# Modified defect relation of Gauss maps on annular ends of minimal surfaces
for hypersuraces of projective varieties in subgeneral position

Si Duc Quang  1Department of Mathematics, Hanoi National University of
Education
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam  2Thang Long Institute of Mathematics
and Applied Sciences
Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be an annular end of a complete minimal surface $S$ in
${\mathbb{R}}^{m}$ and let $V$ be a $k$-dimension projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})\ (n=m-1)$. Let $g$ be the generalized Gauss
map of $S$ into $V\subset{\mathbb{P}}^{n}({\mathbb{C}})$. In this paper, we
establish a modified defect relation of $g$ on the annular end $A$ for $q$
hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$ of ${\mathbb{P}}^{n}({\mathbb{C}})$ in
$N$-subgeneral position with respect to $V$. Our result implies that the image
$g(A)$ cannot omit all $q$ hypersurfaces $Q_{1},\ldots,Q_{q}$ if $g$ is
nondegenerate over $I_{d}(V)$ and $q>\frac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)}$,
where $M=H_{V}(d)-1$ and $d$ is the least of common multiple of $\deg
Q_{1},\ldots,\deg Q_{q}$.

††footnotetext:  2010 Mathematics Subject Classification: Primary 53A10;
Secondary 53C42.
Key words and phrases: Gauss map, value distribution, holomorphic curve,
modified defect relation, ramification, hypersurface.

##  1\. Introduction and Main results

The value distribution of the Gauss maps of minimal surfaces was initially
studied by Osserman in a series of his papers [14, 15, 16]. Later on, this
problem had been deepen by many authors such as F. Xavier [21], H. Fujimoto
[5, 6, 8, 9, 10], M. Ru [19], X. Mo [13], G. Dethloff and P. H. Ha [2, 4] and
others. In 1988, H. Fujimoto [7] showed that the Gauss map of a complete non-
flat minimal surface in ${\mathbb{R}}^{3}$ can omit at most four points, and
this number four is the best possible number. For the case of higher
dimension, in 1991, based on an earlier result of H. Fujimoto, M. Ru [19]
showed that the generalized Gauss map of a complete non-flat minimal surface
$S$ in ${\mathbb{R}}^{m}$ cannot omit more than $\frac{m(m+1)}{2}$ hyperplanes
of ${\mathbb{P}}^{m-1}({\mathbb{C}})$ in general position. In the same year,
S. J. Kao [12] extended H. Fujimoto’s method and showed that the Gauss map of
an end of a non-flat complete minimal surface in $R^{3}$ that is conformally
an annulus $\\{z;0<1/r<|z|<r\\}$ must also assume every value, with at most
four exceptions. Later on, L. Jin and M. Ru in [20] extended the result of Kao
to the case of higher dimension by showing that the generalized Gauss map on
an annular end of a complete minimal surface $S$ in ${\mathbb{R}}^{m}$ cannot
omit more than $\frac{m(m+1)}{2}$ hyperplanes of
${\mathbb{P}}^{m-1}({\mathbb{C}})$ in general position. Recently, G. Dethloff,
P. H. Ha and P. D. Thoan [3] have improved the result of L. Jin and M. Ru by
considering the ramification of the generalized Gauss map on an annular end
over a family of hyperplanes of ${\mathbb{P}}^{n}({\mathbb{C}})$ in subgeneral
position.

In this paper, we will give a modified defect relation for the generalized
Gauss map on an annular end of a complete minimal surface in
${\mathbb{R}}^{m}$ into a projective subvariety
$V\subset{\mathbb{P}}^{n}({\mathbb{C}})\ (n=m-1)$ for hypersurfaces of
${\mathbb{P}}^{n}({\mathbb{C}})$ in subgeneral position with respect to $V$.
Our result extends and generalizes all above mentioned results. In order to
state the main result of this paper, we will introduce the notion of modified
defect for hypersurfaces in a projective subvariety
$V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ of a holomorphic curve $f$ from an
open Riemann surface into a projective variety, which is a generalization of
the notion of modified defect for hypersurplanes in
${\mathbb{P}}^{n}({\mathbb{C}})$ introduced by H. Fujimoto [8, 9, 10]), as
follows.

Let $S$ be an open complete Riemann surface in ${\mathbb{R}}^{m}$. By a
divisor on $S$ we mean a map $\nu$ of $S$ into ${\mathbb{R}}$ whose support
$\mathrm{Supp}\,(\nu):=\\{p\in S;\nu(p)\neq 0\\}$ has no accumulation points
in $S$. Let $A$ be an annular ends on $S$ and let $f$ be a holomorphic curve
from $A$ to a $k$-dimension projective subvariety $V$ of
${\mathbb{P}}^{n}({\mathbb{C}})$ and let $Q$ be a hypersurface in
${\mathbb{P}}^{n}({\mathbb{C}})$ of degree $d$. By $\nu_{Q(f)}$ we denote the
pull-back of the divisor $Q$ by $f$. Let $F=(f_{0},\ldots,f_{n})$ be a global
reduced representation of $f$ on $A$. Assume that, the hypersurface $Q$ has a
defining polynomial, denoted again by the same notation $Q$ (throughout this
paper) if there is no confusion, as follows

$Q(x_{0},\ldots,x_{n})=\sum_{I\in\mathcal{T}_{d}}a_{I}x^{I},$

where
$\mathcal{T}_{d}=\\{(i_{0},\ldots,i_{n})\in\mathbf{Z}^{n+1}_{+};i_{0}+\cdots+i_{n}=d\\}$,
$a_{I}\in{\mathbb{C}}$ are not all zero for $I\in\mathcal{T}_{d}$ and
$x^{I}=x_{0}^{i_{0}}\ldots x_{n}^{i_{n}}$ for each $i=(i_{0},\ldots,i_{n})$.
We set

$Q(F)=\sum_{I\in\mathcal{T}_{d}}a_{I}f^{I},$

where $f^{I}=f_{0}^{i_{0}}\ldots f_{n}^{i_{n}}$ for each
$I\in\mathcal{T}_{d}$. Throughout this paper, for each given hypersurface $Q$
we assume that $\|Q\|=(\sum_{I\in\mathcal{T}_{d}}|a_{I}|^{2})^{1/2}=1$. The
$H$-defect truncated to level $m$ for $Q$ of $f$ on $A$ is defined by

$\delta^{H,m}_{f,A}(Q):=1-\mathrm{inf}\\{\eta;\eta\text{ satisfies Condition
}(*)_{H}\\}.$

Here, condition $(*)_{H}$ means that there exists a $[-\infty,\infty)$-valued
continuous subharmonic function $u\ (\not\equiv\infty)$ on $A$, harmonic on
$A\setminus f^{-1}(H)$ and satisfies conditions:

  * $(D_{1})$

$e^{u}\leq\|F\|^{d\eta}$,

  * $(D_{2})$

for each $\xi\in f^{-1}(H)$ there exists the limit

$\underset{z\rightarrow\xi}{\lim}(u(z)-\min(\nu_{Q(f)}(z),m)\log|z-\xi|)\in[-\infty,\infty).$

Denote by $I(V)$ the ideal of homogeneous polynomials in
${\mathbb{C}}[x_{0},...,x_{n}]$ defining $V$ and by
${\mathbb{C}}[x_{0},...,x_{n}]_{d}$ the vector space of all homogeneous
polynomials in ${\mathbb{C}}[x_{0},...,x_{n}]$ of degree $d$ including the
zero polynomial. Define

$I_{d}(V):=\dfrac{{\mathbb{C}}[x_{0},...,x_{n}]_{d}}{I(V)\cap{\mathbb{C}}[x_{0},...,x_{n}]_{d}}\text{
and }H_{V}(d):=\dim I_{d}(V).$

For an element $D\in H_{d},$ we denote by $[D]$ its equivalent class in
$I_{d}(V)$.

Let $Q_{1},...,Q_{q}\ (q\geq k+1)$ be $q$ hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$. The hypersurfaces $Q_{1},...,Q_{q}$ are said
to be in $N$-subgeneral position with respect to $V$ if

$V\cap\left(\bigcap_{j=1}^{N+1}Q_{i_{j}}\right)=\varnothing\ \forall\ 1\leq
i_{1}<\cdots<i_{N+1}\leq q.$

If $Q_{1},...,Q_{q}$ are in $k$-subgeneral position with respect to $V$ then
we say that they are in general position with respect to $V.$

For $f$ as above, we say that $f$ is algebraic degenerate over $I_{d}(V)$ if
there is $[Q]\in I_{d}(V)\setminus\\{0\\}$ such that $Q(F)\equiv 0$ for some
reduced representation $F$ of $f$. Otherwise, we say that $f$ is nondegenerate
over $I_{d}(V)$. It is clear that if $f:A\rightarrow V$ is algebraic
nondegenerate, then $f$ is nondegenerate over $I_{d}(V)$ for every $d\geq 1.$
Our main result is stated as follows.

######  Theorem 1.1.

Let $A$ be an annular end of a non-flat complete minimal surface $S$ in
${\mathbb{R}}^{m}$. Let $V$ be a $k$-dimension projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})\ (n=m-1)$. Let $Q_{1},\ldots,Q_{q}$ be
hypersurfaces of ${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position
with respect to $V$. Let $d$ be the least common multiple of $\deg Q_{j}\
(1\leq j\leq q)$ . Assume that the generalized Gauss map $g:S\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ is nondegenerate over $I_{d}(V)$. Then

$\sum_{j=1}^{q}\delta^{M}_{g,A}(Q_{j})\leq\dfrac{(2N-k+1)(M+1)}{k+1}+\dfrac{(2N-k+1)M(M+1)}{2d(k+1)},$

where $M=H_{V}(d)-1$.

From Theorem 1.1 we immediately have the following corollary about the
ramification of $f$ over a family of hypersurface on an annular end.

######  Corollary 1.2.

Let $S,A,V,\\{Q_{j}\\}_{j=1}^{q},d,M$ be as in Theorem 1.1. Assume that the
generalized Gauss map $g:S\rightarrow V\subset{\mathbb{P}}^{n}({\mathbb{C}})$
is nondegenerate over $I_{d}(V)$ and ramified over $Q_{j}$ with multiplicity
at least $m_{j}\ (1\leq j\leq q)$. Then

$\sum_{j=1}^{q}\left(1-\frac{M}{m_{j}}\right)\leq\dfrac{(2N-k+1)(M+1)}{k+1}+\dfrac{(2N-k+1)M(M+1)}{2d(k+1)}.$

In particular, $g$ cannot omit $Q_{1},\ldots,Q_{q}$ if
$q>\frac{(2N-k+1)(M+1)}{k+1}+\frac{(2N-k+1)M(M+1)}{2d(k+1)}.$

###### Proof.

Indeed, since $g$ is ramified over $Q_{j}$ with multiplicity at least $m_{j}$
on $A$ for each $j$, we have

$\delta^{H,M}_{g,S}(Q_{j})\geq 1-\dfrac{M}{m_{j}}\ \ (1\leq j\leq q).$

Then by Theorem 1.1, we have

$\sum_{j=1}^{q}\left(1-\frac{M}{m_{j}}\right)\leq\dfrac{(2N-k+1)(M+1)}{k+1}+\dfrac{(2N-k+1)M(M+1)}{2d(k+1)}.$

The theorem is proved. ∎

We note that, in the case of hyperplanes in ${\mathbb{P}}^{n}({\mathbb{C}})$,
i.e., $d=1$ and $V={\mathbb{P}}^{n}({\mathbb{C}})$, then $M=n$. Therefore,
this corollary immediately implies the results mentioned above of S. J. Kao
[12], L. Jin-M. Ru [19] and G. Dethloff - P. H. Ha- P. D. Thoan [3].

