# Modified defect relation for Gauss maps of minimal surfaces with
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.

In this paper, we establish some modified defect relations for the Gauss map
$g$ of a complete minimal surface $S\subset\mathbb{R}^{m}$ into a
$k$-dimension projective subvariety $V\subset\mathbb{P}^{n}(\mathbb{C})\
(n=m-1)$ with hypersurfaces $Q_{1},\ldots,Q_{q}$ of
$\mathbb{P}^{n}(\mathbb{C})$ in $N$-subgeneral position with respect to $V\
(N\geq k)$. In particular, we give the bound above for the number $q$ if the
image $g(S)$ intersects a finite number of times each hypersurfaces
$Q_{1},\ldots,Q_{q}$ and $g$ is nondegenerate over $I_{d}(V)$, where
$d=lcm(\deg Q_{1},\ldots,\deg Q_{q})$, i.e., the image of $g$ is not contained
in any hypersurface $Q$ of degree $d$ with $V\not\subset Q$. Our results
extend and generalise previous results for the case of the Gauss map and
hyperplanes in the projective spaces.

††footnotetext:  2010 Mathematics Subject Classification: Primary 53A10,
53C42; Secondary 30D35, 32H30.
Key words and phrases: Gauss map, value distribution, holomorphic curve,
modified defect relation, ramification, hypersurface.

##  1\. Introduction and Main results

In a series of papers [19, 20, 21, 22], Osserman began studying the value
distribution of the Gauss maps of minimal surfaces. We summarize his results
as the following theorem.

Theorem A (R. Osserman). Let $S$ be a complete minimal surface in
${\mathbb{R}}^{3}$. Then:

  * (1)

$S$ has infinite total curvature $\Leftrightarrow$ the Gauss map of $S$ takes
on all directions infinitely often with the exception of at most a set of
logarithmic capacity zero,

  * (2)

$S$ has finite non-zero total curvature $\Leftrightarrow$ the Gauss map of $S$
takes on all directions a finite number of times, omitting at most three
directions,

  * (3)

$S$ has zero total curvature $\Leftrightarrow$ $S$ is a plane.

In 1981, Xavier [27] improved the assertion (1) of Theorem A by showing that
the Gauss map of a complete minimal surface $S$ in ${\mathbb{R}}^{3}$ can omit
at most six directions unless it is a plane. Later on, Fujimoto [9] showed
that the result of Xavier still valid if the Gauss map omits only four
directions, and the number four is the best possible number.

Combining the early method of Osserman with the method of H. Fujimoto, X. Mo
and R. Osserman [15] proved the following theorem.

Theorem B (X. Mo and R. Osserman [15]). Let $S$ be a complete minimal surface
in ${\mathbb{R}}^{3}$ with infinite total curvature. Then its Gauss map must
take every direction infinitely often except at most four directions.

For the case of higher dimension, H. Fujimoto proved the following theorem.

Theorem C (H. Fujimoto [11]). Let $S$ be a complete minimal surface in
${\mathbb{R}}^{m}$ with nondegenerate Gauss map. Then the image of $S$ under
the Gauss map cannot fail to intersect more than $m(m+1)/2$ hyperplanes in
general position in ${\mathbb{P}}^{m-1}({\mathbb{C}})$.

And then, X. Mo in [16] improved the above theorem of H. Fujimoto by replacing
the condition “fail to intersect more than $m(m+1)/2$ hyperplanes” by the
condition “intersects only a finite number of times with these hyperplanes”.

However, in all the above mentioned results, the assumptions imply directly
that there is a compact subset $K$ of $S$ such that the image of $S\setminus
K$ under the Gauss map does not intersect these hyperplanes, i.e., the Gauss
map at near “the boundary of $S$” will omit these hyperplanes. In order to
study the general case where such compact subset $K$ may not exist, H.
Fujimoto have defined the notion of modified defect for holomorphic curves
from $S$ into ${\mathbb{P}}^{m-1}({\mathbb{C}})$ and proved some modified
defect relation for such curves with hyperplanes (cf. [10, 11, 12]). Here a
modified defect relation is an above bound estimate of the sum of some
modified defects for the curve with hyperplanes. In those works, H. Fujimoto
constructed a metric of negative curvature under certain conditions on $S$ and
then using Schwarz-Pick’s Lemma to derive an inequality which is the essential
key to the study of the value distribution of the Gauss map. Based on that
method, X. Mo [16] proved the following.

