha}$ is a Jordan angle between $P$ and $Q_{0}$. Hence

(2.26)
$\displaystyle\langle\mathcal{P}_{0}u_{\alpha},\mathcal{P}_{0}u_{\beta}\rangle$
$\displaystyle=\langle\mathcal{P}_{0}u_{\alpha},u_{\beta}\rangle=\langle(\mathcal{P}\circ\mathcal{P}_{0})u_{\alpha},u_{\beta}\rangle$
$\displaystyle=\cos^{2}\theta_{\alpha}\langle
u_{\alpha},u_{\beta}\rangle=\cos^{2}\theta_{\alpha}\delta_{\alpha\beta}.$

If $\pi/2\in\text{Arg}(P,Q_{0})$, then there exists $\alpha$, such that
$\theta_{\alpha}=\pi/2$, which implies $\mathcal{P}_{0}u_{\alpha}=0$, and
moreover

(2.27)
$w(P,Q_{0})=\langle\mathcal{P}_{0}u_{1}\wedge\cdots\wedge\mathcal{P}_{0}u_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle=0.$

Otherwise, by (2.26),
$\\{\varepsilon^{\prime}_{\alpha}=\frac{\mathcal{P}_{0}u_{\alpha}}{|\mathcal{P}_{0}u_{\alpha}|}=\sec\theta_{\alpha}(\mathcal{P}_{0}u_{\alpha}):1\leq\alpha\leq
m\\}$ is an orthonormal basis of $Q_{0}$, thus
$Q_{0}=\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}$ or
$-\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}$. If
$Q_{0}=\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}$,
then

(2.28) $\displaystyle w(P,Q_{0})$
$\displaystyle=\langle\mathcal{P}_{0}u_{1}\wedge\cdots\wedge\mathcal{P}_{0}u_{m},\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}\rangle$
$\displaystyle=\langle\cos\theta_{1}\varepsilon^{\prime}_{1}\wedge\cdots\wedge\cos\theta_{m}\varepsilon^{\prime}_{m},\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}\rangle$
$\displaystyle=\prod_{\alpha}\cos\theta_{\alpha}\langle\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m},\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}\rangle=\prod_{\alpha}\cos\theta_{\alpha};$

otherwise
$Q_{0}=-\varepsilon^{\prime}_{1}\wedge\cdots\wedge\varepsilon^{\prime}_{m}$
and a similar calculation shows

(2.29) $w(P,Q_{0})=-\prod_{\alpha}\cos\theta_{\alpha}.$

In summary,

(2.30) $|w(P,Q_{0})|=\prod_{\alpha}\cos\theta_{\alpha}$

and $w(P,Q_{0})>0$ if and only if $\mathcal{P}_{0}|_{P}$ is an orientation
preserving map from $P$ onto $Q_{0}$.

######  Lemma 2.2.

If $w(P,Q_{0})>0$, let

$\pi/2>\theta_{1}\geq\cdots\geq\theta_{r}>\theta_{r+1}=\cdots=\theta_{m}=0$

be the Jordan angles between $P$ and $Q_{0}$, then there exist an orthonormal
basis $\\{\varepsilon_{1},\cdots,\varepsilon_{m}\\}$ of $Q_{0}$, an
orthonormal basis $\\{u_{1},\cdots,u_{m}\\}$ of $P$ and an orthonormal basis
$\\{v_{1},\cdots,v_{n}\\}$ of $P^{\bot}$, such that
$Q_{0}=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}$,
$P=u_{1}\wedge\cdots\wedge u_{m}$,
$\mathcal{P}_{0}u_{\alpha}=\cos\theta_{\alpha}\varepsilon_{\alpha}$ for each
$1\leq\alpha\leq m$ and $\mathcal{P}_{0}v_{i}=-\sin\theta_{i}\varepsilon_{i}$
for each $1\leq i\leq n$. Here we additionally assume $\theta_{i}:=0$ and
$\varepsilon_{i}:=0$ for any $m+1\leq i\leq n$. In particular, for each
$1\leq\alpha\leq r$, $v_{\alpha}=\Phi_{\theta_{\alpha}}(u_{\alpha})$, with
$\Phi_{\theta_{\alpha}}$ denoting the anti-involutive automorphism associated
to $\theta_{\alpha}$.

###### Proof.

Let $\\{u_{1},\cdots,u_{m}\\}$ be an orthonormal basis of $P$, such that
$P=u_{1}\wedge\cdots\wedge u_{m}$ and
$(\mathcal{P}\circ\mathcal{P}_{0})u_{\alpha}=\cos^{2}\theta_{\alpha}u_{\alpha}$.
Putting

(2.31)
$\varepsilon_{\alpha}:=\frac{\mathcal{P}_{0}u_{\alpha}}{|\mathcal{P}_{0}u_{\alpha}|}=\sec\theta_{\alpha}(\mathcal{P}_{0}u_{\alpha})\in
Q_{0}\qquad\forall 1\leq\alpha\leq m,$

then (2.26) gives
$\langle\varepsilon_{\alpha},\varepsilon_{\beta}\rangle=\delta_{\alpha\beta}$
and hence $\\{\varepsilon_{1},\cdots,\varepsilon_{m}\\}$ is an orthonormal
basis of $Q_{0}$. $w(P,Q_{0})>0$ implies that $\mathcal{P}_{0}|_{P}$ is an
orientation preserving map, thus
$Q_{0}=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}$.

For each $1\leq\alpha\leq r$, let
$v_{\alpha}:=\Phi_{\theta_{\alpha}}(u_{\alpha})$. By Lemma 2.1,

(2.32)
$\varepsilon_{\alpha}=\sec\theta_{\alpha}(\mathcal{P}_{0}u_{\alpha})=\cos\theta_{\alpha}u_{\alpha}-\sin\theta_{\alpha}v_{\alpha}$

and

(2.33) $\langle v_{\alpha},v_{\beta}\rangle=\delta_{\alpha\beta}\qquad\forall
1\leq\alpha,\beta\leq r.$

Let $\\{v_{r+1},\cdots,v_{n}\\}$ be an orthonormal basis of
$P_{0}^{\bot}=P^{\bot}\cap Q_{0}^{\bot}$, then $\\{v_{1},\cdots,v_{n}\\}$ is
an orthonormal basis of $P^{\bot}$. Applying (2.32) gives

$\displaystyle\varepsilon_{\alpha}$
$\displaystyle=\mathcal{P}_{0}\varepsilon_{\alpha}=\cos\theta_{\alpha}(\mathcal{P}_{0}u_{\alpha})-\sin\theta_{\alpha}(\mathcal{P}_{0}v_{\alpha})$
$\displaystyle=\cos^{2}\theta_{\alpha}\varepsilon_{\alpha}-\sin\theta_{\alpha}(\mathcal{P}_{0}v_{\alpha}),$

i.e.

(2.34)
$\mathcal{P}_{0}v_{\alpha}=-\sin\theta_{\alpha}\varepsilon_{\alpha}\qquad\forall
1\leq\alpha\leq r.$

On the other hand, $v_{i}\in Q_{0}^{\bot}$ implies $\mathcal{P}_{0}v_{i}=0$
for each $r+1\leq i\leq n$. Hence

(2.35) $\mathcal{P}_{0}v_{i}=-\sin\theta_{i}\varepsilon_{i}\qquad\forall 1\leq
i\leq n.$

∎

Let $M$ be an $n$-dimensional submanifold in ${\tenmsb R}^{n+m}$ and $Q_{0}$
be an fixed $m$-dimensional subspace in ${\tenmsb R}^{n+m}$. Denote by $TM$
and $NM$ the tangent bundle and the normal bundle along $M$, respectively. Let
$\theta$ be a $[0,\pi/2]$-valued smooth function on $M$, if
$\theta(p)\in\text{Arg}(N_{p}M,Q_{0})$
($\theta(p)\in\text{Arg}(T_{p}M,Q_{0}^{\bot})$), we say $\theta$ is a normal
(tangent) Jordan angle function of $M$ relative to $Q_{0}$. Denote by
$\text{Arg}^{N}$ ($\text{Arg}^{T}$) the set consisting of all the normal
(tangent) Jordan angle functions of $M$ relative to $Q_{0}$. If $\theta$ is a
smooth function on $M$ that is nonzero everywhere, then Lemma 2.1 implies
$\theta\in\text{Arg}^{N}$ if and only if $\theta\in\text{Arg}^{T}$.