##  2\. Main lemmas

######  Lemma 2.1 (cf. [17, Lemma 3]).

Let $V$ be a $k$-projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$.
Let $Q_{1},...,Q_{q}$ be $q\ (q>2N-k+1)$ hypersurfaces of
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position with respect to
$V$ of the same degree. Then, there are positive rational constants
$\omega_{i}\ (1\leq i\leq q)$ satisfying the following:

i) $0<\omega_{i}\leq 1\ \forall i\in\\{1,...,q\\}$,

ii) Setting $\tilde{\omega}=\max_{j\in Q}\omega_{j}$, one gets
$\sum_{j=1}^{q}\omega_{j}=\tilde{\omega}(q-2N+k-1)+k+1.$

iii) $\dfrac{k+1}{2N-k+1}\leq\tilde{\omega}\leq\dfrac{k}{N}.$

iv) For each $R\subset\\{1,...,q\\}$ with $\sharp R=N+1$, then $\sum_{i\in
R}\omega_{i}\leq k+1$.

v) Let $E_{i}\geq 1\ (1\leq i\leq q)$ be arbitrarily given numbers. For each
$R\subset\\{1,...,q\\}$ with $\sharp R=N+1$, there is a subset $R^{o}\subset
R$ such that $\sharp R^{o}=\mathrm{rank}\\{Q_{i}\\}_{i\in R^{o}}=k+1$ and

$\prod_{i\in R}E_{i}^{\omega_{i}}\leq\prod_{i\in R^{o}}E_{i}.$

Let $S$ be an open Riemann surface and let $ds^{2}$ be a pseudo-metric on $S$
which is locally written as $ds^{2}=\lambda^{2}|dz|^{2}$, where $\lambda$ is a
nonnegative real-value function with mild singularities and $z$ is a
holomorphic local coordinate. The divisor of $ds^{2}$ is defined by
$\nu_{ds}:=\nu_{\lambda}$ for each local expression
$ds^{2}=\lambda^{2}|dz|^{2}$, which is globally well-defined on $S$. We say
that $ds^{2}$ is a continuous pseudo-metric if $\nu_{ds}\geq 0$ everywhere.

The Ricci of $ds^{2}$ is defined by

$\mathrm{Ric}_{ds^{2}}=dd^{c}\log\lambda^{2}$

for each local expression $ds^{2}=\lambda^{2}|dz|^{2}$. This definition is
globally well-defined on $S$.

The continuous pseudo-metric $ds^{2}$ is said to have strictly negative
curvature on $S$ if there is a positive constant $C$ such that

$-\mathrm{Ric}_{ds^{2}}\geq C\Omega_{ds^{2}},$

where $\Omega_{ds^{2}}=\lambda^{2}\cdot\dfrac{\sqrt{-1}}{2}\cdot dz\wedge
d\bar{z}$.

Let $f$ be a holomorphic map of $S$ into a projective subvariety $V$ of
dimension $k$ of ${\mathbb{P}}^{n}({\mathbb{C}})$. Let $d$ be a positive
integer and assume that $f$ is nondegenerate over $I_{d}(V)$. We fix a
${\mathbb{C}}$-ordered basis $\mathcal{V}=([v_{0}],\ldots,[v_{M}])$ of
$I_{d}(V)$, where $v_{i}\in H_{d}$ and $M=H_{V}(d)-1$.

Let $F=(f_{0},\ldots,f_{n})$ be a local reduced representation of $f$ (in a
local chat $(U,z)$ of $S$). Consider the holomorphic map

$F_{\mathcal{V}}=(v_{0}(F),\ldots,v_{M}(F))$

and

$F_{\mathcal{V},p}:=F_{\mathcal{V}}^{(0)}\wedge
F_{\mathcal{V}}^{(1)}\wedge\cdots\wedge
F_{\mathcal{V}}^{(p)}:S\rightarrow\bigwedge_{p+1}{\mathbb{C}}^{M+1}$

for $0\leq p\leq M$, where

  * •

$F_{\mathcal{V}}^{(0)}:=F_{\mathcal{V}}=(v_{0}(F),\ldots,v_{M}(F))$,

  * •

$F_{\mathcal{V}}^{(l)}=F_{\mathcal{V}}^{(l)}:=\left(v_{0}(F)^{(l)},\ldots,v_{M}(F)^{(l)}\right)$
for each $l=0,1,\ldots,p$,

  * •

$v_{i}(F)^{(l)}\ (i=0,\ldots,M)$ is the $l^{th}$\- derivatives of $v_{i}(F)$
taken with respect to $z$.

The norm of $F_{\mathcal{V},p}$ is given by

$|F_{\mathcal{V},p}|:=\left(\sum_{0\leq i_{0}<i_{1}<\cdots<i_{p}\leq
M}\left|W_{z}(v_{i_{0}}(F),\ldots,v_{i_{p}}(F))\right|^{2}\right)^{1/2},$

where

$W(v_{i_{0}}(F),\ldots,v_{i_{p}}(F)):=\det\left(v_{i_{j}}(F)^{(l)}\right)_{0\leq
l,j\leq p}$

We use the same notation $\langle,\rangle$ for the canonical hermitian product
on $\bigwedge^{l+1}{\mathbb{C}}^{M+1}\ (0\leq l\leq M)$. For two vectors
$A\in\bigwedge^{k+1}{\mathbb{C}}^{M+1}\ (0\leq k\leq M)$ and
$B\in\bigwedge^{p+1}{\mathbb{C}}^{M+1}\ (0\leq p\leq k)$, there is one and
only one vector $C\in\bigwedge^{k-p}{\mathbb{C}}^{M+1}$ satisfying

$\langle C,D\rangle=\langle A,B\wedge D\rangle\ \forall
D\in\bigwedge^{k-p}{\mathbb{C}}^{M+1}.$

The vector $C$ is called the interior product of $A$ and $B$, and defined by
$A\vee B$.

Now, for a hypersurface $Q$ of degree $d$ in ${\mathbb{P}}^{n}({\mathbb{C}})$,
we have

$[Q]=\sum_{i=0}^{M}a_{i}[v_{i}].$

Hence, we associate $Q$ with the vector
$(a_{0},\ldots,a_{M})\in{\mathbb{C}}^{M+1}$ and define
$F_{\mathcal{V},p}(Q)=F_{\mathcal{V},p}\vee H$. Then, we may see that

$\displaystyle F_{\mathcal{V},0}(Q)$
$\displaystyle=a_{0}v_{0}(F)+\cdots+a_{M}v_{M}(F)=Q(F),$
$\displaystyle|F_{\mathcal{V},p}(Q)|$ $\displaystyle=\left(\sum_{0\leq
i_{0}<i_{1}<\cdots<i_{p}\leq M}\sum_{l\neq
i_{0},\ldots,i_{p}}a_{l}\left|W_{z}(v_{l}(F),v_{i_{0}}(F),\ldots,v_{i_{p}}(F))\right|^{2}\right)^{1/2}$

Finally, for $0\leq p\leq M$, the $p^{th}$-contact function of $f$ for $Q$
with respect to $\mathcal{V}$ is defined (not depend on the choice of the
local coordinate) by

$\varphi_{\mathcal{V},p}(Q):=\dfrac{|F_{\mathcal{V},p}(Q)|}{|F_{\mathcal{V},p}|}=\dfrac{|F_{\mathcal{V},p}(Q)|}{|F_{\mathcal{V},p}|}.$

For each $p\ (0\leq p\leq M-1)$, let $M_{p}=\binom{M+1}{p+1}-1$ and $\pi_{p}$
be the canonical projection from
$\bigwedge^{p+1}{\mathbb{C}}^{M+1}\sim{\mathbb{C}}^{M_{p}+1}$ onto
${\mathbb{P}}^{M_{p}}({\mathbb{C}})$. Denote by $\Omega_{p}$ the pull-back of
the Fubini-Study form on ${\mathbb{P}}^{M_{p}}({\mathbb{C}})$ by the map
$\pi\circ F_{\mathcal{V},p}$, i.e.,
$\Omega_{p}=dd^{c}\log|F_{\mathcal{V},p}|^{2}$.

######  Proposition 2.2 (cf. [11, Proposition 2.5.1]).

Let $S,V,d,\mathcal{V}$ and $M$ be as above. For each positive $\epsilon$
there exists a constant $\delta_{0}(\epsilon)$, depending only on $\epsilon$,
such that for any hypersurface $Q$ of degree $d$ in
${\mathbb{P}}^{n}({\mathbb{C}})$ and any constant
$\delta>\delta_{0}(\epsilon)$

$dd^{c}\log\dfrac{1}{\log(\delta/\varphi_{\mathcal{V},p}(Q))}\geq\dfrac{\varphi_{\mathcal{V},p}(Q)}{\varphi_{\mathcal{V},p+1}(Q)\log(\delta/\varphi_{\mathcal{V},p}(Q))}\Omega_{p}-\epsilon\Omega_{p}.$

Actually, [11, Proposition 2.5.1] states only for hyperplanes. However, by
using Serge embedding to embed ${\mathbb{P}}^{n}({\mathbb{C}})$ into
${\mathbb{P}}^{M}({\mathbb{C}})$, we automatically deduce the above
proposition.

######  Theorem 2.3 (cf. [11, Proposition 2.5.7]).

Set $\sigma_{p}=p(p+1)/2$ for $0\leq p\leq M+1$ and
$\tau_{m}=\sum_{p=1}^{m}\sigma_{m}$. Then, we have

$dd^{c}\log(|F_{\mathcal{V},0}|^{2}\cdots|F_{\mathcal{V},M-1}|^{2})\geq\dfrac{\tau_{M}}{\sigma_{M}}\left(\dfrac{|F_{\mathcal{V},0}|^{2}\cdots|F_{\mathcal{V},M}|^{2}}{|F_{\mathcal{V},0}|^{2\sigma_{M+1}}}\right)^{1/\tau_{M}}dd^{c}|z|^{2}.$

######  Theorem 2.4.

With the notations and the assumption in Theorem 2.7, we have

$\nu_{\phi}+\sum_{j=1}^{q}\omega_{j}\cdot\min\\{\nu_{Q_{j}(f)},M\\}\geq 0$

where
$\phi=\dfrac{|F_{\mathcal{V},M}|}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}}}$ and
$\nu_{\phi}$ is the divisor of the function $\phi$.

###### Proof.

Fix a point $z\in S$. Since $\\{Q_{i}\\}_{i=1}^{q}$ is in $N$-subgeneral
position, $z$ is not zero of more than $N$ functions $Q_{i}(F)$. Suppose that
$z$ is zero of $Q_{i}(F)$ for each $1\leq i\leq k\leq N)$ and $z$ is not zero
of $Q_{i}(F)$ for each $i>N$. Put $R=\\{1,...,N+1\\}.$ Choose $R^{1}\subset R$
such that $\sharp R^{1}=\mathrm{rank}\\{Q_{i}\\}_{i\in R^{1}}=k+1$ and $R^{1}$
satisfies Lemma 2.1 v) with respect to numbers
$\bigl{\\{}e^{\max\\{\nu_{Q_{i}(F)}(z)-M,0\\}}\bigl{\\}}_{i=1}^{q}.$ Then we
have

$\displaystyle\nu_{F_{\mathcal{V},M}}(z)$ $\displaystyle\geq\sum_{i\in
R^{1}}\max\\{\nu_{Q_{i}(f)}(z)-M,0\\}$ $\displaystyle\geq\sum_{i\in
R}\omega_{i}\max\\{\nu_{Q_{i}(f)}(z)-M,0\\}.$

Hence

$\displaystyle\sum_{i=1}^{q}\omega_{i}\nu_{Q_{i}(f)}(z)-\nu_{F_{\mathcal{V},M}}(z)$
$\displaystyle=\sum_{i\in
R}\omega_{i}\nu_{Q_{i}(f)}(z)-\nu_{F_{\mathcal{V},M}}(z)$
$\displaystyle\leq\sum_{i\in R}\omega_{i}\nu_{Q_{i}(f)}(z)-\sum_{i\in
R}\omega_{i}\max\\{\nu_{Q_{i}(f)}(z)-M,0\\}$
$\displaystyle=\sum_{i=1}^{q}\omega_{i}\min\\{\nu_{Q_{i}(f)}(z),M\\}.$

The theorem is proved. ∎

######  Lemma 2.5 (Generalized Schwarz’s Lemma [1]).

Let $v$ be a non-negative real-valued continuous subharmonic function on
$\Delta(R)=\\{z\in{\mathbb{C}};\|z\|<R\\}$. If $v$ satisfies the inequality
$\Delta\log v\geq v^{2}$ in the sense of distribution, then

$v(z)\leq\dfrac{2R}{R^{2}-|z|^{2}}.$

Next, we give the following two of the key theorems in this paper, Theorems
2.6 and 2.7. These theorems have been initially proved in another recent work
of the author [18]. For sake of completeness, we also give their detail proofs
here.