Theorem D (X. Mo [16]). Let $S$ be a complete nondegenerate minimal surface in
${\mathbb{R}}^{m}$ such that the Gauss map $f=(f_{0}:\cdots:f_{n})$ (here
$n=m-1$) intersects only a finite number of times the hyperplanes
$H_{1},\ldots,H_{q}$ of ${\mathbb{P}}^{n}({\mathbb{C}})$ in general position.
If $q>m(m+l)/2=(n+1)+n(n+l)/2$, then $S$ must have finite total curvature.

Our aim in this paper is to generalize the notion of modified defect for
holomorphic curves and hyperplanes of a projective space of H. Fujimoto to the
case of hypersurfaces in projective varieties, and then study the relations
between those modified defect relations for the Gauss maps and hypersurfaces
of a projective variety in subgeneral position with the total curvature of
minimal surfaces in ${\mathbb{R}}^{m}$. In order to do so, firstly we have to
establish the defect relation for holomorphic curves from punctured disks into
projective varieties with hypersurfaces in subgeneral position. Secondly, we
reconstructed all notions, functions and estimates introduced by H. Fujimoto
in [10, 11, 12] for the case of hypersurfaces. To state our results, we
introduce the following notions.

Let $S$ be an open complete Riemann surface. In this paper, 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 $f:S\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic curve and
let $Q$ be a hypersurface in ${\mathbb{P}}^{n}({\mathbb{C}})$ of degree $d$.
By $\nu_{Q(f)}$ we denote the pullback divisor $Q$ by $f$. Let
$F=(f_{0},\ldots,f_{n})$ be a global reduced representation of $f$ and $K$ be
a compact subset of $F$. Let $A=S\setminus K$.

The $S$-defect truncated to level $m$ of the hyperpsurface $Q$ for $f$ is
defined by

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

Here, condition $(*)_{S}$ means that there exists a $[-\infty,\infty)$-valued
continuous subharmonic function $u\ (\not\equiv\infty)$ on $A$ satisfying the
following conditions:

  * $(D_{1})$

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

  * $(D_{2})$

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

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

where $z$ is a holomorphic local coordinate around $\xi$.

The $H$-defect truncated to level $m$ of $Q$ for $f$ 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
$A\setminus f^{-1}(Q)$ and satisfies conditions $(D_{1}),(D_{2})$.

We say a function $u$ with mild singularities on a domain $D\subset S$ if $u$
is a complex-valued function of the class $\mathcal{C}^{\infty}$ on $D$
outside a discrete set $E$ and every point $a\in D$ has a neighborhood $U$
such that for a holomorphic local coordinate $z$ with $z(a)=0$ on $U$ we can
write

$|u(z)|=|z|^{\alpha}u^{*}(z)\prod_{j=1}^{q}\left|\log\dfrac{1}{|g_{j}(z)|v_{j}(z)}\right|^{r_{j}}$

on $U\setminus E$ with some real number $\sigma$, non-positive real numbers
$r_{j}$, nonzero holomorphic functions $g_{j}$ with $g_{j}(0)=0$ and some
positive $\mathcal{C}^{\infty}$ functions $u^{*}$ and $v_{j}$ on $U$, where
$0\leq q<\infty$.

The modified $H$-defect truncated to level $m$ of $Q$ for $f$ is defined by

$D^{m}_{f}(Q):=1-\mathrm{inf}\\{1-\eta\ |\ \eta\text{ satisfies the condition
(*)}.\\}$

Here, the condition (*) means that: there are a compact subset $K$ of $S$, a
divisor $\nu$ and a continuous real-valued bounded function $k$ with mild
singularities on $S\setminus K$, a positive constant $c$ such that $\nu(z)>c$
for each $z\in\mathrm{Supp}\,(\nu)$ and

$[\min\\{m,\nu_{Q(f)}\\}]+[\nu]=d\eta f^{*}\Omega+dd^{c}[\log|k|^{2}]$

in the sense of current, where $\Omega$ is the Fubini-Study form of
${\mathbb{P}}^{n}({\mathbb{C}})$. Hence, if we set $h=k\|F_{z}\|^{\eta}$ for a
local reduced representation $F_{z}$ (on a local holomorphic chat $U,z$) of
$f$ then $\log h$ is harmonic on $U\setminus(K\cup\mathrm{Supp}\,(\nu_{h})\cup
f^{-1}(Q))$ and $\nu_{h}-\min\\{m,\nu_{H(f)}\\}=v$.