Denote

(2.36) $\displaystyle N_{\theta}M$ $\displaystyle:=\\{\nu\in N_{p}M:p\in
M,\nu\text{ is an angle direction associated to }\theta(p)\\},$ $\displaystyle
T_{\theta}M$ $\displaystyle:=\\{v\in T_{p}M:p\in M,v\text{ is an angle
direction associated to }\theta(p)\\}.$

Let $\mathcal{P}_{0}$ and $\mathcal{P}_{0}^{\bot}$ be orthogonal projections
onto $Q_{0}$ and $Q_{0}^{\bot}$, $(\cdot)^{T}$ and $(\cdot)^{N}$ denote
orthogonal projections onto $T_{p}M$ and $N_{p}M$, respectively. Then $\nu\in
N_{\theta}M\cap N_{p}M$ if and only if

(2.37) $(\mathcal{P}_{0}\nu)^{N}=\cos^{2}\theta(p)\nu$

and similarly $u\in T_{\theta}M\cap T_{p}M$ if and only if

(2.38) $(\mathcal{P}_{0}^{\bot}u)^{T}=\cos^{2}\theta(p)u.$

Let $m_{\theta}^{N}(p):=\dim(N_{\theta}M\cap N_{p}M)$,
$m_{\theta}^{T}(p):=\dim(T_{\theta}M\cap T_{p}M)$ for every $p\in M$, then
$m_{\theta}^{N}$ and $m_{\theta}^{T}$ are both ${\tenmsb Z}^{+}$-valued
functions on $M$.

######  Lemma 2.3.

Let $\theta$ be a normal (tangent) Jordan angle function of $M$ relative to
$Q_{0}$. If $m_{\theta}^{N}$ ($m_{\theta}^{T}$) is a constant function on $M$,
then $N_{\theta}M$ ($T_{\theta}M$) is a smooth subbundle of $NM$ ($TM$). In
this case, $N_{\theta}M$ ($T_{\theta}M$) is said to be a normal (tangent)
angle space distribution associated to $\theta$.

###### Proof.

We only give the proof for the tangential case, because it is similar for the
normal case.

For any $p_{0}\in M$, let $U$ be a coordinate chart around $p_{0}$ and
$\\{e_{1},\cdots,e_{n}\\}$ be a orthonormal tangent frame field on $U$. Denote

(2.39) $A_{ij}(p)=\langle\mathcal{P}_{0}^{\bot}e_{i},e_{j}\rangle\qquad\forall
1\leq i,j\leq n,$

then

$\displaystyle A_{ij}(p)$
$\displaystyle=\langle\mathcal{P}_{0}^{\bot}e_{i},e_{j}\rangle=\langle\mathcal{P}_{0}^{\bot}e_{i},\mathcal{P}_{0}^{\bot}e_{j}\rangle$
$\displaystyle=\langle\mathcal{P}_{0}^{\bot}e_{j},\mathcal{P}_{0}^{\bot}e_{i}\rangle=\langle\mathcal{P}_{0}^{\bot}e_{j},e_{i}\rangle=A_{ji}(p)$

and hence $p\mapsto A(p)$ is a smooth mapping from $U$ into $S_{n}$, the set
of all $n\times n$ real symmetric matrices. For any $u=\sum_{i}u_{i}e_{i}\in
TU$,

$\displaystyle(\mathcal{P}_{0}^{\bot}u)^{T}$
$\displaystyle=\sum_{j}\langle\mathcal{P}_{0}^{\bot}u,e_{j}\rangle
e_{j}=\sum_{i,j}\langle u_{i}(\mathcal{P}_{0}^{\bot}e_{i}),e_{j}\rangle e_{j}$
$\displaystyle=\sum_{i,j}u_{i}A_{ij}e_{j}.$

Thus $(\mathcal{P}_{0}^{\bot}u)^{T}=\cos^{2}\theta(p)u$ if and only if
$A(p)\xi=\cos^{2}\theta(p)\xi$, with $\xi:=(u_{1}\ \cdots\ u_{n})^{T}$.
Furthermore, $\theta\in\text{Arg}^{T}$ if and only if
$\lambda(p):=\cos^{2}\theta(p)$ is a characteristic root of $A(p)$ for every
$p\in U$, and the multiplicity of $\lambda(p)$ equals $m_{\theta}^{T}(p)$. If
$m_{\theta}^{T}\equiv k$, the main theorem in [16] enable us to find smooth
mappings $\xi_{1},\cdots,\xi_{k}:V\rightarrow{\tenmsb R}^{n}$, where $V$ is a
neighborhood of $p_{0}$, such that for every $p\in V$,
$A(p)\xi_{i}=\lambda(p)\xi_{i}$ holds for $1\leq i\leq k$ and
$\\{\xi_{1}(p),\cdots,\xi_{m}(p)\\}$ is linear independent. Denote
$\xi_{i}=(u_{i1},\cdots,u_{in})^{T}$ and let

$X_{i}=\sum_{j}u_{ij}e_{j},$

then $X_{1},\cdots,X_{m}$ are smooth tangent vector fields on $V$ and
$T_{\theta}M\cap T_{p}M$ is spanned by $X_{1}(p),\cdots,X_{m}(p)$ for any
$p\in V$. Finally the arbitrariness of $p_{0}$ ensures that $T_{\theta}M$ is a
smooth distribution on $M$.

∎

##  3\. Subharmonic functions

Let $\bar{M}^{n+m}$ be a Riemannian manifold, and
$M^{n}\rightarrow\bar{M}^{n+m}$ be an isometric immersion. The second
fundamental form $B$ is a pointwise symmetric bilinear form on $T_{p}M$ ($p\in
M$) with values in $N_{p}M$ defined by

$B_{XY}=(\overline{\nabla}_{X}Y)^{N}$

with $\overline{\nabla}$ the Levi-Civita connection on $\bar{M}$. The induced
connections on $TM$ and $NM$ are defined by

$\nabla_{X}Y=(\overline{\nabla}_{X}Y)^{T},\qquad\nabla_{X}\nu=(\overline{\nabla}_{X}\nu)^{N}.$

Here $X,Y$ are smooth sections of $TM$ and $\nu$ denotes a smooth sections of
$NM$. The second fundamental form, the curvature tensor of the submanifold,
the curvature tensor of the normal bundle and the curvature tensor of the
ambient manifold satisfy the Gauss equations, the Codazzi equations and the
Ricci equations (see [20] for details).

The trace of the second fundamental form gives a normal vector field $H$ on
$M$, which is called the mean curvature vector field. If $\nabla H\equiv 0$,
then we say $M$ has parallel mean curvature. Moreover if $H\equiv 0$, $M$ is
called a minimal submanifold in $\bar{M}$.

Now we consider an $n$-dimensional oriented submanifold $M$ in ${\tenmsb
R}^{n+m}$. Let $\\{e_{1},\cdots,e_{n}\\}$ be a local orthonormal tangent frame
field, $\\{\nu_{1},\cdots,\nu_{m}\\}$ be a local orthonormal normal frame
field on $M$, and

$h_{\alpha,ij}:=\langle B_{e_{i}e_{j}},\nu_{\alpha}\rangle$

be the coefficients of the second fundamental form $B$. We shall use the
summation convention and agree on the ranges of indices

$1\leq i,j,k,l\leq n,\qquad 1\leq\alpha,\beta,\gamma,\delta\leq m.$

The normal Gauss map $\gamma:M\rightarrow\mathbb{G}_{m,n}$ is defined by

$\gamma(p)=N_{p}M\in\mathbb{G}_{m,n}$

via parallel translation in ${\tenmsb R}^{n+m}$ for every $p\in M$. Let
$Q_{0}$ be a fixed point in $\mathbb{G}_{m,n}$ and define

(3.1) $\displaystyle
w:=w(\cdot,Q_{0})\circ\gamma=\langle\nu_{1}\wedge\cdots\wedge\nu_{m},Q_{0}\rangle.$

By the Codazzi equations, it is not hard to get basic formulas for the
function $w$ as follows.