######  Theorem 2.6.

Let $S,V,d,\mathcal{V}$ and $M$ be as above. Let $f:S\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic curve and let
$Q_{1},\ldots,Q_{q}$ be hypersurfaces of ${\mathbb{P}}^{n}({\mathbb{C}})$ in
$N$-subgeneral position with respect to $V$ of the same degree $d$, where
$q>\frac{(2N-k+1)(M+1)}{k+1}$. Assume that $f$ is nondegenerate over
$I_{d}(V)$ and have a local reduced representation $F=(f_{0},\ldots,f_{n})$ on
a local holomorphic chat $(U,z)$. Let $\omega_{j}\ (1\leq j\leq q)$ be the
Nochka weights for these hypersurfaces (defined in Lemma 2.1). For an
arbitrarily given $\delta>1$ and $0\leq p\leq M-1$, we set

$\Phi_{\mathcal{V},jp}:=\dfrac{\varphi_{\mathcal{V},p+1}(Q_{j})}{\varphi_{\mathcal{V},p}(Q_{j})\log^{2}\left(\delta/\varphi_{\mathcal{V},p}(Q_{j})\right)}.$

Then, there exists a positive constant $C_{\mathcal{V},p}$ depending only in
$\mathcal{V},p$ and $Q_{j}\ (1\leq j\leq q)$ such that

$\sum_{j=1}^{q}\omega_{j}\Phi_{\mathcal{V},jp}\geq
C_{\mathcal{V},p}\left(\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{1/(M-p)}$

holds on $S-\bigcup_{1\leq j\leq
q}\\{z;\varphi_{\mathcal{V},p}(Q_{j})(z)=0\\}$.

###### Proof.

Suppose that

$[Q_{j}]=\sum_{i=0}^{M}a_{ji}[v_{i}]\ (1\leq j\leq q).$

We define $a_{j}:=(a_{j0},\ldots,a_{jM})\in{\mathbb{C}}^{M+1}$ and consider
the set $\mathcal{R}_{p}$ of all subsets $R$ of $Q=\\{1,2,\ldots,q\\}$ such
that $\mathrm{rank}_{{\mathbb{C}}}\\{a_{j};j\in R\\}\leq M-p$. For each $P\in
G(M,p)$ (the Grassmannian manifold of all $(p+1)$-dimension linear subspaces
of ${\mathbb{C}}^{M+1}$), take a decomposable $(p+1)$-vector $E$ such that

$P=\\{X\in{\mathbb{C}}^{M+1};E\wedge X=0\\}$

and set

$\phi_{p}(P)=\underset{R\in\mathcal{R}_{p}}{\max}\min\left\\{\dfrac{\left|E\vee
a_{j}\right|^{2}}{|E|^{2}};\ j\not\in R\right\\}.$

Then $\phi_{p}(P)$ depends only on $P$ and may be regarded as a function on
the Grassmannian manifold $G(M,p)$. For each nonzero $(k+1)$-vector
$E=E_{0}\wedge E_{1}\wedge\ldots\wedge E_{k}$ we set

$R=\\{j\in Q;\ E\vee a_{j}=0\\}.$

Note that, $E\vee a_{j}=0$ if and only if $a_{j}$ is contained in the
orthogonal complement of the vector space $\mathrm{Span}(E_{0},\ldots,E_{k})$.
Then we see

$\mathrm{rank}_{{\mathbb{C}}}\\{a_{j};\ j\in R\\}=\dim\mathrm{Span}(a_{j};\
j\in R)\leq M-p,$

namely, $R\in\mathcal{R}_{p}$. Hence $\phi_{p}$ is positive everywhere on
$G(M,p)$. Since $\phi_{p}$ is obviously continuous and $G(M,p)$ is compact, we
can take a positive constant $\delta$ such that $\phi_{p}(P)>\delta$ for each
$P\in G(M,p)$.

Take a point $z$ with $F_{\mathcal{V},p}(z)\neq 0$. The vector space generated
by $F^{(0)}_{\mathcal{V}}(z),F^{(1)}_{\mathcal{V}}(z)$,…,
$F^{(p)}_{\mathcal{V}}(z)$ determines a point in $G(M,p)$. Therefore, there is
a set $R$ in $\mathcal{R}_{p}$ with

$\mathrm{rank}_{{\mathbb{C}}}\\{a_{j};j\in R\\}\leq M-p$

such that $\varphi_{\mathcal{V},p}(Q_{j})(z)\geq\delta$ for all $j\not\in R$.
Then, we can choose a finite positive constant $K$ depending only on
$\mathcal{V},Q_{j}\ (1\leq j\leq q)$ such that $\Phi_{\mathcal{V},jp}(z)\leq
K$ for all $j\not\in R$. Set

$T:=\\{j;\ \Phi_{\mathcal{V},jp}(z)>K\\},l:=\sum_{j\in T}\omega_{j}.$

We distinguish the following two cases.

Case 1: Assume that $T$ is an empty set. We have

$\displaystyle\sum_{j=1}^{q}\omega_{j}\Phi_{\mathcal{V},jp}$
$\displaystyle\geq\left(\sum_{j=1}^{q}\omega_{j}\right)\left(\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{\frac{1}{\sum_{j}\omega_{j}}}$
$\displaystyle\geq(p+1)K\left(\prod_{j=1}^{q}\left(\dfrac{\Phi_{\mathcal{V},jp}}{K}\right)^{\frac{\omega_{j}}{M-p}}\right)^{\frac{M-p}{\sum_{j}\omega_{j}}}$
$\displaystyle\geq(p+1)K\left(\prod_{j=1}^{q}\left(\frac{\Phi_{\mathcal{V},jp}}{K}\right)^{\omega_{j}}\right)^{\frac{1}{M-p}},$

because
$\sum_{j=1}^{q}\omega_{j}=\tilde{\omega}(q-2N+k-1)+k+1\geq\dfrac{q(k+1)}{2N-k+1}\geq
M+1>M-p$.

Case 2: Assume that $T\neq\emptyset$. We see that $T\subset R$ and so
$\mathrm{rank}_{{\mathbb{C}}}\\{a_{j};\ j\in T\\}\leq M-p$ holds. Since the
intersection of $V$ and arbitrary $N+1$ hyperplanes $Q_{j}\ (1\leq j\leq q)$
is empty, there are at most $N$ indexes $j$ satisfying $\langle
F^{(0)}_{\mathcal{V}}(z),a_{j}\rangle=0$ (i.e., $Q_{j}(F)(z)=0$). It yields
that $\sharp T<N+1$, and hence $l\leq\mathrm{rank}_{{\mathbb{C}}}\\{a_{j};\
j\in T\\}\leq M-p$. Then, we have

$\displaystyle\sum_{j=1}^{q}\omega_{j}\Phi_{\mathcal{V},jp}$
$\displaystyle\geq\sum_{j\in T}\omega_{j}\Phi_{\mathcal{V},jp}\geq
Kl\prod_{j\in
T}\left(\frac{\Phi_{\mathcal{V},jp}}{K}\right)^{\frac{\omega_{j}}{l}}$
$\displaystyle\geq Kl\prod_{j\in
T}\left(\frac{\Phi_{\mathcal{V},jp}}{K}\right)^{\omega_{j}/(M-p)}\geq
C\prod_{j\in
S}\left(\frac{\Phi_{\mathcal{V},jp}}{K}\right)^{\omega_{j}/(M-p)},$

for some positive constant $C>0$, which depends only on $Q_{1},\ldots,Q_{q}$.

From the above two cases, we get the conclusion of the theorem. ∎

######  Theorem 2.7.

Let the notations and the assumption be as in Theorem 2.6 and let
$\tilde{\omega}$ be the Nochka constant for these hypersurfaces (defined in
the Lemma 2.1). Then, for every $\epsilon>0$, there exist a positive number
$\delta\ (>1)$ and $C$, depending only on $\mathcal{V},\epsilon$ and $Q_{j}$
such that

$\displaystyle dd^{c}$
$\displaystyle\log\dfrac{\prod_{p=0}^{M-1}|F_{\mathcal{V},p}|^{2\epsilon}}{\prod_{1\leq
j\leq q,0\leq p\leq
M-1}\log^{2\omega_{j}}\left(\delta/\varphi_{\mathcal{V},p}(Q_{j})\right)}$
$\displaystyle\geq
C\left(\dfrac{|F_{\mathcal{V},0}|^{2\left(\tilde{\omega}(q-(2N-k+1))-M+k\right)}|F_{\mathcal{V},M}|^{2}}{\prod_{j=1}^{q}(|F_{\mathcal{V},0}(Q_{j})|^{2}\prod_{p=0}^{M-1}\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{2}{M(M+1)}}dd^{c}|z|^{2}.$

###### Proof.

We denote by $A$ the left hand side. Thus

$A=\epsilon\sum_{p=0}^{M-1}\Omega_{p}+\sum_{j=1}^{q}\omega_{j}\sum_{p=0}^{M-1}dd^{c}\log\dfrac{1}{\log^{2}\left(\delta/\varphi_{\mathcal{V},p}(Q_{j})\right)}.$

Choose a positive number $\delta_{0}(\epsilon/l)$ with the properties as in
Proposition 2.2, where $l=\sum_{j=1}^{q}\omega_{j}$. For an arbitrarily fixed
$\delta\geq\delta_{0}(\epsilon/l)$, we obtain

$\displaystyle A$
$\displaystyle\geq\epsilon\sum_{p=0}^{M-1}\Omega_{p}+\sum_{j=1}^{q}\omega_{j}\sum_{p=0}^{M-1}\left(\dfrac{2\varphi_{\mathcal{V},p+1}(Q_{j})}{\varphi_{\mathcal{V},p}(Q_{j})\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}-\dfrac{\epsilon}{l}\right)\Omega_{p}$
$\displaystyle=\sum_{p=0}^{M-1}2\left(\sum_{j=1}^{q}\omega_{j}\Phi_{\mathcal{V},jp}\right).$

From Theorem 2.6, it follows that

$A\geq
C_{1}\sum_{p=0}^{M-1}2\left(\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{\frac{1}{M-p}}\Omega_{p},$

for some positive constant $C_{1}>0$. Let $\Omega_{p}=h_{p}dd^{c}|z|^{2}$, we
have

$\displaystyle A$ $\displaystyle\geq
2C_{1}\sum_{p=0}^{M-1}\left(h_{p}^{M-p}\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{\frac{1}{M-p}}dd^{c}|z|^{2}$
$\displaystyle\geq
C^{\prime}_{1}\sum_{p=0}^{M-1}(M-p)\left(h_{p}^{M-p}\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{\frac{1}{M-p}}dd^{c}|z|^{2}$
$\displaystyle\geq
C_{2}\prod_{p=0}^{M-1}\left(h_{p}^{M-p}\prod_{j=1}^{q}\Phi_{\mathcal{V},jp}^{\omega_{j}}\right)^{\frac{2}{M(M+1)}}dd^{c}|z|^{2},$

for some positive constants $C^{\prime}_{1}>0,C_{2}>0$. On the other hand, we
note that

$\displaystyle\prod_{p=0}^{M-1}\Phi_{\mathcal{V},jp}$
$\displaystyle=\prod_{p=0}^{M-1}\dfrac{\varphi_{\mathcal{V},p+1}(Q_{j})}{\varphi_{\mathcal{V},p}(Q_{j})}\dfrac{1}{\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}$
$\displaystyle=\dfrac{|F_{0}|^{2}}{|F_{0}(Q_{j})|^{2}}\prod_{p=0}^{M-1}\dfrac{1}{\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}$

and

$\prod_{p=0}^{M-1}h_{p}^{M-p}=\prod_{p=0}^{M-1}\left(\dfrac{|F_{\mathcal{V},p-1}|^{2}|F_{\mathcal{V},p+1}|^{2}}{|F_{\mathcal{V},p}|^{4}}\right)^{M-p}=\dfrac{|F_{\mathcal{V},M}|^{2}}{|F_{\mathcal{V},0}|^{2(M+1)}},$

because
$\varphi_{\mathcal{V},0}(Q_{j})=|F_{\mathcal{V},0}(Q_{j})|/|F_{\mathcal{V},0}|,\varphi_{\mathcal{V},M}(Q_{j})=1$.
Therefore, we get

$A\geq
C\left(\dfrac{|F_{\mathcal{V},0}|^{2(l-M-1)}|F_{\mathcal{V},M}|^{2}}{\prod_{j=1}^{q}(|F_{\mathcal{V},0}(Q_{j})|^{2}\prod_{p=0}^{M-1}\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{2}{M(M+1)}}dd^{c}|z|^{2}.$

Since $l-M-1=\tilde{\omega}(q-2N+k-1)-M+k$, we have the conclusion of the
theorem. ∎

Applying Theorems 2.6 and 2.7, we now give the following main lemmas of this
paper (Lemmas 2.8 and 2.10) as follows.