By [11, Proposition 2.3] and [12, Remark 5.3] we see that

(1.1) $\displaystyle 0\leq\delta^{H,m}_{f,A}(Q)\leq\delta^{S,m}_{f,A}(Q)\leq
1\text{ and }0\leq\delta^{H,m}_{f,A}(Q)\leq D^{m}_{f}(Q)\leq 1.$

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$. Let $d$ be a positive integer. We 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

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

Then $H_{V}(d)$ is called the Hilbert function of $V$. Each element of
$I_{d}(V)$ which is an equivalent class of an element $Q\in H_{d},$ will be
denoted by $[Q]$.

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

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

If $\\{Q_{i}\\}_{i=1}^{q}$ is in $n$-subgeneral position then we say that it
is in general position with respect to $V.$ Throughout this paper, sometime we
will denote by the same notation $Q$ for the defining homogeneous polynomial
of the hypersurface $Q$ if there is no confusion.

Let $f:S\longrightarrow V$ be a holomorphic curve. We say that $f$ is
degenerate over $I_{d}(V)$ if there is $[Q]\in I_{d}(V)\setminus\\{0\\}$ such
that $Q(F)\equiv 0$ for some local reduced representation $F$ of $f$.
Otherwise, we say that $f$ is nondegenerate over $I_{d}(V)$. It is clear that
if $f:S\longrightarrow V$ is algebraically nondegenerate, then $f$ is
nondegenerate over $I_{d}(V)$ for every $d\geq 1.$ Our first main result is
stated as follows.

######  Theorem 1.2.

Let $S$ be a complete minimal surface in ${\mathbb{R}}^{m}$. Let $V$ be a
projective subvariety of dimension $k$ 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 $\deg Q_{j}\ (1\leq j\leq q)$,
i.e., $d=lcm(\deg Q_{1},\ldots,\deg Q_{q})$. Assume that the Gauss map
$g:S\rightarrow V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ is nondegenerate over
$I_{d}(V)$ and

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

where $M=H_{V}(d)-1$. Then $S$ has finite total curvature.

By the inequality (1.1), from Theorem 1.2 we immediately have the following
corollary about $H$-defect relation.

######  Corollary 1.3.

Let $S,V,\\{Q_{i}\\}_{i=1}^{q},d,M$ be as in Theorem 1.2. Let $K$ be a compact
subset of $S$ and $A=S\setminus K$. Assume that the Gauss map $g:S\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ is nondegenerate over $I_{d}(V)$ and

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

Then $S$ has finite total curvature.

We note that, a complete minimal surface has a universal covering
biholomorphic to ${\mathbb{C}}$ or a ball in ${\mathbb{C}}$. Then, from
Theorem 4.2 in [24], we have the following $S$-defect relation.

Theorem E (D. D. Thai and S. D. Quang [24]). Let $S$ be a complete minimal
surface in ${\mathbb{R}}^{m}$. Let $V,\\{Q_{i}\\}_{i=1}^{q},d,M$ be as in
Theorem 1.2. Assume that the Gauss map $g:S\rightarrow
V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ is nondegenerate over $I_{d}(V)$. Then
we have

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

On the other hand, for the case of minimal surface with finite total curvature
and the Gauss map is ramified at least $m_{j}$ over $Q_{j}\ (1\leq j\leq q)$,
in a recent work [25] the author with D. D. Thai and S. D. Quang showed a
sharper defect relation (cf. [25, Theorem 1.2]) as follows

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

Without the condition “ramified over hyperplanes”, for the case of linear
nondegenerate Gauss map and hyperplanes $\\{H_{i}\\}_{i=1}^{q}$ of
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position, P. H. Ha [3]
showed that

$\sum_{j=1}^{q}\delta^{S,n}_{g,S}(H_{j})\leq 2N-n+1+\dfrac{(2N-n+1)n}{2}.$

Our second purpose in this paper is to generalize all above results to the
case of hypersurfaces of projective varieties in subgeneral position and
without the condition “ramified over hypersurfaces”. Our second main result is
stated as follows.

######  Theorem 1.4.

Let $S$ be a complete minimal surface with finite total curvature in
${\mathbb{R}}^{m}$. Let $V,\\{Q_{i}\\}_{i=1}^{q},d,M$ be as in Theorem 1.2.
Assume that the Gauss map $g:S\rightarrow V$ is nondegenerate over $I_{d}(V)$.
Then we have

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

From the above theorems, we get a corollary on the ramification of the Gauss
map as follows.

######  Corollary 1.5.

Let $S,V,\\{Q_{i}\\}_{i=1}^{q},d,M,g,K,A$ be as in Theorem 1.3. Assume that
$g$ is ramified over $Q_{j}$ with multiplicity at least $m_{j}$ on $A$ for
each $j$ (i.e., $\nu_{Q_{j}(f)}\geq m_{j}$ on
$\mathrm{Supp}\,\nu_{Q_{j}(f)}\cap A$) and

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

Then $S$ has finite total curvature.