######  Proposition 3.1.

([6][21]) Let $M$ be a submanifold in ${\tenmsb R}^{n+m}$, then

(3.2) $\nabla_{e_{i}}w=-h_{\alpha,ij}\langle\nu_{\alpha j},Q_{0}\rangle$

with

(3.3) $\nu_{\alpha j}:=\nu_{1}\wedge\cdots\wedge
e_{j}\wedge\cdots\wedge\nu_{m}$

that is obtained by replacing $\nu_{\alpha}$ by $e_{j}$ in
$\nu_{1}\wedge\cdots\wedge\nu_{m}$. Moreover if $M$ has parallel mean
curvature, then

(3.4) $\Delta w=-|B|^{2}w+\sum_{i}\sum_{\alpha\neq\beta,j\neq
k}h_{\alpha,ij}h_{\beta,ik}\langle\nu_{\alpha j,\beta k},Q_{0}\rangle.$

with

(3.5) $\nu_{\alpha j,\beta k}:=\nu_{1}\wedge\cdots\wedge
e_{j}\wedge\cdots\wedge e_{k}\wedge\cdots\wedge\nu_{m}$

that is obtained by replacing $\nu_{\alpha}$ by $e_{j}$ and $\nu_{\beta}$ by
$e_{k}$ in $\nu_{1}\wedge\cdots\wedge\nu_{m}$, respectively.

Let $p\in M$ satisfying $w(p)=w(N_{p}M,Q_{0})>0$. Denote by

$\pi/2>\theta_{1}\geq\cdots\geq\theta_{r}>\theta_{r+1}=\cdots=\theta_{m}=0$

the Jordan angles between $N_{p}M$ and $Q_{0}$. By Lemma 2.2, one can find an
oriented orthonormal basis $\\{\varepsilon_{1},\cdots,\varepsilon_{m}\\}$ of
$Q_{0}$, an oriented orthonormal basis $\\{\nu_{1},\cdots,\nu_{m}\\}$ of
$N_{p}M$ and an orthonormal basis $\\{e_{1},\cdots,e_{n}\\}$ of $T_{p}M$, such
that $\mathcal{P}_{0}\nu_{\alpha}=\cos\theta_{\alpha}\varepsilon_{\alpha}$ and
$\mathcal{P}_{0}e_{i}=-\sin\theta_{i}\varepsilon_{i}$. Especially

(3.6) $e_{\alpha}=\Phi_{\theta_{\alpha}}(\nu_{\alpha})\qquad\forall
1\leq\alpha\leq r.$

Putting

(3.7) $\lambda_{\alpha}:=\tan\theta_{\alpha},$

then

(3.8) $\displaystyle\langle\nu_{\alpha j},Q_{0}\rangle$
$\displaystyle=\langle\nu_{1}\wedge\cdots\wedge
e_{j}\wedge\cdots\wedge\nu_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=\langle\mathcal{P}_{0}\nu_{1}\wedge\cdots\wedge\mathcal{P}_{0}e_{j}\wedge\cdots\wedge\mathcal{P}_{0}\nu_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=\langle\cos\theta_{1}\varepsilon_{1}\wedge\cdots\wedge(-\sin\theta_{j}\varepsilon_{j})\wedge\cdots\wedge\cos\theta_{m}\varepsilon_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=-\delta_{\alpha
j}\tan\theta_{\alpha}\big{(}\prod_{\gamma}\cos\theta_{\gamma})$
$\displaystyle=-\delta_{\alpha j}\lambda_{\alpha}w(p)$

and

(3.9) $\displaystyle\langle\nu_{\alpha j,\beta k},Q_{0}\rangle$
$\displaystyle=$ $\displaystyle\langle\nu_{1}\wedge\cdots\wedge
e_{j}\wedge\cdots\wedge
e_{k}\wedge\cdots\wedge\nu_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=$
$\displaystyle\langle\mathcal{P}_{0}\nu_{1}\wedge\cdots\wedge\mathcal{P}_{0}e_{j}\wedge\cdots\wedge\mathcal{P}_{0}e_{k}\wedge\cdots\wedge\mathcal{P}_{0}\nu_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=$
$\displaystyle\langle\cos\theta_{1}\varepsilon_{1}\wedge\cdots\wedge(-\sin\theta_{j}\varepsilon_{j})\wedge\cdots\wedge(-\sin\theta_{k}\varepsilon_{k})\wedge\cdots\wedge\cos\theta_{m}\varepsilon_{m},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{m}\rangle$
$\displaystyle=$ $\displaystyle(\delta_{\alpha j}\delta_{\beta
k}-\delta_{\alpha k}\delta_{\beta
j})\tan\theta_{\alpha}\tan\theta_{\beta}\big{(}\prod_{\gamma}\cos\theta_{\gamma})$
$\displaystyle=$ $\displaystyle(\delta_{\alpha j}\delta_{\beta
k}-\delta_{\alpha k}\delta_{\beta j})\lambda_{\alpha}\lambda_{\beta}w(p).$

Substituting (3.8) and (3.9) into (3.2) and (3.4), respectively, we obtain

(3.10)
$w^{-1}\nabla_{e_{i}}w=\sum_{\alpha}\lambda_{\alpha}h_{\alpha,i\alpha},$
(3.11) $w^{-1}\Delta
w=-|B|^{2}+\sum_{i}\sum_{\alpha\neq\beta}\lambda_{\alpha}\lambda_{\beta}(h_{\alpha,i\alpha}h_{\beta,i\beta}-h_{\alpha,i\beta}h_{\beta,i\alpha}).$

Let

(3.12) $v:=w^{-1},$

then

(3.13) $\displaystyle v^{-1}\Delta v$ $\displaystyle=w\big{(}-w^{-2}\Delta
w+2w^{-3}|\nabla w|^{2}\big{)}$
$\displaystyle=|B|^{2}+2\sum_{i,\alpha}\lambda_{\alpha}^{2}h_{\alpha,i\alpha}^{2}+\sum_{i}\sum_{\alpha\neq\beta}\lambda_{\alpha}\lambda_{\beta}(h_{\alpha,i\alpha}h_{\beta,i\beta}+h_{\alpha,i\beta}h_{\beta,i\alpha})$

Observing that $\lambda_{\alpha}\lambda_{\beta}=0$ whenever $\alpha>r$ or
$\beta>r$, we group the terms of (3.14) according to the different types of
the indices of the coefficients of the second fundamental form as follows.