######  Lemma 2.8.

Let $V,\mathcal{V},d,M,\\{Q_{i}\\}_{i=1}^{q}$ and $\\{\omega_{i}\\}_{i=1}^{q}$
be as in Theorem 2.6. Let $f:\Delta(R)\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic map with a reduced
representation $F=(f_{0},\ldots,f_{n})$, which is nondegenerate over
$I_{d}(V)$. Assume that there are positive real numbers $\eta_{j}$, functions
$u_{j}$ satisfying two conditions $(D_{1}),(D_{2})$ with respect to $F,Q_{j}$
and $\eta_{j}\ (1\leq j\leq q)$. Then for an arbitrarily given $\epsilon$
satisfying

$\gamma=\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1>\epsilon(\sigma_{M+1}+1).$

the pseudo-metric $d\tau^{2}=\eta^{2}|dz|^{2}$, where

$\eta=\left(\dfrac{|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}|F_{\mathcal{V},M}|\prod_{p=0}^{M}|F_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}\cdot(\prod_{j,p}\log(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}$

and $\delta$ is the number satisfying the conclusion of Theorem 2.7, is
continuous and has strictly negative curvature.

###### Proof.

We see that the function $\eta$ is continuous at every point $z$ with
$\prod_{j=1}^{q}Q_{j}(F)(z)\neq 0$. For a point $z_{0}\in S$ such that
$\prod_{j=1}^{q}Q_{j}(F)(z_{0})=0$, from Theorem 2.4 and the property of
$u_{j}$, we have

$\displaystyle\nu_{\eta}(z_{0})$
$\displaystyle\geq\frac{1}{\sigma_{M}+\epsilon\tau_{M}}\biggl{(}\nu_{F_{\mathcal{V},M}}(z_{0}))+\sum_{j=1}^{q}\min(\nu_{Q_{j}(F)}(z_{0}),M)-\sum_{j=1}^{q}\omega_{j}\nu_{Q_{j}(F)}(z_{0})$
$\displaystyle+\sum_{j=1}^{q}\omega_{j}\nu_{e^{u_{j}}|z-z_{0}|^{-\min(M,\nu_{Q_{j}(F)}(z_{0}))}}(z_{0})\biggl{)}\geq
0.$

This implies that $d\tau^{2}$ is a continuous pseudo-metric on $\Delta(R)$.

We now prove that $d\tau^{2}$ has strictly negative curvature on $\Delta$. Let
$\Omega$ be the Fubini-Study form of ${\mathbb{P}}^{n}({\mathbb{C}})$ and
denote by $\Omega_{f}$ the pull-back of $\Omega$ by $f$. By Theorem 2.4, we
have

$dd^{c}\log\dfrac{|F_{\mathcal{V},M}|}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}+\sum_{j=1}^{q}\omega_{j}dd^{c}[u_{j}]\geq
0.$

Then by Theorems 2.3 and 2.7 and the definition of $\eta$, we have

(2.9)
$\displaystyle\begin{split}dd^{c}\log\eta&\geq\dfrac{\gamma-\epsilon(\sigma_{M+1}+1)}{\sigma_{M}+\epsilon\tau_{M}}d\Omega_{f}+\dfrac{\epsilon}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\left(|F_{\mathcal{V},0}|\cdots|F_{\mathcal{V},M}|\right)\\\
&+\dfrac{1}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\dfrac{\prod_{p=0}^{M-1}|F_{\mathcal{V},p}|^{2\epsilon}}{\prod_{p=0}^{M-1}\log^{4\omega_{j}}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}\\\
&\geq\dfrac{\epsilon\tau_{M}}{2\sigma_{M}(\sigma_{M}+\epsilon\tau_{M})}\left(\dfrac{|F_{\mathcal{V},0}|^{2}\cdots|F_{\mathcal{V},M}|^{2}}{|F_{\mathcal{V},0}|^{2\sigma_{M+1}}}\right)^{\frac{1}{\tau_{M}}}dd^{c}|z|^{2}\\\
&+C_{0}\left(\dfrac{|F_{\mathcal{V},0}|^{2\left(\tilde{\omega}(q-2N+k-1)-M+k\right)}|F_{\mathcal{V},M}|^{2}}{\prod_{j=1}^{q}(|Q_{j}(F)|^{2}\prod_{p=0}^{M-1}\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}}}dd^{c}|z|^{2}\\\
&\geq
C_{1}\left(\dfrac{|F_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}|F_{\mathcal{V},M}|\prod_{p=0}^{M}|F_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(F)|\prod_{p=0}^{M-1}\log(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{2}{\sigma_{M}+\epsilon\tau_{M}}}dd^{c}|z|^{2}\end{split}$

for some positive constant $C_{0},C_{1}$, where the last inequality comes from
Hölder’s inequality. On the other hand, we have
$e^{u_{j}}\leq\|F\|^{d\eta_{j}}$ and $|Q_{j}(F)|\leq\|F\|^{d\eta_{j}}$,

$\displaystyle|F_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}=|F_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}+\sum_{j=1}^{q}\omega_{j}\eta_{j}}$
$\displaystyle\hskip
20.0pt\geq|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}e^{\omega_{1}u_{1}}|Q_{1}(F)|^{\frac{\epsilon}{q}}\cdots
e^{\omega_{q}u_{q}}|Q_{q}(F)|^{\frac{\epsilon}{q}}.$

This implies that $\Delta\log\eta^{2}\geq C_{2}\eta^{2}$ for some positive
constant $C_{2}$. Therefore, $d\tau^{2}$ has strictly negative curvature. ∎

######  Lemma 2.10 (Main Lemma).

Let $V,\mathcal{V},d,M,\\{Q_{i}\\}_{i=1}^{q}$ and $\\{\omega_{i}\\}_{i=1}^{q}$
be as in Theorem 2.6. Let $f:\Delta(R)\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic map with a reduced
representation $F=(f_{0},\ldots,f_{n})$, which is nondegenerate over
$I_{d}(V)$. Assume that there are positive real numbers $\eta_{j}$, functions
$u_{j}$ satisfying two conditions $(D_{1}),(D_{2})$ with respect to $F,Q_{j}$
and $\eta_{j}\ (1\leq j\leq q)$. Then for every $\epsilon>0$ satisfying

$\gamma=\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1>\epsilon(\sigma_{M+1}+1),$

there exists a positive constant $C$, depending only on
$\mathcal{V},Q_{j},\eta_{j},\omega_{j}\ (1\leq j\leq q)$, such that

$\eta=\dfrac{|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\leq
C\left(\dfrac{2R}{R^{2}-|z|^{2}}\right)^{\sigma_{M}+\epsilon\tau_{M}}.$

###### Proof.

As in the proof of Lemma 2.8, we have

$dd^{c}\log\eta\leq C_{2}\eta^{2}dd^{c}|z|^{2}.$

According to Lemma 2.5, this implies that

$\eta\leq C_{3}\dfrac{2R}{R^{2}-|z|^{2}},$

for some positive constant $C_{3}$. Then we have

$\left(\dfrac{|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}|F_{\mathcal{V},M}|\prod_{p=0}^{M}|F_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}\cdot(\prod_{j,p}\log(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}\leq
C_{3}\dfrac{2R}{R^{2}-|z|^{2}}.$

It follows that

$\left(\dfrac{|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\frac{\epsilon}{q}}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{i}-\frac{\epsilon}{q}}\prod_{j=1}^{q}\prod_{p=0}^{M-1}(\varphi_{\mathcal{V},p}(Q_{j}))^{\frac{\epsilon}{2q}}(\log(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}\leq
C_{3}\dfrac{2R}{R^{2}-|z|^{2}}.$

Note that the function
$x^{\frac{\epsilon}{q}}\log^{\omega}\left(\dfrac{\delta}{x^{2}}\right)\
(\omega>0,0<x\leq 1)$ is bounded. Then we have

$\left(\dfrac{|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+1)}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\frac{\epsilon}{q}}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{i}-\frac{\epsilon}{q}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}\leq
C_{4}\dfrac{2R}{R^{2}-|z|^{2}},$

for a positive constant $C_{4}$. The lemma is proved. ∎

##  3\. Modified defect relation for Gauss map with hypersurfaces

In this section, we will prove the main theorem of the paper. Firstly, we need
some following preparation.

######  Lemma 3.1 (cf. [11, Lemma 1.6.7]).

Let $d\sigma^{2}$ be a conformal flat metric on an open Riemann surface $S$.
Then for every point $p\in S$, there is a holomorphic and locally
biholomorphic map $\Phi$ of a disk $\Delta(R_{0}):=\\{w:|w|<R_{0}\\}\
(0<R_{0}\leq\infty)$ onto an open neighborhood of $p$ with $\Phi(0)=p$ such
that $\Phi$ is a local isometric, namely the pull-back $\Phi^{*}(d\sigma^{2})$
is equal to the standard (flat) metric on $\Delta(R_{0})$, and for some point
$a_{0}$ with $|a_{0}|=1$, the curve $\Phi(\overline{0,R_{0}a_{0}})$ is
divergent in $S$ (i.e. for any compact set $K\subset S$, there exists an
$s_{0}<R_{0}$ such that $\Phi(\overline{0,s_{0}a_{0}})$ does not intersect
$K$).

Let $x=(x_{0},\ldots,x_{n}):S\rightarrow{\mathbb{R}}^{m}$ be the immersion of
the minimal surface $S$ into ${\mathbb{R}}^{m}$. Let $(x,y)$ be a isothermal
coordinate of $S$. Then $z=x+iy$ is a holomorphic coordinate of $S$. The
generalized Gauss map $g$ of $x$ is defined by

$g:S\rightarrow{\mathbb{P}}^{n}({\mathbb{C}}),g:=\left(\dfrac{\partial
x_{0}}{\partial z}:\cdots:\dfrac{\partial x_{n}}{\partial z}\right),\ \
(n=m-1).$

Denote by $G$ the reduced representation of $g$ defined by

$G_{z}=\left(\dfrac{\partial x_{0}}{\partial z},\ldots,\dfrac{\partial
x_{n}}{\partial z}\right).$

Also, the metric $ds^{2}$ on $S$ induced by the canonical metric on
${\mathbb{R}}^{m}$ satisfies

$ds^{2}=2\|G_{z}\|^{2}|dz|^{2}.$

We note that, $g$ is holomorphic since $S$ is minimal.

###### Proof of Theorem 1.1.

Let $A$ be an annular end of $S$, that is, $A=\\{z;\
0<1/r\leq|z|<r<\infty\\}$, where $z$ is the conformal coordinate. Denote by
$G$ the reduced representation of $g$ defined with respect to the coordinate
$z$.