In particular, if $g$ intersects only a finite number of times the
hypersurfaces $Q_{1},\ldots,Q_{q}$ and

$q>\dfrac{(2N-k+1)(M+1)}{k+1}+\dfrac{(2N-k+1)M(M+1)}{2d(k+1)}$

then $S$ must have finite total curvature.

Then, this corollary immediately implies Theorem C mentioned above of X. Mo.

###### Proof.

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

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

Then

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

By Theorem 1.3, $S$ has finite total curvature. ∎

Now, combining Theorems 1.3 and 1.4, we may prove the following.

######  Corollary 1.6.

Let $S,V,\\{Q_{i}\\}_{i=1}^{q},d,M,g$ be as in Theorem 1.3. Then we have

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

In particular, the image of $S$ under $g$ cannot intersects only a finite
number of times with hyperplanes $Q_{1},\ldots,Q_{q}$ if
$q>\dfrac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)}$.

###### Proof.

Suppose contrarily that

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

By Theorem 1.3, this yields that $S$ has finite total curvature. Applying
Theorem 1.4, we have

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

Since $\delta^{H,M}_{g,S}(Q_{j})\leq\delta^{S,M}_{g,S}(Q_{j})$, we obtain

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

This contradicts the supposition. The desired inequality of the corollary must
hold. ∎

As we known, the first results on the value distribution of the Gauss map with
hypersurfaces are given by S. D.Quang, D. D. Thai, P. D. Thoan in [25].
However, they only consider the case of algebraic complete minimal surfaces.
Our results in this paper are the initial results for the general case of
complete minimal surfaces. The results of this paper generalize for previous
results in this area, for instance see [2, 3, 4, 6, 11, 16, 26].

##  2\. Defect relation of holomorphic curves from a punctured disk into a
projective variety with a family of hypersurfaces

A. Nevanlinna functions. For each $s>0$, we define the punctured disk

$\Delta_{s,\infty}=\\{z\in{\mathbb{C}}\ |\ s\leq|z|<\infty\\}.$

In this section, we always assume that functions and mappings from
$\Delta_{s,\infty}$ are defined on an open neighborhood of $\Delta_{s,\infty}$
in ${\mathbb{C}}$.

For a divisor $\nu$ on an open neighborhood of $\Delta_{s,\infty}$ and for a
positive integer $p$ or $p=\infty$, we define the counting function of $\nu$
by

$\displaystyle\nu^{[p]}(z)$ $\displaystyle=\min\ \\{M,\nu(z)\\},$
$\displaystyle n(t,s)$ $\displaystyle=\sum\limits_{s\leq|z|\leq t}\nu(z).$

Similarly, we define $n^{[p]}(t,s).$ The counting function of $\nu$ is defined
by

$N(r,s,\nu)=\int\limits_{s}^{r}\dfrac{n(t,s)}{t}dt\quad(s<r<\infty).$

Similarly, define $N(r,s,\nu^{[p]})$ and denote it by $N^{[p]}(r,s,\nu)$.

Let $\varphi:\Delta_{s,\infty}\longrightarrow{\mathbb{C}}$ be a meromorphic
function. Denote by $\nu^{0}_{\varphi}$ the zero divisor of $\varphi$. Define

$N_{\varphi}(r,s)=N(r,s,\nu^{0}_{\varphi}),\
N_{\varphi}^{[p]}(r,s)=N^{[p]}(r,s,\nu^{0}_{\varphi}).$

For brevity, we will omit the character [p] if $M=\infty$. The proximity
function of $\varphi$ (with respect to the point $\infty$) is defined by

$m(r,s,\varphi)=\int\limits_{0}^{2\pi}\log^{+}|\varphi(re^{i\theta})|\dfrac{d\theta}{2\pi}-\int\limits_{0}^{2\pi}\log^{+}|\varphi(se^{i\theta})|\dfrac{d\theta}{2\pi}\
(s<r<\infty),$

where $\log^{+}x=\log(\max\\{0,x\\})$ for a real number $x$. The Nevanlinna’s
characteristic function of $\varphi$ is defined by