(3.14) $\displaystyle v^{-1}\Delta v=$
$\displaystyle\sum_{\alpha}\sum_{i,j>r}h_{\alpha,ij}^{2}+\sum_{i>r}I_{i}+\sum_{i>r,1\leq\alpha<\beta\leq
r}II_{i\alpha\beta}$ $\displaystyle+\sum_{1\leq\alpha<\beta<\gamma\leq
r}III_{\alpha\beta\gamma}+\sum_{1\leq\alpha\leq r}IV_{\alpha}$

where

(3.15) $I_{i}=\sum_{1\leq\alpha\leq
r}(2+2\lambda_{\alpha}^{2})h_{\alpha,i\alpha}^{2}+\sum_{1\leq\alpha,\beta\leq
r,\alpha\neq\beta}\lambda_{\alpha}\lambda_{\beta}h_{\alpha,i\alpha}h_{\beta,i\beta},$
(3.16)
$II_{i\alpha\beta}=2h_{\alpha,i\beta}^{2}+2h_{\beta,i\alpha}^{2}+2\lambda_{\alpha}\lambda_{\beta}h_{\alpha,i\beta}h_{\beta,i\alpha},$
(3.17) $\displaystyle III_{\alpha\beta\gamma}=$ $\displaystyle
2h_{\alpha,\beta\gamma}^{2}+2h_{\beta,\gamma\alpha}^{2}+2h_{\gamma,\alpha\beta}^{2}$
$\displaystyle+2\lambda_{\alpha}\lambda_{\beta}h_{\alpha,\beta\gamma}h_{\beta,\gamma\alpha}+2\lambda_{\beta}\lambda_{\gamma}h_{\beta,\gamma\alpha}h_{\gamma,\alpha\beta}+2\lambda_{\gamma}\lambda_{\alpha}h_{\gamma,\alpha\beta}h_{\alpha,\beta\gamma}$

and

(3.18) $\displaystyle IV_{\alpha}=$
$\displaystyle(1+2\lambda_{\alpha}^{2})h_{\alpha,\alpha\alpha}^{2}+\sum_{1\leq\beta\leq
r,\beta\neq\alpha}\big{(}h_{\alpha,\beta\beta}^{2}+(2+2\lambda_{\beta}^{2})h_{\beta,\alpha\beta}^{2}\big{)}$
$\displaystyle+\sum_{1\leq\beta,\gamma\leq
r,\beta\neq\gamma}\lambda_{\beta}\lambda_{\gamma}h_{\beta,\alpha\beta}h_{\gamma,\alpha\gamma}+2\sum_{1\leq\beta\leq
r,\beta\neq\alpha}\lambda_{\alpha}\lambda_{\beta}h_{\alpha,\beta\beta}h_{\beta,\alpha\beta}.$

Note that the first term vanishes whenever $n=r$, the second term vanishes
whenever $n=r$ or $r\leq 1$ and the third term vanishes whenever $r\leq 2$.

It is easily seen that

(3.19) $I_{i}=\big{(}\sum_{1\leq\alpha\leq
r}\lambda_{\alpha}h_{\alpha,i\alpha}\big{)}^{2}+\sum_{1\leq\alpha\leq
r}(2+\lambda_{\alpha}^{2})h_{\alpha,i\alpha}^{2}\geq 2\sum_{1\leq\alpha\leq
r}h_{\alpha,i\alpha}^{2}.$

Hence $I_{i}=0$ if and only if $h_{\alpha,i\alpha}=0$ for any $1\leq\alpha\leq
r$.

######  Lemma 3.1.

If $v(p)\leq 3$, then $II_{i\alpha\beta}\geq 0$ for any $i>r$,
$1\leq\alpha<\beta\leq r$, and $\sum_{i>r,1\leq\alpha<\beta\leq
r}II_{i\alpha\beta}=0$ if and only if one of the following two cases occurs:
(a) $h_{\alpha,i\beta}=h_{\beta,i\alpha}=0$ for every $i,\alpha,\beta$; (b)
$r=2$, $\lambda_{1}=\lambda_{2}=\sqrt{2}$ and $h_{1,i2}=-h_{2,i1}$ for each
$i$.

###### Proof.

In conjunction with (3.12), (2.30) and (3.7) we have

$\displaystyle v^{2}$
$\displaystyle=w^{-2}=\prod_{\alpha}\sec^{2}\theta_{\alpha}=\prod_{1\leq\alpha\leq
r}(1+\tan^{2}\theta_{\alpha})$ $\displaystyle=\prod_{1\leq\alpha\leq
r}(1+\lambda_{\alpha}^{2})\geq(1+\lambda_{1}^{2})(1+\lambda_{2}^{2})$
$\displaystyle=1+\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{1}^{2}+\lambda_{2}^{2}\geq
1+\lambda_{1}^{2}\lambda_{2}^{2}+2\lambda_{1}\lambda_{2}$
$\displaystyle=(1+\lambda_{1}\lambda_{2})^{2}\geq(1+\lambda_{\alpha}\lambda_{\beta})^{2}.$

Thus $v\leq 3$ implies $\lambda_{\alpha}\lambda_{\beta}\leq 2$, and equality
holds if and only if $r=2$ and $\lambda_{1}=\lambda_{2}=\sqrt{2}$.

By (3.16),

(3.20)
$II_{i\alpha\beta}=\lambda_{\alpha}\lambda_{\beta}(h_{\alpha,i\beta}+h_{\beta,i\alpha})^{2}+(2-\lambda_{\alpha}\lambda_{\beta})(h_{\alpha,i\beta}^{2}+h_{\beta,i\alpha}^{2})\geq
0.$

Equality holds whenever one and only one of the two cases occurs: (i)
$h_{\alpha,i\beta}=h_{\beta,i\alpha}=0$; (ii)
$\lambda_{\alpha}\lambda_{\beta}=2$ and
$h_{\alpha,i\beta}=-h_{\beta,i\alpha}\neq 0$. Thereby the conclusion follows.

∎

######  Lemma 3.2.

If $v(p)\leq 3$, then $III_{\alpha\beta\gamma}\geq 0$ for any
$1\leq\alpha<\beta<\gamma\leq r$ and the equality holds if and only if
$h_{\alpha,\beta\gamma}=h_{\beta,\gamma\alpha}=h_{\gamma,\alpha\beta}=0$.

###### Proof.

Let

(3.21) $x:=h_{\alpha,\beta\gamma},\ y:=h_{\beta,\gamma\alpha},\
z:=h_{\gamma,\alpha\beta}$

and

(3.22) $a:=\lambda_{\alpha},\ b:=\lambda_{\beta},\ c:=\lambda_{\gamma}.$

Then it is sufficient to show that the quadratic form

(3.23) $III_{\alpha\beta\gamma}=2x^{2}+2y^{2}+2z^{2}+2abxy+2bcyz+2cazx$

is positive definite whenever $a\geq b\geq c>0$ and

(3.24) $\displaystyle(1+a^{2})(1+b^{2})(1+c^{2})$
$\displaystyle=(1+\lambda_{\alpha}^{2})(1+\lambda_{\beta}^{2})(1+\lambda_{\gamma}^{2})$
$\displaystyle\leq\prod_{1\leq\delta\leq
r}(1+\lambda_{\delta}^{2})=v(p)^{2}\leq 9.$

It is easily-seen that

(3.25)
$III_{\alpha\beta\gamma}=(ax+by+cz)^{2}+(2-a^{2})x^{2}+(2-b^{2})y^{2}+(2-c^{2})z^{2}.$

If $a^{2}\leq 2$, then (3.24) implies $2>b^{2}\geq c^{2}>0$. Hence
$III_{\alpha\beta\gamma}\geq 0$ and equality holds if and only if
$ax+by+cz=y=z=0$, which is equivalent to $x=y=z=0$.

If $a^{2}>2$, again using (3.24) gives $2>b^{2}\geq c^{2}>0$. Putting
$s:=by+cz$, then by the Cauchy-Schwarz inequality,

(3.26) $\displaystyle s^{2}$ $\displaystyle=(by+cz)^{2}$
$\displaystyle=\Big{(}\frac{b}{\sqrt{2-b^{2}}}\sqrt{2-b^{2}}y+\frac{c}{\sqrt{2-c^{2}}}\sqrt{2-c^{2}}z\Big{)}^{2}$
$\displaystyle\leq\Big{(}\frac{b^{2}}{2-b^{2}}+\frac{c^{2}}{2-c^{2}}\Big{)}\Big{[}(2-b^{2})y^{2}+(2-c^{2})z^{2}\Big{]}$

and equality holds if and only if
$\big{(}\frac{b}{\sqrt{2-b^{2}}},\frac{c}{\sqrt{2-c^{2}}}\big{)}$ and
$(\sqrt{2-b^{2}}y,\sqrt{2-c^{2}}z)$ is linear independent, i.e.