We fix a ${\mathbb{C}}$-ordered basis $\mathcal{V}=([v_{0}],\ldots,[v_{M}])$
of $I_{d}(V)$ as in the Section 3. By replacing $Q_{i}$ with $Q_{i}^{d/\deg
Q_{i}}\ (1\leq i\leq q)$ if necessary, we may assume that all $Q_{i}\ (1\leq
i\leq q)$ are of the same degree $d$. Suppose that

$[Q_{j}]=\sum_{i=0}^{M}a_{ji}[v_{i}],$

where $\sum_{i=0}^{M}|a_{ji}|^{2}=1$.

Since $g$ is nondegenerate over $I_{d}(V)$, the contact function satisfying

$G_{\mathcal{V},p}(Q_{j})\not\equiv 0,\forall 1\leq j\leq q,0\leq p\leq M.$

Then, for each $j,p\ (1\leq j\leq q,0\leq p\leq M)$, we may choose
$i_{1},\ldots,i_{p}$ with $0\leq i_{1}<\cdots<i_{p}\leq M$ such that

$\psi(G)_{jp}=\sum_{l\neq
i_{1},\ldots,i_{p}}a_{jl}W(v_{l}(G),v_{i_{1}}(G_{z}),\ldots,v_{i_{p}}(G))\not\equiv
0.$

We also note that $\psi(G)_{j0}=G_{\mathcal{V},0}(Q_{j})=Q_{j}(G)$ and
$\psi(G)_{jM}=G_{\mathcal{V},M}$.

Suppose contrarily that

$\sum_{j=1}^{q}\delta^{H,M}_{g,A}(Q_{j})>\dfrac{(2N-k+1)(M+1)}{k+1}+\dfrac{(2N-k+1)M(M+1)}{2d(k+1)}.$

Then, there exist numbers $\eta_{j}>0\ (1\leq j\leq q)$ and functions $u_{j}$
satisfying two conditions $(D_{1}),(D_{2})$ with respect to $G$ and
$\\{Q_{j}\\}_{j=1}^{q}$ such that

$\sum_{j=1}^{q}(1-\eta_{j})>\dfrac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)},$

From Theorem 2.1, we have

$(q-2N+k-1)\tilde{\omega}=\sum_{j=1}^{q}\omega_{j}-k-1;\
\tilde{\omega}\geq\omega_{j}>0\text{ and
}\tilde{\omega}\geq\dfrac{k+1}{2N-k+1}.$

Therefore,

(3.2)
$\displaystyle\begin{split}\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1&\geq\tilde{\omega}(q-2N+k-1-\sum_{j=1}^{q}\eta_{j})-M+k\\\
&\geq\dfrac{k+1}{2N-k+1}\left(\sum_{j=1}^{q}(1-\eta_{j})-2N+k-1\right)-M+k\\\
&=\dfrac{k+1}{2N-k+1}\left(\sum_{j=1}^{q}(1-\eta_{j})-\dfrac{(2N+k-1)(M+1)}{k+1}\right)\\\
&>\dfrac{k+1}{2N-k+1}\cdot\dfrac{(2N+k-1)M(M+1)}{2d(k+1)}=\dfrac{\sigma_{M}}{d}.\end{split}$

Then, we can choose a rational number $\epsilon\ (>0)$ such that

$\dfrac{d(\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1)-\sigma_{M}}{d(\sigma_{M+1}+1)+\tau_{M}}>\epsilon>\dfrac{d(\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1)-\sigma_{M}}{\frac{1}{q}+d(\sigma_{M+1}+1)+\tau_{M}}.$

We define the following numbers

$\displaystyle h$
$\displaystyle:=d(\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1)-\epsilon
d(\sigma_{M+1}+1)>\sigma_{M}+\epsilon\tau_{M},$ $\displaystyle\rho$
$\displaystyle:=\dfrac{1}{h}(\sigma_{M}+\epsilon\tau_{M}),$
$\displaystyle\rho^{*}$
$\displaystyle:=\dfrac{1}{(1-\rho)h}=\dfrac{1}{d(\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1)-\sigma_{M}-\epsilon(d(\sigma_{M+1}+1)+\tau_{M})}.$

It is clear that $0<\rho<1$ and $\frac{\epsilon\rho^{*}}{q}>1.$

We consider a set

$A_{1}=\\{a\in A;\psi(G_{z})_{jp}(a)\neq 0,u_{j}(a)\neq-\infty\ \forall
j=1,\ldots,q;p=0,\ldots,M\\}$

and define a new pseudo-metric on $A_{1}$ as follows

$d\tau^{2}=\left(\dfrac{\prod_{j=1}^{q}\|G(Q_{j})\|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}}\prod_{j,p}|\psi(G)_{jp}|^{\frac{\epsilon}{q}}}\right)^{2\rho^{*}}|dz|^{2}.$

Here and throughout this section, for simplicity we will write $\prod_{j,p}$
for $\prod_{j=1}^{q}\prod_{p=1}^{M-1}$.

Since $Q_{j}(G),G_{\mathcal{V},M},\psi(G)_{jp}$ are all holomorphic functions
and $u_{j}\ (1\leq j\leq q)$ are harmonic functions on $A_{1}$, $d\tau$ is
flat on $A_{1}$.

######  Claim 3.3.

$d\tau$ is complete on the set $\\{z;\
|z|=r\\}\cup\\{z|\psi(G)_{jp}(z)=0\\}\cup\\{z|\ G(Q_{j})(z)=0\\}$, i.e., set
$\\{z;\ |z|=r\\}\cup\\{z|\ \psi(G)_{jp}(z)=0\\}\cup\\{z|\ G(Q_{j})(z)=0\\}$ is
at infinite distance from any interior point.

Indeed, if $a\in A$ such that
$\prod_{j=1}^{q}\left(G(Q_{j})(a)\cdot\prod_{i=0}^{M}\psi(G)_{jp}(a)\right)=0$
then by Theorem 2.4, we have

$\displaystyle\nu_{d\tau}(a)$
$\displaystyle\leq-\biggl{[}\nu_{G_{\mathcal{V},M}}(a)-\sum_{j=1}^{q}\omega_{j}\nu_{G(Q_{j})}(a)+\sum_{j=1}^{q}\omega_{j}\min\\{\nu_{G(Q_{j})}(a),M\\}$
$\displaystyle\ \
+\sum_{j=1}^{q}\omega_{j}\nu_{e^{u_{j}|z-z_{0}|^{-\min\\{\nu_{G(Q_{j})}(a),M\\}}}}(a)+\epsilon\nu_{G_{\mathcal{V},M}}(a)$
$\displaystyle\ \
+\left(\dfrac{\epsilon}{q}\sum_{j=1}^{q}\big{(}\min\\{\nu_{G(Q_{j})}(a),M\\}+\sum_{p=0}^{M-1}\nu_{\psi(G)_{jp}}(a)\big{)}\right)\biggl{]}\rho^{*}$
$\displaystyle\leq-\epsilon\rho^{*}\nu_{G_{\mathcal{V},M}}(a)-\dfrac{\epsilon\rho^{*}}{q}\sum_{j=1}^{q}\big{(}\min\\{\nu_{G(Q_{j})}(a),M\\}+\sum_{p=0}^{M-1}\nu_{\psi(G)_{jp}}(a)\big{)}\leq-\dfrac{\epsilon\rho^{*}}{q}$

(since $\min\\{\nu_{G(Q_{j})}(a),M\\}>1$ on
$\mathrm{Supp}\,\min\\{\nu_{G(Q_{j})}(a),M\\}$). Then, there is a positive
constant $C$ such that

$|d\tau|\geq\dfrac{C}{|z-z(a)|^{\frac{\epsilon\rho^{*}}{q}}}|dz|$

in a neighborhood of $a$, where $\frac{\epsilon\rho^{*}}{q}>1$. Then $d\tau$
is complete on the set

$\\{z|\psi(G)_{jp}(z)=0\\}\cup\\{z|\ G(Q_{j})(z)=0\\}.$

Now suppose that $d\tau^{2}$ is not complete on the set $\\{z;\ |z|=r\\}$.
Then, there exists a curve $\gamma:[0,1)\rightarrow A_{1}$, where
$\gamma(1)\in\\{z;\ |z|=r\\}$ so that the length of $\gamma$ in the metric
$d\tau$ satisfying $l_{d\tau}(\gamma)<\infty$. Furthermore, we may also assume
that $dist(\gamma(0),\\{z;\ |z|=1/r\\})>3l_{d\tau}(\gamma)$.

Take a disk $\Delta$ (in the metric induced by $d\tau$) with centered at
$\gamma(0)$. Since $d\tau$ is flat, by Lemma 3.1, $\Delta$ is isometric to an
ordinary disk in the plane. Let
$\Phi:\Delta(r)=\\{|\omega|<r\\}\rightarrow\Delta$ be this isometric with
$\Phi(0)=\gamma(0)$. Extend $\Phi$ as a local isometric into $A_{1}$ to a the
largest disk possible $\Delta(R)=\\{|\omega|<R\\}$, and denoted again by
$\Phi$ this extension (for simplicity, we may consider $\Phi$ as the
exponential map). Since $\int_{\gamma|_{[0,1)}}d\tau<{\rm
dist}(\gamma(0),\\{z;\ |z|=1/r\\})/3$, we have $R\leq{\rm
dist}(\gamma(0),\\{z;\ |z|=1/r\\})/3$. Hence, the image of $\Phi$ is bounded
away from $\\{z;\ |z|=1/r\\}$ by a distance at least ${\rm
dist}(\gamma(0),\\{z;\ |z|=1/r\\})/3$. Since $\Phi$ cannot be extended to a
larger disk, it must be hold that the image of $\Phi$ goes to the boundary
$A_{1}$. But, this image cannot go to points $a$ of $A$ with
$\prod_{j=1}^{q}G(Q_{j})(a)\prod_{p=0}^{M}|\psi(G)_{ip}|(a)=0$, since we have
already shown that $\gamma(0)$ is infinitely far away in the metric with
respect to these points. Then the image of $\Phi$ must go to the set $\\{z;\
|z|=r\\}$. Hence, by again Lemma 3.1, there exists a point $w_{0}$ with
$|w_{0}|=R$ so that $\Gamma=\Phi(\overline{0,w_{0}})$ is a divergent curve on
$S$.

We now show that $\Gamma$ has finite length in the original metric $ds^{2}$ on
$S$, contradicting the completeness of $S$. Let
$f:=g\circ\Phi:\Delta(R)\rightarrow V\subset{\mathbb{P}}^{n}({\mathbb{C}})$,
$\omega\in\Delta(R)\mapsto f(\omega)\in V$, be a holomorphic curve which is
nondegenerate over $I_{d}(V)$. Then $f$ have a reduced representation

$F=(f_{0},\ldots,f_{n}),$

where $f_{i}=G\circ\Phi\ (0\leq i\leq n).$ Hence, we have:

$\displaystyle\Phi^{*}ds^{2}$
$\displaystyle=2\|G\circ\Phi\|^{2}|\Phi^{*}dz|^{2}=2\|F\|^{2}\left|\dfrac{d(z\circ\Phi)}{dw}\right|^{2}|dw|^{2},$
$\displaystyle F_{\mathcal{V},M}$
$\displaystyle=(G\circ\Phi)_{\mathcal{V},M}=G_{\mathcal{V},M}\circ\Phi\cdot\left(\dfrac{d(z\circ\Phi)}{dw}\right)^{\sigma_{M}},$
$\displaystyle\psi(F)_{jp}$
$\displaystyle=\psi(G\circ\Phi)_{jp}=\psi(G)_{jp}\cdot\left(\dfrac{d(z\circ\Phi)}{dw}\right)^{\sigma_{p}},(0\leq
p\leq M).$

On the other hand, since $\Phi$ is locally isometric,

$\displaystyle|dw|=|\Phi^{*}d\tau|$
$\displaystyle=\left(\dfrac{\prod_{j=1}^{q}|Q_{j}(G)\circ\Phi|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}\circ\Phi|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|\psi(G)_{jp}\circ\Phi|^{\epsilon/q}}\right)^{\rho^{*}}\left|\dfrac{d(z\circ\Phi)}{dw}\right|\cdot|dw|$
$\displaystyle=\left(\dfrac{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}{|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|\psi(F)_{jp}|^{\epsilon/q}}\right)^{\rho^{*}}\left|\dfrac{d(z\circ\Phi)}{dw}\right|^{1+h\rho\rho^{*}}\cdot|dw|$

(because $1+\rho^{*}(\sigma_{M}+\epsilon\tau_{M})=1+h\rho\rho^{*}$). This
implies that

$\displaystyle\left|\dfrac{d(z\circ\Phi)}{dw}\right|$
$\displaystyle=\left(\dfrac{|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|\psi(F)_{jp}|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\right)^{\frac{\rho^{*}}{1+h\rho\rho^{*}}}$
$\displaystyle\leq\left(\dfrac{|(F_{\mathcal{V},M})_{w}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\right)^{\frac{\rho^{*}}{1+h\rho\rho^{*}}}$
$\displaystyle=\left(\dfrac{|(F_{\mathcal{V},M})_{w}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\right)^{1/h}.$

Hence, we have

$\displaystyle\Phi^{*}ds$
$\displaystyle\leq\sqrt{2}\|F\|\left(\dfrac{|F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\right)^{1/h}|dw|$
$\displaystyle=\sqrt{2}\left(\dfrac{|F_{\mathcal{V},0}|^{\frac{h}{d}}F_{\mathcal{V},M}|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}\circ\Phi}\prod_{j,p}|F_{\mathcal{V},p}(Q_{j})|^{\epsilon/q}}{\prod_{j=1}^{q}|Q_{j}(F)|^{\omega_{j}-\frac{\epsilon}{q}}}\right)^{1/h}|dw|.$

Here, note that
$\frac{h}{d}=\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1-\epsilon(\sigma_{M+1}+1)$.
Then the inequality (3.2) yields that the conditions of Lemma 2.5 are
satisfied. Applying Lemma 2.5 we have

$\Phi^{*}ds\leq C\left(\dfrac{2R}{R^{2}-|w|^{2}}\right)^{\rho}|dw|,$

for some positive constant $C$. Also, since $0<\rho<1$,

$l_{ds^{2}}(\Gamma)\leq\int_{\Gamma}ds=\int_{\overline{0,w_{0}}}\Phi^{*}ds\leq
C\cdot\int_{0}^{R}\left(\dfrac{2R}{R^{2}-|w|^{2}}\right)^{\rho}|dw|<+\infty.$

This contradicts the assumption of completeness of $S$ with respect to
$ds^{2}$. Thus, Claim 3.3 is proved.

Now, we set $d\tilde{\tau}^{2}=\lambda|dz|^{2}$ where

$\displaystyle\lambda(z)$
$\displaystyle=\left(\dfrac{\prod_{j=1}^{q}\|G(Q_{j})(z)\|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}(z)|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}(z)}\prod_{j,p}|\psi(G)_{jp}(z)|^{\frac{\epsilon}{q}}}\right)^{2\rho^{*}}$
$\displaystyle\times\left(\dfrac{\prod_{j=1}^{q}\|G(Q_{j})(\frac{1}{z})\|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}(\frac{1}{z})|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}(\frac{1}{z})}\prod_{j,p}|\psi(G)_{jp}(\frac{1}{z})|^{\frac{\epsilon}{q}}}\right)^{2\rho^{*}}.$

Also, we note that $\\{u_{j}=-\infty\\}=Q_{j}(G)^{-1}\\{0\\}$. This yields
that the function

$\dfrac{\prod_{j=1}^{q}\|G(Q_{j})(z)\|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}(z))|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}(z)}\prod_{j,p}|\psi(G)_{jp}(z)|^{\frac{\epsilon}{q}}}$

is continuous and non-vanishing on $A_{1}$. Similarly the function

$\dfrac{\prod_{j=1}^{q}\|G(Q_{j})(\frac{1}{z})\|^{\omega_{j}-\frac{\epsilon}{q}}}{|G_{\mathcal{V},M}(\frac{1}{z}))|^{1+\epsilon}\prod_{j=1}^{q}e^{\omega_{j}u_{j}(\frac{1}{z})}\prod_{j,p}|\psi(G)_{jp}(\frac{1}{z})|^{\frac{\epsilon}{q}}}$

is continuous and non-vanishing on $A_{1}^{\prime}$, where

$A_{1}^{\prime}=\left\\{a\in A;\psi(G_{z})_{jp}\left(\frac{1}{a}\right)\neq
0,u_{j}\left(\frac{1}{a}\right)\neq-\infty\ \forall
j=1,\ldots,q;p=0,\ldots,M\right\\}.$

We set $A^{\prime}=A_{1}\cap A_{1}^{\prime}$. These facts yields that the
metric $d\tilde{\tau}^{2}$ is complete and flat on $A^{\prime}$.

From Lemma 3.1, there exist a local isometric $\Phi:\Delta(R_{0})\rightarrow
A^{\prime}$. Since $(A^{\prime},d\tilde{\tau}^{2})$ is a complete Riemann
surface, it is necessary that $R_{0}=\infty$. So $\Phi:{\mathbb{C}}\rightarrow
A^{\prime}\subset\\{z;\ |z|<r\\}$ is a non-constant holomorphic map, which
contradicts to Liouville’s theorem. Then, the supposition was wrong. Hence,
Theorem 1.1 is proved. ∎

## References

  * [1] L. V. Ahlfors, An extension of Schwarz’s lemma, Trans. Am. Math. Soc. 43 (1938) 359–364.
  * [2] G. Dethloff and P. H. Ha, Ramification of the Gauss map of complete minimal surfaces in ${\mathbb{R}}^{3}$ and ${\mathbb{R}}^{4}$ on annular ends, Ann. Fac. Sci. Toulouse Math. 23 (2014), 829–846.
  * [3] G. Dethloff, P. H. Ha and P. D. Thoan, Ramification of the Gauss map of complete minimal surfaces in ${\mathbb{R}}^{m}$ on annular ends, Colloquium Mathematicae 142 (2016), 149–167.
  * [4] P. H. Ha, An estimate for the Gaussian curvature of minimal surfaces in ${\mathbb{R}}^{m}$ whose Gauss map is ramified over a set of hyperplanes, Differential Geom. Appl. 32 (2014), 130–138.
  * [5] H. Fujimoto, Value distribution of the Gauss maps of complete minimal surfaces in $R^{m}$, J. Math. Soc. Japan, 35, no. 4 (1983), 663–681.
  * [6] H. Fujimoto, On the Gauss map of a complete minimal surface in $R^{m}$ , J. Math. Soc. Japan, 35 (1983), 279–288.
  * [7] H. Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan, 40 (1988), 235–247.
  * [8] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geom., 29 (1989), 245–262.
  * [9] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces II, J. Differential Geom., 31 (1990), 365–385.
  * [10] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces III, Nagoya Math. J., 124 (1991), 13–40.
  * [11] H. Fujimoto, Value Distribution Theory of the Gauss map of Minimal Surfaces in $R^{m}$, Aspect of Math., Vol. E21, Vieweg, Wiesbaden (1993).
  * [12] S. J. Kao, On values of Gauss maps of complete minimal surfaces on annular ends, Math. Ann. 291 (1991) 315–318.
  * [13] X. Mo, Value distribution of the Gauss map and the total curvature of complete minimal surface in ${\mathbb{R}}^{m}$, Pacific. J. Math., 163 (1994), 159–174.
  * [14] R. Osserman, Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math., 12 (1959), 229–232.
  * [15] R. Osserman, On complete minimal surfaces, Arch. Rational Mech. Anal., 13 (1963), 392–404.
  * [16] R. Osserman, Global properties of minimal surfaces in ${\mathbb{E}}^{3}$ and $E^{n}$, Ann. of Math., 80 (1964), 340–364.
  * [17] S. D. Quang and D. P. An, Second main theorem and unicity of meromorphic mappings for hypersurfaces in projective varieties, Acta Math. Vietnam. 42 (2017), no. 3, 455–470.
  * [18] S. D. Quang, Modified defect relation for Gauss maps of minimal surfaces with hypersuraces of projective varieties in subgeneral position, preprint.
  * [19] M. Ru, On the Gauss map of minimal surfaces immersed in ${\mathbb{R}}^{n}$, J. Diff. Geometry, 34 (1991) 411–423.
  * [20] L. Jin and M. Ru, Values of Gauss maps of complete minimal surfaces in ${\mathbb{R}}^{m}$ on annular ends, Trans. Amer. Math. Soc. 359 (2007), 1547–1553.
  * [21] F. Xavier, The Gauss map of a complete non-flat minimal surface cannot omit 1 points on the sphere, Ann. of Math., 113 (1981), 211-214.


# Semantics of the Black-Box: Can knowledge graphs help make deep learning
systems more interpretable and explainable?

Manas Gaur, Keyur Faldu, Amit Sheth Manas Gaur is a PhD student at the
Artificial Intelligence Institute, University of South Carolina, e-mail:
<EMAIL_ADDRESS>Faldu is Chief Data Scientist at Embibe, e-mail:
<EMAIL_ADDRESS>Sheth is the Founding Director at the Artificial Intelligence
Institute, University of South Carolina, e-mail<EMAIL_ADDRESS>accepted to IEEE Internet Computing, October 30, 2020

###### Abstract

The recent series of innovations in deep learning (DL) have shown enormous
potential to impact individuals and society, both positively and negatively.
The DL models utilizing massive computing power and enormous datasets have
significantly outperformed prior historical benchmarks on increasingly
difficult, well-defined research tasks across technology domains such as
computer vision, natural language processing, signal processing, and human-
computer interactions. However, the Black-Box nature of DL models and their
over-reliance on massive amounts of data condensed into labels and dense
representations poses challenges for interpretability and explainability of
the system. Furthermore, DLs have not yet been proven in their ability to
effectively utilize relevant domain knowledge and experience critical to human
understanding. This aspect is missing in early data-focused approaches and
necessitated knowledge-infused learning and other strategies to incorporate
computational knowledge. This article demonstrates how knowledge, provided as
a knowledge graph, is incorporated into DL methods using knowledge-infused
learning, which is one of the strategies. We then discuss how this makes a
fundamental difference in the interpretability and explainability of current
approaches, and illustrate it with examples from natural language processing
for healthcare and education applications.

###### Index Terms:

Knowledge Graphs, Knowledge Infusion, Neuro-Symbolic AI, Explainability,
Interpretability, Black-Box Deep Learning, Mental Healthcare, Education
Technology

##  I Black-Box Nature of DL Models

DLmodels are complex and opaque. The cascading sequences of linear and non-
linear mathematical transformations learned by models comprising millions of
parameters are beyond human comprehension and reasoning. This renders them as
“Black-Box” models for decision-making. For example, in a trivial case of
question answering: Context: I sometimes wonder how many alcoholics are
relapsing under the lockdowns; Question: Does the person have an addiction?
Response generated from a pre-trained seq2seq model: Yes;

Comparing this to a non-trivial case of question answering: Context: Then
others that insisted that what I have is depression even though manic episodes
aren’t characteristic to depression. I dread having to retread all this again
because the clinic where I get my mental health addressed is closing down due
to loss in business caused by the pandemic; Question: Does the person suffer
from depression?; Response generated from a pre-trained seq2seq model: Yes
(the correct answer is no).

In such scenarios, it is exceptionally challenging to probe the model’s
mechanism without the support of background knowledge [1]. Although Neural
Attention Models (NAM) [2] are endowed with a certain degree of
interpretability in visualizing parts of the sentence which are focused on
answering the question, they cannot provide further explanations to answer the
question in a human-understandable format. Furthermore, recent research has
highlighted challenges concerning the model behavior in question answering
domains, particularly towards queries with disjunctive clauses (questions that
contain or) [3]. We have observed unpredictable responses to questions like
Are you feeling nervous or anxious or on edge? Is the feeling of restlessness
due to stress or anxiety? Does an employee own a company or work for a
company?