(3.27) $\frac{2-b^{2}}{b}y=\frac{2-c^{2}}{c}z.$

Hence

(3.28) $\displaystyle III_{\alpha\beta\gamma}$
$\displaystyle=(ax+s)^{2}+(2-a^{2})x^{2}+(2-b^{2})y^{2}+(2-c^{2})z^{2}$
$\displaystyle\geq(ax+s)^{2}+(2-a^{2})x^{2}+\Big{(}\frac{b^{2}}{2-b^{2}}+\frac{c^{2}}{2-c^{2}}\Big{)}^{-1}s^{2}$
$\displaystyle=2x^{2}+2axs+\Big{[}1+\Big{(}\frac{b^{2}}{2-b^{2}}+\frac{c^{2}}{2-c^{2}}\Big{)}^{-1}\Big{]}s^{2}.$

It is well-known that the quadratic form $\sum_{1\leq i,j\leq 2}a_{ij}u_{ij}$
is positive definite if and only if $a_{11}>0$ and $\det(a_{ij})>0$.
Therefore, to show that the right hand side of (3.28) is positive definite, it
is sufficient to prove

$2\Big{[}1+\Big{(}\frac{b^{2}}{2-b^{2}}+\frac{c^{2}}{2-c^{2}}\Big{)}^{-1}\Big{]}-a^{2}>0,$

which is equivalent to

(3.29) $\frac{1}{2-a^{2}}+\frac{1}{2-b^{2}}+\frac{1}{2-c^{2}}<1.$

Denoting $u:=1+a^{2},v:=1+b^{2},w:=1+c^{2}$,

$\Omega=\\{(u,v,w)\in{\tenmsb R}^{3}:u>3>v\geq w>1,uvw\leq 9\\}$

and

$f(u,v,w)=\frac{1}{3-u}+\frac{1}{3-v}+\frac{1}{3-w}=\frac{1}{2-a^{2}}+\frac{1}{2-b^{2}}+\frac{1}{2-c^{2}},$

then it suffices to show $f|_{\Omega}<1$. For a sufficiently small positive
constant $\varepsilon$, let

$\Omega_{\varepsilon}=\\{(u,v,w)\in{\tenmsb R}^{3}:u\geq
3+\varepsilon,3-\varepsilon\geq v,w\geq 1+\varepsilon,uvw\leq 9\\},$

then $\Omega_{\varepsilon}$ is compact and there exists
$(u_{0},v_{0},w_{0})\in\Omega_{\varepsilon}$, such that

(3.30) $f(u_{0},v_{0},w_{0})=\max_{\Omega_{\varepsilon}}f.$

Fix $v_{0}$, then (3.30) implies for any $(u,v_{0},w)\in\Omega_{\varepsilon}$
such that $uw=u_{0}w_{0}$,

$f_{v_{0}}(u,w):=\frac{1}{3-u}+\frac{1}{3-v_{0}}+\frac{1}{3-w}\leq\frac{1}{3-u_{0}}+\frac{1}{3-v_{0}}+\frac{1}{3-w_{0}}.$

Differentiating both sides of $uw=u_{0}w_{0}$ yields
$\frac{du}{u}+\frac{dw}{w}=0$. Hence

$\displaystyle df_{v_{0}}$
$\displaystyle=\frac{du}{(3-u)^{2}}+\frac{dw}{(3-w)^{2}}$
$\displaystyle=\Big{[}-\frac{u}{(3-u)^{2}}+\frac{w}{(3-w)^{2}}\Big{]}\frac{dw}{w}$
$\displaystyle=\frac{(w-u)(9-uw)}{(3-u)^{2}(3-w)^{2}}\frac{dw}{w}$

which means that $f_{v_{0}}(u,w)$ is strictly decreasing in $w$. Therefore
$w_{0}=1+\varepsilon$. Similarly one can derive $v_{0}=1+\varepsilon$.
Therefore $u_{0}=9(1+\varepsilon)^{-2}$ and

$\displaystyle\max_{\Omega_{\varepsilon}}f$
$\displaystyle=f(u_{0},v_{0},w_{0})=\frac{1}{3-9(1+\varepsilon)^{-2}}+\frac{1}{3-(1+\varepsilon)}+\frac{1}{3-(1+\varepsilon)}$
$\displaystyle<\frac{1}{3-9}+\frac{1}{3-1}+\frac{1}{3-1}=\frac{5}{6}<1.$

Then $f|_{\Omega}<1$ follows from
$\Omega\subset\bigcup_{\varepsilon}\Omega_{\varepsilon}$.

Thus $III_{\alpha\beta\gamma}\geq 0$ and equality holds if and only if $x=0$,
$s=by+cz=0$ and (3.27) holds true, which imply $x=y=z=0$ and the conclusion
follows.

∎

In [11], we obtained an estimate for the fourth term as follows.

######  Lemma 3.3.

[11] There exists a positive constant $\varepsilon_{0}$ with the following
property. If $v(p)\leq 3$, then

$IV_{\alpha}\geq\varepsilon_{0}\big{(}h_{\alpha,\alpha\alpha}^{2}+\sum_{1\leq\beta\leq
r,\beta\neq\alpha}(h_{\alpha,\beta\beta}^{2}+2h_{\beta,\alpha\beta}^{2})\big{)}.$

In conjunction with (3.14), (3.15), Lemma 3.1-3.3, $\Delta v(p)\geq 0$
whenever $v(p)\leq 3$. Moreover if $\Delta v(p)=0$ and $|B|^{2}(p)>0$, then
$r=2$, $\lambda_{1}=\lambda_{2}=\sqrt{2}$, $h_{1,i2}=-h_{2,i1}$ for any $i\geq
3$ and the coeffients of $B$ belonging to the other types all vanish. In other
words, $\text{Arg}(N_{p}M,Q_{0})\subset\\{0,\theta_{0}\\}$,
$\text{Arg}(T_{p}M,Q_{0}^{\bot})\subset\\{0,\theta_{0}\\}$ and the
multiplicity of the Jordan angle $\theta_{0}$ equals $2$, where
$\theta_{0}:=\arctan\sqrt{2}$. Let $N_{p,\theta_{0}}M$, $N_{p,0}M$
($T_{p,\theta_{0}}M$, $T_{p,0}M$) denote angle spaces of $N_{p}M$ ($T_{p}M$)
relative to $Q_{0}$ ($Q_{0}^{\bot}$), then
$N_{p,\theta_{0}}M=\text{span}\\{\nu_{1},\nu_{2}\\}$ and
$T_{p,\theta_{0}}M=\text{span}\\{e_{1},e_{2}\\}$ with
$e_{\alpha}=\Phi_{\theta_{0}}(\nu_{\alpha})$ for each $\alpha=1,2$,
$N_{p,0}M=\text{span}\\{\nu_{3},\cdots,\nu_{m}\\}$ and
$T_{p,0}M=\text{span}\\{e_{3},\cdots,e_{n}\\}$.

Denote

(3.31) $S_{\mu\nu}(v):=\langle
B_{v,\Phi_{\theta_{0}}(\mu)},\nu\rangle\qquad\forall\mu,\nu\in
N_{p,\theta_{0}}M,v\in T_{p,0}M,$

Then $(\mu,\nu)\mapsto S_{\mu\nu}$ is a $T_{p,0}^{*}M$-valued bilinear form on
$N_{p,\theta_{0}}M$, where $V^{*}$ denotes the dual space of $V$. In other
words, $S\in N_{p,\theta_{0}}^{*}M\otimes N_{p,\theta_{0}}^{*}M\otimes
T_{p,0}^{*}M$. For any $1\leq\alpha,\beta\leq 2$ and $3\leq i\leq n$,

$S_{\nu_{\alpha}\nu_{\beta}}(e_{i})=\langle
B_{e_{i},\Phi_{\theta}(\nu_{\alpha})},\nu_{\beta}\rangle=\langle
B_{e_{i}e_{\alpha}},\nu_{\beta}\rangle=h_{\beta,i\alpha},$

which implies $S_{\nu_{1}\nu_{1}}=S_{\nu_{2}\nu_{2}}=0$ and
$S_{\nu_{1}\nu_{2}}=-S_{\nu_{2}\nu_{1}}\neq 0$, i.e. $S$ is antisymmetric.