The ability to fine-tune pre-trained models trained in unsupervised settings
to a downstream task has alleviated the need for an extensive labeled dataset.
This introduces the challenge of explaining decisions using insights from
limited labeled data. Research practices have often resorted to using frozen
datasets, intending to surpass benchmark test-data-accuracy. As a result, DL
models have outperformed human baselines on specific datasets in terms of
‘test-data-accuracy’ but still fail to generalize intuitive cases in real-
world scenarios. Ribeiro et al. introduced CHECKLIST, a task agnostic
framework to test the generalisability of NLP models, and found that these
models are vulnerable to test data prepared with linguistically intuitive
rules [4]. The bottom line is that most state-of-the-art DL approaches are not
integrated with prior knowledge, a necessary condition for explaining the
predictions, and interpreting the model mechanism for appropriate probing.

##  II Need for Explainability and Interpretability

The Black-Box nature of DL models needs to be addressed to foster trust among
users and domain experts to be used with higher confidence. This can also
facilitate broader assimilation in a variety of domains. In healthcare,
clinicians routinely choose methods that allow them to understand how an
outcome was derived compared to an objectively superior method that cannot be
explained. In education, tracing students’ learning outcomes with attribution
to weak academic and behavioral areas is a better tool for teachers compared
with the ability to predict only a student’s performance. As a result,
explainability and interpretability for the DL models have focused on recent
key research areas.

Prior research has attempted to establish a generic definition of
explainability, which is the ability to generate human-comprehensible
explanations around the decision-making process [5][6][7]. In contrast,
interpretability is the ability to discern the internal mechanisms of any
module. Key reasons we need explainability and interpretability are to: (a)
Trace and verify the fine-grained supporting information in safety-critical
systems (e.g., autonomous driving vehicles) (b) Support evolving events and
discern necessary context - as the underlying facts (data) may not be static
(e.g. newer findings replace earlier findings), and meta-information such as
time and space/location may be critical in understanding, interpreting, and
explaining results (e.g., identifying emerging sub-events in natural disasters
[8]) (c) Discover inherent bias in the model’s predictive strategy (e.g.,
contextual modeling [2][9]) (d) Prevent prediction errors in unintuitive
scenarios (e.g. adversarial examples [10], CHECKLIST [4]) (e) Make sure minor
perturbations in inputs are handled in robust ways (f) After enhancement of
training data through knowledge. graph (KG), proactive detection of outliers
is possible. (g) Gather new knowledge insights to further enable research and
ensure that acquired domain knowledge is getting leveraged for the decision
making process. The enumerated list combines the need for “explainability and
interpretability” as it applies to DL systems and enhancements obtained by
integrating a knowledge graph-based approach. Addressing these concerns would
foster confidence among domain experts and trust among end-users. The terms
explainability and interpretability are often used interchangeably in the
prior research without clear distinctions and the different roles that they
play [10][11][12].

##  III Defining Explainability and Interpretability with Knowledge-infusion

Making explanations on the model behavior is subjective to the problem from
the stakeholders. A set of privileged knowledge (e.g., domain expertise,
advice specific to the situation) must be infused to comprehend the model
outcomes and interpret its functioning. The methods for infusing knowledge in
DL models occurs through a set of neuro-symbolic procedures that enrich the
dataset with concepts and relationships from multiple KGs or ontologies, which
assists end-users in decision making. For instance: Taxonomic Knowledge (e.g.,
Columbia Suicide Severity Risk Scale (C-SSRS)), Relational Knowledge (e.g.,
ConceptNet), Metaphorical Knowledge, and Behavioral Knowledge (e.g., LIWC) are
necessary forms of external contextual information required to understand
online conversations, mainly when the problem is a low resource (insufficient
benchmark datasets and unlabeled corpus for transfer learning) [13]. Also,
knowledge-infusion during the model learning using information-theoretic loss
function (e.g., KL divergence [2]) can check conceptual drifting at the
representational level through weak-supervision from KG. Alternatively, the
loss of information during learning can be supplemented by augmenting the
abstract information from KG to layered representation in DL models through
various mathematical operations (e.g., pointwise multiplication,
concatenation). Fusing the relevant information from KG to hidden
representations in DL allows quantitative and qualitative assessment of its
functioning, which we define as knowledge-infused interpretability. One
possible way to do knowledge infusion for discerning model behavior is through
an architecture having layers of DL connected to KG through a function. Such a
function can be a “knowledge-aware loss function” that computes the loss in
information at each layer per epoch. Or a function can be “knowledge-aware
propagation function,” which computes the loss information and transfers
missing information through mathematical operations. The computing of loss in
information is performed by tracing the KG through the embedding of hidden
layers and printing the concepts and relationships that the model is learning
through pattern recognition. On the other hand, having a ground-truth subgraph
of KG would help in calculating the distance between the concepts learned by
hidden layers and actual concepts in the subgraph (all done at the embedding
level), which can be used to modulate the hidden embeddings and thus leading
to faster convergence with model interpretations. In achieving
interpretability and explainability through knowledge-infusion, these two
terminologies can be differentiated as Explainability would cater to why the
prediction has been made. In contrast, interpretability would unfold the
ability to understand patterns learned or knowledge acquired by the system.
Any explainable system has to be interpretable, but the reverse is not valid.
Explainability can be evaluated with its interpretability and completeness.

“Empirically, an explainable system would comprise of collectively exhaustive
interpretable subsystems and orchestration among them. More often than not,
explanations would be in natural language explaining the decision, while
interpretations can be statistical or conceptual (using either generic or
domain-specific KG [14, 9]) in nature pertaining to its inner functioning.”

Explanations can be thought of as answers to cascading “why questions” in the
context of model prediction until there is no longer a need to ask another
“why question”. Explanations are expected to be faithful and plausible.
Faithfulness defines how well the explanation correlates with the model’s
decision making. It is considered plausible when it has a human-understandable
justification for the prediction [15][16]. Such systems are considered to be
potentially useful for real-world decision making in various domains,
especially healthcare and education technology.

Focus Area |  Approach |  Methods |  Interpretable outcomes
---|---|---|---
Input Features |  Proxy Functions |  LIME, Gradients, Saliency maps, SmoothGrad |  Linear Model Coefficients, Relative Features Importance
Input Features |  Occlusion or Perturbations |  SHAP, Integrated Gradients, DeepLift, PDA, Activation Maximization |  Relative Features Importance
Input Features |  Tracing Positive Contributions |  LRP, Deconvolutions, Guided Backpropagation |  Relative Features Importance
Representations |  Projections |  PCA, ICA, NMF, Conicity |  Interpretable Vector Representations Space
Attentions |  Transformations |  Effective attention, attention flow |  Interpretable Attentions Distribution
Neurons, Layers Contribution |  Transformations |  Conductance, Layer Relevance Propagation, DIFFPOOL |  Neurons, Layers Contribution
Encoded Knowledge in Representations, Attentions, Layers |  Probes |  Knowledge Probes, Diagnostic Classifiers, Auxiliary Tasks |  Interpretable Vector Representations Space, Attentions Distribution, Relative Feature Importance
TABLE I: Prior approaches and methods to discover interpretable subsystems.
Though it forms the superset of the taxonomy outlined in Gilpin et al., these
methods are not sufficient to achieve explainability and interpretability of
the level required for decision making in high stakes problems, including
mental healthcare and education technology.

##  IV Limitations of Learning using Distributional Semantics

The recent success of DL language models in NLP has been attributed to self-
supervised objectives over a large volume of unlabeled data such as BERT,
RoBERTa, and T5. These models learn distributional semantics about the text
and relationships between phrases. It is also an active research area to probe
whether these models learn linguistic knowledge like part of speech and
dependency tree. Many probing experiments have given insights into linguistic
patterns discovered by different layers and internal components like attention
heads of neural networks. However, it is still an open question, how much
semantic knowledge is learned by these models, specifically when the semantics
are not expressed with statistically significant patterns in the data. This
becomes evident when these models fail to learn facts around concepts.

Furthermore, there is a growing trend of fine-tuning the pre-trained models
with limited labeled data. These have given success when (a) the distribution
of the labeled dataset is similar to unlabeled data used for pre-training, (b)
tasks are relatively straightforward like natural language entailment and span
extractive question answering. However, real-world scenarios are often more
complex, which poses the following challenges: (a) Fine-tuning such models for
domain-specific tasks with limited labeled data may not be sufficient to learn
domain knowledge as these data may not be able to capture domain knowledge.
(b) Similarly, self-supervised training objectives over unlabeled data is not
attempting to learn the domain knowledge required for real-world adoption. (c)
Generating explanations and interpretations would be even more challenging as
internal layers or mechanisms of distributional semantics would not represent
specific domain knowledge.

##  V Knowledge Graph Infusion for Better Explainability

Aforementioned concerns in distributional semantics calls for a specific need
to infuse domain knowledge in deep neural models for better predictive
performance and the descriptive performance of generating explanations and
interpretations. Infusing domain knowledge in DL models can be categorized
into the shallow infusion, semi-deep infusion, and deep infusion [2]. In
shallow infusion, both the external information and method of knowledge
infusion is shallow, i.e., Word2Vec or GloVE embeddings are reconstructed
using domain knowledge as features (see Figure 1 [17]). On the other hand,
deep infusion of knowledge is a paradigm that couples the latent
representation learned by deep neural networks with the KGs exploiting the
semantic relationships between entities [18].

Recently, there has been a flurry of research to experiment with different
techniques to infuse knowledge graphs or domain rules to deep neural models
for different needs. Mainly they can be categorized as follows: (a) Fusion of
relevant external knowledge as an additional context in the query or as
external knowledge-based features [19],[20], (b) Infusing KG embedding to
hidden layers of neural models for explainable decision making [21], (c)
Infusing contextual representations from relevant subgraph or paths of KG
through either concatenation or pooling or non-linear transformations [22],
(d) Leveraging KG to alter attention mechanism and pre-training of DL model
[23], and (e) Developing approaches that uses domain-specific KG using Integer
Linear Programming for non-trivial natural language inference [24].

Figure 1: Illustration of shallow infusion wherein the Google BERT model is
reconstructed after concepts from Drug Abuse Ontology (DAO) were appended to
existing vocabulary to assess the negative media exposure during COVID-19.
This weakly supervised approach with knowledge infusion is practical when it
is difficult to annotate millions of news articles and procure labeled dataset
for depression and drug abuse

##  VI Explanations and Interpretations for Healthcare Domain Use Cases

Consider an application domain of summarization, which is at the intersection
of natural language understanding and DL. An extensive body of research has
investigated summarizing news articles, meeting logs, and financial/legal
contracts that are often templatized. On the other hand, patient-clinician
conversations are open-ended as the clinician tries to reflect on the
patient’s responses. Consequently, the interview forms an inherent structure,
that raises challenges in natural language understanding: (1) Anaphora- where
sentences are purposefully paraphrased to elicit meaningful responses from an
agent (or user); (2) The clinical conversation contains implicit references to
healthcare conditions, developing sparsity in the clinically-relevant
concepts. Such a problem scenario requires a model to capture the context of
the conversation (see Figure 2).

The summarization of clinical conversation requires meaningful responses to be
associated with clinically-relevant questions while resolving anaphora
problem. State-of-the-art language models (e.g., BERT) trained over the large-
scale corpus of news articles fail to capture the question’s context and
assess the importance of a response from the clinician perspective. In
addition, BERT-like models’ fine-tuning is not helpful because updating the
model parameter needs to be governed by a structure that can abstractly
describe a clinical conversation with stratified knowledge (see Figure 3).
Hence, recently, Integer Linear Programming (ILP)-based summarization
approaches have gained sufficient traction from the NLP
community111https://www.jmir.org/preprint/20865. The optimization framework is
interpretable as knowledge is incorporated as constraints. The outcome is
explainable because optimization criteria reflect on the end-users
requirement.