Moreover, observing that $h_{\alpha,ij}=0$ whenever $j\geq 3$ or $\alpha\geq
3$, $h_{\alpha,\beta\gamma}=0$ for any $1\leq\alpha,\beta,\gamma\leq 2$, we
can derive the following result.

######  Proposition 3.2.

Let $M^{n}$ be a submanifold in ${\tenmsb R}^{n+m}$ with parallel mean
curvature, $\gamma:M\rightarrow\mathbb{G}_{m,n}$ be the normal Gauss map and
$Q_{0}$ be a fixed point in $\mathbb{G}_{m,n}$. Put
$v:=w^{-1}(\cdot,Q_{0})\circ\gamma$, then for any $p\in M$, $v(p)\leq 3$
implies $\Delta v\geq 0$ at $p$. Furthermore, if $\Delta v(p)=0$, then one and
only one of following 2 cases must occur: (a) $|B|^{2}(p)=0$. (b)
$|B|^{2}(p)\neq 0$, $\text{Arg}(N_{p}M,Q_{0})\subset\\{0,\theta_{0}\\}$ and
$\text{Arg}(T_{p}M,Q_{0}^{\bot})\subset\\{0,\theta_{0}\\}$ with
$\theta_{0}:=\arctan\sqrt{2}$, and the multiplicity of $\theta_{0}$ is $2$;
let $N_{p,\theta_{0}}M,N_{p,0}M,T_{p,\theta_{0}}M,T_{p,0}M$ be angle spaces
and $\Phi_{\theta_{0}}$ denote the anti-involutive automorphism associated to
$\theta_{0}$, then there exists a nonzero $T_{p,0}^{*}M$-valued antisymmetric
bilinear form $S$ on $N_{p,\theta_{0}}M$, such that

(3.32) $B=\sum_{1\leq\alpha,\beta\leq
2}(S_{\nu_{\alpha}\nu_{\beta}}\odot\omega_{\alpha})\otimes\nu_{\beta},$

where $\\{\nu_{1},\nu_{2}\\}$ is an arbitrary orthonormal basis of
$N_{p,\theta_{0}}M$, $\omega_{\alpha}(v):=\langle
v,\Phi_{\theta_{0}}(\nu_{\alpha})\rangle$ for every $v\in T_{p}M$ and
$\omega\odot\sigma:=\omega\otimes\sigma+\sigma\otimes\omega$.

Remark. Let $V$ be an $n$-dimensional real linear space equipped with an inner
product $\langle\cdot,\cdot\rangle$. A linear subspace $\mathcal{Q}\subset
S^{2}(V^{*})$ of the quadratic functions on $V$ is said to be austere if the
odd symmetric functions of the eigenvalues of any element of $\mathcal{Q}$
with respect to $\langle\cdot,\cdot\rangle$ are all zero, i.e. the nonzero
eigenvalues occur in pairs of opposite signs. Let $\mathcal{Q}$ be an austere
subspace of $S^{2}(V^{*})$, if there is a nonzero vector $v_{0}\in V$, such
that $\varphi(v_{0},v_{0})=0$ and $\varphi(v,w)=0$ for any
$\varphi\in\mathcal{Q}$ and $v,w\in v_{0}^{\bot}$, and $\dim\mathcal{Q}\geq
2$, then we say $\mathcal{Q}$ is simple (see [4]). Let $M^{n}$ be a
submanifold of ${\tenmsb R}^{n+m}$, then for any $p\in M$ and $\nu\in N_{p}M$,
$B_{p}^{\nu}:(v,w)\mapsto\langle B_{v,w},\nu\rangle$ is a quadratic function
on $T_{p}M$ and hence $\mathcal{B}_{p}:=\\{B_{p}^{\nu}:\nu\in N_{p}M\\}$ is a
linear subspace of $S^{2}(T_{p}^{*}M)$. If $\mathcal{B}_{p}$ is (simple)
austere for every $p\in M$, then $M$ is said to be a (simple) austere
submanifold. The concept of austere submanifolds was introduced by Harvey-
Lawson [8] in connection with their foundational work on calibrations. Assume
$M$ to be a submanifold with parallel mean curvature. Let $p\in M$, if
$v(p)\leq 3$, $\Delta v(p)=0$ and $|B|^{2}(p)\neq 0$, then
$\text{Arg}(N_{p}M,Q_{0})\subset\\{0,\theta_{0}\\}$ with
$\theta_{0}:=\arctan\sqrt{2}$ and the multiplicity of $\theta_{0}$ is $2$. Now
we choose $v_{0}\in T_{p,0}M$, such that $\langle
v_{0},v\rangle=S_{\nu_{1}\nu_{2}}(v)$ for any $v\in T_{p,0}M$, then (3.32)
implies $\langle B_{v_{0}v_{0}},\nu\rangle=0$ and $\langle
B_{vw},\nu\rangle=0$ for any $v,w\in v_{0}^{\bot}$, i.e. $\mathcal{B}_{p}$ is
a simple austere subspace of $S^{2}(T^{*}M)$. Moreover, if $v\leq 3$ and
$\Delta v\equiv 0$ on $M$, then $M$ is a simple austere submanifold, which is
congruent to an open subset of a generalized helicoids, due to Bryant’s
structure theorem for simple austere submanifolds (see [4]).

##  4\. Bernstein theorems

We will primarily study a submanifold $M^{n-1}$ in $S^{n+m-1}$, the standard
unit sphere in ${\tenmsb R}^{n+m}$. The cone $CM$ over $M$ is the image of the
map $M\times[0,\infty)\rightarrow{\tenmsb R}^{n+m}$ defined by
$(\mathbf{x},t)\mapsto t\mathbf{x}$, namely

(4.1) $CM=\\{t\mathbf{x}\in{\tenmsb R}^{n+m}:t\in[0,\infty),\mathbf{x}\in
M\\}.$

Obviously $CM$ has a singularity $t=0$ unless $M$ is a subsphere. To avoid the
singularity we consider the truncated cone $CM_{\varepsilon}$ defined by

(4.2) $CM_{\varepsilon}=\\{t\mathbf{x}\in{\tenmsb
R}^{n+m}:t\in(\varepsilon,\infty),\mathbf{x}\in M\\}$

with $\varepsilon>0$.

$M$ and $CM_{\varepsilon}$ share similar geometric properties. At First, the
comparison of the second fundamental form $B$ of $M$ and $B^{c}$ of
$CM_{\varepsilon}$ immediately yields the following result.

######  Proposition 4.1.

([19] p.64) If $CM_{\varepsilon}$ has parallel mean curvature in ${\tenmsb
R}^{n+m}$, then $M$ is a minimal submanifold in $S^{n+m-1}$. Conversely, if
$M$ is a minimal submanifold in $S^{n+m-1}$, then $CM_{\varepsilon}$ is a
minimal submanifold in ${\tenmsb R}^{n+m}$.

There is a natural map from ${\tenmsb R}^{n+m}/\\{0\\}$ to $S^{n+m-1}$ by

(4.3) $\psi(\mathbf{x})=\frac{\mathbf{x}}{|\mathbf{x}|}.$

Hence for any map $F_{1}$ from $M$ to an arbitrary Riemannian manifold, the
map

(4.4) $F:=F_{1}\circ\psi$

on $CM_{\varepsilon}$ is called a cone-like map (see [19] p.66). A direct
calculation shows that $F$ is a harmonic map if and only if $F_{1}$ is
harmonic ([19] p.67). Especially, when $F_{1}$ is a function, $F$ is a
harmonic (subharmonic, superharmonic) function if and only if $F_{1}$ is
harmonic (subharmonic, superharmonic).