Figure 2:  Overview of Knowledge-infused Abstractive Summarization (KiAS) for
an interview snippet of a patient’s responses to the question asked by Ellie
(virtual interviewer). Phrases relevant to mental health are identified using
the PHQ-9 lexicon. The contextual similarity between utterances is calculated
through a retrofitted embedding model. The resulting summaries contain
relevant questions and meaningful responses. KiAS strategy recognizes the
semantic similarity between Anxiety disorder [SNOMEDCT ID: 197480006] and PTSD
[SNOMEDCT ID: 47505003] as described in the SNOMED-CT hierarchy through an
”is-a” relationship and recorded in the PHQ-9 lexicon. It associates responses
of anxiety and PTSD interchangeably Figure 3: In an experiment to summarize a
transcript of a 12 minute recorded conversation between a patient and
clinician (https://bit.ly/patid313), ILP with PHQ-9 knowledge could align
appropriate responses to the question much better than simple ILP based
abstractive summarization and pre-trained BERT. KiAS generates summaries (7
sentences on an average) that capture informative questions and responses
exchanged during long (58 sentences on an average), ambiguous, and sparse
clinical diagnostic interviews Figure 4: Interpretations and Explanations for
Students Performance Prediction. Student’s attempt stream in practice or test
session is fed as “Inputs”. Domain knowledge present in knowledge graphs and
data lakes can be “shallow infused” in features, or “deep infused” in the
model. Explanations and Interpretations are generated along with prediction of
student score.

##  VII Explanations and Interpretations for Education Domain Use Cases

Let’s take an example of a use case in the education domain (see Figure 4). We
illustrate this figure with respect to two types of knowledge infusion -
shallow and deep infusion. Shallow infusion covers the scenario of using
knowledge at the start of the DL pipeline and deep infusion refers to using
knowledge at every layer of the pipeline 222http://bit.ly/kidl2020. Student’s
score prediction is one of the key problems to assess the student’s true
potential for a goal. In the context of this problem, explanations of this
prediction are really important as they reveal what governs the student score,
both strengths, and weaknesses on the academic and behavioral aspects. These
explanations can potentially induce remedial actions from the student and
mentor.

The domain knowledge for student performance prediction can be thought of as
(a) Academic knowledge graph, set of concepts, it’s metadata and relations
between concepts that plays an important role in tracing student’s concepts
mastery [25][18], and (b) Engagement and learning patterns from historical
student’s data [15]. Students’ historical knowledge state could be derived
from this data [26].

The domain knowledge infused model would be able to explain student’s
performance by tracing the predicted score to weak concepts, furthermore, to
the cause of these weak concepts. Infusing academic knowledge graphs and a
student’s historical knowledge state makes it possible to trace academic
weakness until the deepest pre-requisite concept impacts student score [27].
As shown in Figure 4, the student has got an attempt wrong on a question of
concept “Projectile Motion”, using a knowledge graph and student’s knowledge
state it could be traced deeper to the previous grade concept “Quadratic
Equation”.

Students’ behavior is an equally important part of tracing learning outcomes
[26]. Assessments or diagnostic tests are also created using such behavioral
profiling [25]. Student’s tendency to answer a question without comprehending
or thinking through can lead to “Careless Mistakes”, which could be the result
of guessing the answer or expressing over-confidence. Similarly, the inability
to leave a question unanswered to focus more on answering other questions
could be termed as a lack of decisiveness. Which could lead to “Overtime
Incorrects” attempts. Domain knowledge is needed to classify attempts to a
“Careless Mistakes”, or an “Overtime Incorrects” attempt. Figure 4 shows an
example of generating explanations with student’s behavioral traits like
“Careless Mistakes”, “Overtime Incorrects” by infusing domain knowledge into
the model.

Interpretations around these explanations could be traced back to the
student’s attempts-stream as input, domain knowledge, and data lake to
understand how these are derived by applying interpretability techniques
mentioned in Table 1. Moreover, the impact of such explanations on student’s
performance prediction could also be derived and it could furnish as a
foundation for recommendations and estimating learning outcomes upon student’s
action for the same [26].

##  VIII Discussion

In this article, we highlighted the need for explanations and interpretations
for the domain adoption of AI models, particularly in healthcare and education
technology. We overviewed existing statistical methods and metrics devised to
quantitatively assess the explainability of the model and interpretability of
its mechanisms. Existing frameworks categorized as post-hoc Interpretability,
counterfactual explanations, and rule-based explanations fall short in
providing answers to the following open questions:

  1. 1.

Can the model mine (varied) relationships from the existing text?

  2. 2.

Can the model reliably classify entities into known ontology?

  3. 3.

Can the model answer the question with trust and transparency?

  4. 4.

Is it possible to measure the model’s “reasonability” and “meaningfulness” of
the response to a question?

  5. 5.

How much context is needed for the model to provide a precise response?

An emerging trend to fine-tune a pre-trained model on limited labeled data for
a downstream task and the inability of distributional semantics learning to
capture domain-specific knowledge pose limitations in addressing the above
questions. We noticed the necessity of KG as an integral component in neuro-
symbolic AI systems with capabilities to generate explainable outcomes and
interpretability through tracing over KG. Future advances in this area would
seek AI systems that can help stakeholders (e.g., instructors, public health
experts) to conceptually understand the working of involved AI systems [28].
There is also a pressing requirement for benchmark datasets which assess the
quality of explanations derived from model outcomes and interpretability of
the algorithms in achieving explanations [4] [29].

## References

  * [1] O.-M. Camburu, T. Rocktäschel, T. Lukasiewicz, and P. Blunsom, “e-snli: Natural language inference with natural language explanations,” in _Advances in Neural Information Processing Systems_ , 2018, pp. 9539–9549.
  * [2] M. Gaur, U. Kursuncu, and R. Wickramarachchi, “Shades of knowledge-infused learning for enhancing deep learning,” _IEEE Internet Computing_ , vol. 23, no. 6, pp. 54–63, 2019.
  * [3] H. Ren, W. Hu, and J. Leskovec, “Query2box: Reasoning over knowledge graphs in vector space using box embeddings,” _arXiv preprint arXiv:2002.05969_ , 2020\.
  * [4] M. T. Ribeiro, T. Wu, C. Guestrin, and S. Singh, “Beyond accuracy: Behavioral testing of nlp models with checklist,” _arXiv preprint arXiv:2005.04118_ , 2020.
  * [5] S. Chari, O. Seneviratne, D. M. Gruen, M. A. Foreman, A. K. Das, and D. L. McGuinness, “Explanation ontology: A model of explanations for user-centered ai,” _arXiv preprint arXiv:2010.01479_ , 2020.
  * [6] A. Oltramari, J. Francis, C. Henson, K. Ma, and R. Wickramarachchi, “Neuro-symbolic architectures for context understanding,” _arXiv preprint arXiv:2003.04707_ , 2020.
  * [7] W. Samek and K.-R. Müller, “Towards explainable artificial intelligence,” in _Explainable AI: interpreting, explaining and visualizing deep learning_. Springer, 2019, pp. 5–22.
  * [8] C. Arachie, M. Gaur, S. Anzaroot, W. Groves, K. Zhang, and A. Jaimes, “Unsupervised detection of sub-events in large scale disasters,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, no. 01, 2020, pp. 354–361.
  * [9] M. Gaur, U. Kursuncu, A. Sheth, R. Wickramarachchi, and S. Yadav, “Knowledge-infused deep learning,” in _Proceedings of the 31st ACM Conference on Hypertext and Social Media_ , 2020, pp. 309–310.
  * [10] L. H. Gilpin, D. Bau, B. Z. Yuan, A. Bajwa, M. Specter, and L. Kagal, “Explaining explanations: An overview of interpretability of machine learning,” in _2018 IEEE 5th International Conference on data science and advanced analytics (DSAA)_. IEEE, 2018, pp. 80–89.
  * [11] B. Mittelstadt, C. Russell, and S. Wachter, “Explaining explanations in ai,” in _Proceedings of the conference on fairness, accountability, and transparency_ , 2019, pp. 279–288.
  * [12] O. Biran and C. Cotton, “Explanation and justification in machine learning: A survey,” in _IJCAI-17 workshop on explainable AI (XAI)_ , vol. 8, no. 1, 2017, pp. 8–13.
  * [13] H. Purohit, V. L. Shalin, and A. P. Sheth, “Knowledge graphs to empower humanity-inspired ai systems,” _IEEE Internet Computing_ , vol. 24, no. 4, pp. 48–54, 2020.
  * [14] S. Bhatt, M. Gaur, B. Bullemer, V. Shalin, A. Sheth, and B. Minnery, “Enhancing crowd wisdom using explainable diversity inferred from social media,” in _2018 IEEE/WIC/ACM International Conference on Web Intelligence (WI)_. IEEE, 2018, pp. 293–300.
  * [15] K. Faldu, A. Avasthi, and A. Thomas, “Adaptive learning machine for score improvement and parts thereof,” Mar. 29 2018, uS Patent App. 15/708,591.
  * [16] S. Goel, M. Gaur, and E. Jain, “Nature inspired algorithms in remote sensing image classification,” _Procedia Computer Science_ , vol. 57, pp. 377–384, 2015.
  * [17] A. Alambo, M. Gaur, and K. Thirunarayan, “Depressive, drug abusive, or informative: Knowledge-aware study of news exposure during covid-19 outbreak,” _arXiv preprint arXiv:2007.15209_ , 2020.
  * [18] U. Kursuncu, M. Gaur, and A. Sheth, “Knowledge infused learning (k-il): Towards deep incorporation of knowledge in deep learning,” _arXiv preprint arXiv:1912.00512_ , 2019.
  * [19] U. Kursuncu, M. Gaur, C. Castillo, A. Alambo, K. Thirunarayan, V. Shalin, D. Achilov, I. B. Arpinar, and A. Sheth, “Modeling islamist extremist communications on social media using contextual dimensions: Religion, ideology, and hate,” _Proceedings of the ACM on Human-Computer Interaction_ , vol. 3, no. CSCW, pp. 1–22, 2019.
  * [20] L. Yao, C. Mao, and Y. Luo, “Kg-bert: Bert for knowledge graph completion,” _arXiv preprint arXiv:1909.03193_ , 2019.
  * [21] X. Wang, D. Wang, C. Xu, X. He, Y. Cao, and T.-S. Chua, “Explainable reasoning over knowledge graphs for recommendation,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, 2019, pp. 5329–5336.
  * [22] B. Yang and T. Mitchell, “Leveraging knowledge bases in lstms for improving machine reading,” _arXiv preprint arXiv:1902.09091_ , 2019.
  * [23] E. Choi, Z. Xu, Y. Li, M. W. Dusenberry, G. Flores, Y. Xue, and A. M. Dai, “Graph convolutional transformer: Learning the graphical structure of electronic health records,” _arXiv preprint arXiv:1906.04716_ , 2019.
  * [24] D. Khashabi, T. Khot, A. Sabharwal, P. Clark, O. Etzioni, and D. Roth, “Question answering via integer programming over semi-structured knowledge,” _arXiv preprint arXiv:1604.06076_ , 2016.
  * [25] S. Dhavala, C. Bhatia, J. Bose, K. Faldu, and A. Avasthi, “Auto generation of diagnostic assessments and their quality evaluation,” in _Proceedings of The 13th International Conference on Educational Data Mining_ , 2020, pp. 730–735.
  * [26] C. Donda, S. Dasgupta, S. S. Dhavala, K. Faldu, and A. Avasthi, “A framework for predicting, interpreting, and improving learning outcomes,” _arXiv preprint arXiv:2010.02629_ , 2020.
  * [27] Z. Wang, X. Feng, J. Tang, G. Y. Huang, and Z. Liu, “Deep knowledge tracing with side information,” in _International Conference on Artificial Intelligence in Education_. Springer, 2019, pp. 303–308.
  * [28] D. Huk Park, L. Anne Hendricks, Z. Akata, A. Rohrbach, B. Schiele, T. Darrell, and M. Rohrbach, “Multimodal explanations: Justifying decisions and pointing to the evidence,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 8779–8788.
  * [29] C. Rudin, “Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead,” _Nature Machine Intelligence_ , vol. 1, no. 5, pp. 206–215, 2019.