For any $p\in M$, $N_{p}M\subset T_{p}S^{n+m-1}$ can be viewed as an
$m$-dimensional affine subspace in ${\tenmsb R}^{n+m}$. Via parallel
translation in the Euclidean space, one can define the normal Gauss map
$\gamma:M\rightarrow\mathbb{G}_{m,n}$

(4.5) $p\mapsto N_{p}M.$

The canonical normal Gauss map on $CM_{\varepsilon}\subset{\tenmsb R}^{n+m}$
is defined by

(4.6) $\gamma^{c}:p\in CM_{\varepsilon}\mapsto
N_{p}(CM_{\varepsilon})\in\mathbb{G}_{n,m}.$

It is easily-seen that $\gamma^{c}$ is a cone-like map; more precisely,

(4.7) $\gamma^{c}=\gamma\circ\psi.$

Therefore, the Gauss image of $CM_{\varepsilon}$ coincides with the Gauss
image of $M$. Applying Proposition 3.2, we can derive the following spherical
Bernstein theorem.

######  Theorem 4.1.

Let $M^{n-1}$ be a compact, oriented minimal submanifold in $S^{n+m-1}$. If
there is a fixed oriented $m$-plane $Q_{0}$, such that

(4.8) $\langle N,Q_{0}\rangle\geq 1/3$

for all normal $m$-planes $N$ of $M$, then $M$ is a totally geodesic subsphere
of $S^{n+m-1}$.

###### Proof.

Let $CM_{\varepsilon}$ be the truncated cone generated by $M$, then by
Proposition 4.1, $CM_{\varepsilon}$ is an $n$-dimensional minimal submanifold
in ${\tenmsb R}^{n+m}$. Denote by $\gamma^{c}$ the normal Gauss map of $M$,
and

(4.9) $v:=w^{-1}(\cdot,Q_{0})\circ\gamma^{c}.$

Since $\gamma^{c}$ is a cone-like map extended by the normal Gauss map
$\gamma:M\rightarrow\mathbb{G}_{m,n}$, $v$ is a cone-like function. Hence the
condition (4.8) implies $v\leq 3$ everywhere on $CM_{\varepsilon}$ and by
applying Proposition 3.2 we know $v$ is a subharmonic function on
$CM_{\varepsilon}$. Since $v$ is a cone-like function, $v|_{M}$ is also a
subharmonic function on $M$. The classical maximum principle implies $v|_{M}$
is constant, therefore $\Delta v\equiv 0$ on $CM_{\varepsilon}$.

(4.10) $U:=\\{p\in CM_{\varepsilon}:|B|^{2}(p)>0\\}$

is an open subset of $CM_{\varepsilon}$. It suffices to show $U=\emptyset$.

We prove it by a reductio ad absurdum. Assume $U\neq\emptyset$. Let $\theta$
be a normal (tangent) Jordan angle function on $U$, then Proposition 3.2
implies $\theta(p)=\theta_{0}$ or $0$ for any $p\in U$, where
$\theta_{0}:=\arctan\sqrt{2}$. The continuity of $\theta$ forces
$\theta\equiv\theta_{0}$ or $0$. Moreover, since
$m_{\theta_{0}}^{N}=m_{\theta_{0}}^{T}\equiv 2$, Lemma 2.3 implies
$N_{\theta_{0}}U$, $N_{0}U$ are smooth subbundles of $NU$, and
$T_{\theta_{0}}U$, $T_{0}U$ are smooth subbundles of $TU$. It is worthy to
note that, if $m=2$ ($n=2$), $\nu\in N_{0}U$ ($u\in T_{0}U$) if and only if
$\nu=0$ ($u=0$). Again applying Proposition 3.2, the normal bundle and the
tangent bundle have the following decomposition

$\displaystyle NU$ $\displaystyle=N_{\theta_{0}}U\oplus N_{0}U,$
$\displaystyle TU$ $\displaystyle=T_{\theta_{0}}U\oplus T_{0}U$

and there exists $S\in\Gamma\big{(}\Lambda^{2}(N_{\theta_{0}}^{*}U)\otimes
T_{0}^{*}U\big{)}$, which is nonzero everywhere on $U$, such that for any
$p_{0}\in U$,

(4.11) $B=\sum_{1\leq\alpha,\beta\leq
2}(S_{\nu_{\alpha}\nu_{\beta}}\odot\omega_{\alpha})\otimes\nu_{\beta}$

holds on a neighborhood of $p_{0}$, where $\\{\nu_{1},\nu_{2}\\}$ is a local
orthonormal frame field of $N_{\theta_{0}}U$ and $\omega_{\alpha}(v):=\langle
v,\Phi_{\theta_{0}}(\nu_{\alpha})\rangle$. Now we put
$e_{\alpha}(p):=\Phi_{\theta_{0}}\big{(}\nu_{\alpha}(p)\big{)}$ for every $p$
and each $\alpha=1,2$, then $\\{e_{1},e_{2}\\}$ is a local orthonormal frame
field of $T_{\theta_{0}}U$. Since $S$ is nonzero everywhere, there are a unit
vector field $e_{n}$ of $T_{0}U$, and a positive function $h$, such that

(4.12) $S_{\nu_{1}\nu_{2}}(v)=h\langle v,e_{n}\rangle\qquad\forall v\in
T_{0}U.$

Now we choose $e_{3},\cdots,e_{n-1}$ ($\nu_{3},\cdots,\nu_{m}$) to be unit
vector fields of $T_{0}U$ ($N_{0}U$), such that $\\{e_{1},\cdots,e_{n}\\}$
($\\{\nu_{1},\cdots,\nu_{m}\\}$) is a local orthonormal tangent (normal) frame
field around $p_{0}$. Then (4.11) and (4.12) shows $B_{e_{i}e_{j}}=0$ for any
$1\leq i,j\leq n-1$, $B_{e_{n}e_{i}}=0$ for every $i\geq 3$,
$B_{e_{n}e_{1}}=h\nu_{2}$ and $B_{e_{n}e_{2}}=-h\nu_{1}$.

Noting that $e_{n}\in Q_{0}^{\bot}$ everywhere, we have

(4.13) $\overline{\nabla}_{e_{i}}e_{n}\in Q_{0}^{\bot}\qquad\forall 1\leq
i\leq n$

with $\overline{\nabla}$ the Levi-Civita connection on ${\tenmsb R}^{n+m}$.

Denote

(4.14) $\Gamma_{ij}^{n}:=\langle\nabla_{e_{i}}e_{j},e_{n}\rangle\qquad\forall
1\leq i,j\leq n-1.$

Differentiating both sides of $B_{e_{i}e_{j}}\equiv 0$ gives

(4.15) $\displaystyle 0$
$\displaystyle=\nabla_{e_{k}}B_{e_{i}e_{j}}=(\nabla_{e_{k}}B)_{e_{i}e_{j}}+B_{\nabla_{e_{k}}e_{i},e_{j}}+B_{e_{i},\nabla_{e_{k}}e_{j}}$
$\displaystyle=(\nabla_{e_{k}}B)_{e_{i}e_{j}}+\Gamma_{ki}^{n}B_{e_{n}e_{j}}+\Gamma_{kj}^{n}B_{e_{i}e_{n}}$

Interchanging the position of $i$ and $k$ in the above formula yields

(4.16)
$(\nabla_{e_{i}}B)_{e_{k}e_{j}}+\Gamma_{ik}^{n}B_{e_{n}e_{j}}+\Gamma_{ij}^{n}B_{e_{k}e_{n}}=0.$

The well-known Codazzi equations tell us
$(\nabla_{e_{k}}B)_{e_{i}e_{j}}=(\nabla_{e_{i}}B)_{e_{k}e_{j}}$, then
combining (4.15) and (4.16) gives

(4.17)
$\Gamma_{ki}^{n}B_{e_{n}e_{j}}+\Gamma_{kj}^{n}B_{e_{i}e_{n}}-\Gamma_{ik}^{n}B_{e_{n}e_{j}}-\Gamma_{ij}^{n}B_{e_{k}e_{n}}=0$

holds for any $1\leq i,j,k\leq n-1$. For any $3\leq i,j\leq n-1$, choosing
$k=1$ in (4.17) implies
$0=-\Gamma_{ij}^{n}B_{e_{1}e_{n}}=-h\Gamma_{ij}^{n}\nu_{2}$, hence
$\Gamma_{ij}^{n}=0$. For any $3\leq i\leq n-1$, taking $j=1,k=2$ in (4.17)
gives

$\displaystyle 0$
$\displaystyle=\Gamma_{2i}^{n}B_{e_{n}e_{1}}-\Gamma_{i2}^{n}B_{e_{n}e_{1}}-\Gamma_{i1}^{n}B_{e_{2}e_{n}}$
$\displaystyle=h(\Gamma_{2i}^{n}-\Gamma_{i2}^{n})\nu_{2}+h\Gamma_{i1}^{n}\nu_{1}$

and hence $\Gamma_{2i}^{n}-\Gamma_{i2}^{n}=\Gamma_{i1}^{n}=0$. Similarly,
choosing $j=2,k=1$ yields $\Gamma_{1i}^{n}-\Gamma_{i1}^{n}=\Gamma_{i2}^{n}=0$.
Therefore $\Gamma_{1i}^{n}=\Gamma_{2i}^{n}=\Gamma_{i1}^{n}=\Gamma_{i2}^{n}=0$.
Now we put $i=j=1$, $k=2$, then (4.17) tells us

$\displaystyle 0$
$\displaystyle=\Gamma_{21}^{n}B_{e_{n}e_{1}}+\Gamma_{21}^{n}B_{e_{1}e_{n}}-\Gamma_{12}^{n}B_{e_{n}e_{1}}-\Gamma_{11}^{n}B_{e_{2}e_{n}}$
$\displaystyle=h(2\Gamma_{21}^{n}-\Gamma_{12}^{n})\nu_{2}+h\Gamma_{11}^{n}\nu_{1}$

and hence $2\Gamma_{21}^{n}-\Gamma_{12}^{n}=\Gamma_{11}^{n}=0$. Similarly,
putting $i=j=2$ and $k=1$ in (4.17) yields
$2\Gamma_{12}^{n}-\Gamma_{21}^{n}=\Gamma_{22}^{n}=0$. Therefore
$\Gamma_{11}^{n}=\Gamma_{12}^{n}=\Gamma_{21}^{n}=\Gamma_{22}^{n}=0$. In
summary

(4.18) $\langle\nabla_{e_{i}}e_{j},e_{n}\rangle=\Gamma_{ij}^{n}=0\qquad\forall
1\leq i,j\leq n-1.$

Thus

(4.19) $\displaystyle\overline{\nabla}_{e_{1}}e_{n}$
$\displaystyle=\sum_{1\leq j\leq n-1}\langle\nabla_{e_{1}}e_{n},e_{j}\rangle
e_{j}+\langle\nabla_{e_{1}}e_{n},e_{n}\rangle+B_{e_{1}e_{n}}$
$\displaystyle=-\sum_{1\leq j\leq n-1}\langle\nabla_{e_{1}}e_{j},e_{n}\rangle
e_{j}+B_{e_{1}e_{n}}$ $\displaystyle=h\nu_{2}$

and then by (4.13), $\nu_{2}$ is orthogonal to $Q_{0}$, i.e.
$\mathcal{P}_{0}\nu_{2}=0$. But on the other hand, $\nu_{2}\in
N_{\theta_{0}}U$ implies $|\mathcal{P}_{0}\nu_{2}|=\cos\theta_{0}$, which is a
contradiction.

Therefore $U=\emptyset$ and $CM_{\varepsilon}$ is totally geodesic in
${\tenmsb R}^{n+m}$. Hence $M$ has to be a totally geodesic subsphere.

∎

With the results of geometric measure theory, we can also prove a Euclidean
Bernstein type theorem.

######  Theorem 4.2.

Let $f:=(f^{1},\cdots,f^{m})$ be a smooth ${\tenmsb R}^{m}$-valued function
defined everywhere on ${\tenmsb R}^{n}$. Suppose its graph $M:=\text{graph
}f=\\{(x,f(x)):x\in{\tenmsb R}^{m}\\}$ is a minimal submanifold in ${\tenmsb
R}^{n+m}$, and

(4.20) $\Delta_{f}:=\Big{[}\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\Big{]}^{\frac{1}{2}}\leq 3,$

then $f^{1},\cdots,f^{m}$ has to be affine linear, representing an affine
$n$-plane in ${\tenmsb R}^{n+m}$.

###### Proof.

As shown in [11], the condition (4.20) says

$v:=w^{-1}(\cdot,Q_{0})\circ\gamma\leq 3.$

Here $\gamma$ denotes the normal Gauss map of $M$ into $\mathbb{G}_{m,n}$, and
$Q_{0}:=\varepsilon_{n+1}\wedge\cdots\wedge\varepsilon_{n+m}$, where
$\varepsilon_{1},\cdots,\varepsilon_{n+m}$ denotes the Cartesian coordinate
vectors in ${\tenmsb R}^{n+m}$. In other words, the Gauss image of $M$ is
contained in a closed region

$\overline{{\tenmsb V}}:=\\{P\in\mathbb{G}_{m,n}:w(P,Q_{0})\geq 1/3\\}.$

Now we consider the tangent cone of $M$ at $\infty$, which is the limit of a
one-parameter family of minimal submanifolds in ${\tenmsb R}^{n+m}$; each one
is obtained by a contracting procedure. More precisely, let

(4.21) $f_{t}=\frac{1}{t}f(tx),\qquad\forall t\in{\tenmsb R}^{+},$

then it is easy to check that $f_{t}$ satisfies the minimal surface equations
and hence $\\{M_{t}=\text{graph }f_{t}:t\in{\tenmsb R}^{+}\\}$ defines a
family of minimal submanifolds in ${\tenmsb R}^{n+m}$. Based on (4.20), one
can proceed as in [6] §5 to show that $\\{f_{t}:t\in{\tenmsb R}^{+}\\}$ is an
equicontinuous family on any compact subset of ${\tenmsb R}^{n}$. Thus the
Arzela-Ascoli Theorem implies that there exists a subsequence
$\\{f_{t_{i}}:i\in{\tenmsb Z}^{+}\\}$ such that
$\lim_{i\rightarrow\infty}t_{i}=+\infty$ and
$\lim_{i\rightarrow\infty}f_{t_{i}}=h$, which is a Lipschitz function. One can
prove that $h$ is a weak solution to the minimal surface equations and its
graph $C(M,\infty):=\\{(x,h(x)):x\in{\tenmsb R}^{n}\\}$ is a cone (see [13]),
which is called the tangent cone of $M$ at $\infty$. In the framework of
geometric measure theory, we can prove that $C(M,\infty)$ is regular except at
$0$, as in [6]. Hence the intersection of $C(M,\infty)$ and the unit sphere
gives an $(n-1)$-dimensional embedded minimal submanifold in $S^{n+m-1}$,
which is denoted by $M^{\prime}$. The image of the normal Gauss map of
$M^{\prime}$ is still contained in $\overline{{\tenmsb V}}$, i.e. $\langle
N,Q_{0}\rangle\geq 1/3$ for all normal $m$-planes of $M^{\prime}$ (the proof
is the same as in [20] §7.3). By Theorem 4.1, $M^{\prime}$ has to be a totally
geodesic subsphere. Then Allard’s regularity estimate [1] implies $f$ is
affine linear and $M$ is an affine $n$-plane. ∎

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