$T(r,s,\varphi):=m(r,s,\varphi)+N_{1/\varphi}(r,s).$

Throughout this paper, we fix a homogeneous coordinates system
$(x_{0}:\cdots:x_{n})$ on ${\mathbb{P}}^{n}({\mathbb{C}})$. Let
$f:\Delta_{0,\infty}\longrightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a
holomorphic curve with a reduced representation $F=(f_{0},\ldots,f_{n})$ on
$\Delta_{s,\infty}$, which means $f_{0},\ldots,f_{n}$ are holomorphic
functions on $\Delta_{s,\infty}$ without common zeros. We set

$\|F\|=(|f_{0}|^{2}+\cdots+|f_{n}|^{2})^{1/2}.$

The characteristic function of $f$ is defined by

$T_{f}(r,s)=\int_{0}^{2\pi}\log\|F(re^{i\theta})\|\dfrac{d\theta}{2\pi}-\int_{0}^{2\pi}\log\|F(se^{i\theta})\|\dfrac{d\theta}{2\pi}\
(s<r<R).$

For a meromorphic function $\varphi$, if we consider $\varphi$ as a
holomorphic curve into ${\mathbb{P}}^{1}({\mathbb{C}})$ the

$T_{\varphi}(r,s)=T(r,s,\varphi)+O(1).$

B. Auxiliary results. We need the following.

######  Proposition 2.1 (cf. [8, Proposition 4.5]).

Let $f_{0},\ldots,f_{n}$ be holomorphic functions on $\Delta_{s,\infty}$ such
that $\\{f_{0},\ldots,f_{n}\\}$ are linearly independent over ${\mathbb{C}}.$
Then the Wronskian

$W(f_{0},\ldots,f_{n})=\det\left(f_{j}^{(i)}\right)_{0\leq i,j\leq n},$

where $f_{j}^{(i)}$ is the $i^{th}$-derivative of $f_{j}$, satisfying:

(i) $W(f_{0},\ldots,f_{n})\not\equiv 0.$

(ii) $W(hf_{0},\ldots,hf_{n})=h^{n+1}W(f_{0},\ldots,f_{n})$ for any
meromorphic function $h$ on $\Delta_{s,\infty}.$

######  Lemma 2.2 (cf. [18, Lemma 2.2]).

Let $f:\Delta_{s,\infty}\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a
holomorphic curve. We have

$\biggl{\|}\quad m\left(r,s,\dfrac{f^{(k)}}{f}\right)\leq O(\log
T_{f}(r,s))+O(\log r).$

######  Lemma 2.3 (cf. [23, Lemma 3]).

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$. Let $Q_{1},...,Q_{q}$ be $q\ (q>2N-k+1)$
hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position
with respect to $V$ of the common degree $d.$ 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

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

We know the following characterization of a removable singularity (cf. [18]).

######  Lemma 2.4.

Let $f$ be a holomorphic curve from $\Delta_{s,\infty}$ into a projective
subvariety $V$ of ${\mathbb{P}}^{n}({\mathbb{C}})$. Then $f$ extends
holomorphically in a neighborhood of $\infty$ if and only if

$\liminf_{r\to\infty}T_{f}(r,s)/(\log r)<\infty.$

Let $\\{Q_{i}\\}_{i\in R}$ be a family of hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ of the common degree $d$. Assume that each
$Q_{i}$ is defined by

$\sum_{I\in\mathcal{I}_{d}}a_{iI}x^{I}=0,$

where $\mathcal{I}_{d}=\\{(i_{0},...,i_{n})\in\mathbb{N}_{0}^{n+1}\ :\
i_{0}+\cdots+i_{n}=d\\}$, $I=(i_{0},...,i_{n})\in\mathcal{I}_{d},$
$x^{I}=x_{0}^{i_{0}}\cdots x_{n}^{i_{n}}$ and $(x_{0}:\cdots:x_{n})$ is
homogeneous coordinates of ${\mathbb{P}}^{n}({\mathbb{C}})$.

Let $f:\Delta_{s,\infty}\rightarrow V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ be
a meromorphic mapping into $V$ with a reduced representation
$F=(f_{0},\ldots,f_{n})$. We define

$Q_{i}(F)=\sum_{I\in\mathcal{I}_{d}}a_{iI}f^{I},$

where $f^{I}=f_{0}^{i_{0}}\cdots f_{n}^{i_{n}}$ for $I=(i_{0},...,i_{n})$.
Since the divisor $\nu_{Q(F)}$ does not depend on the choice of the reduced
representation of $f$, we denote it by $\nu_{Q(f)}$. Then we can consider
$f^{*}Q_{i}=\nu_{Q_{i}(f)}$ as divisors.

######  Lemma 2.5 (cf. [23, Lemma 4]).

Let $\\{Q_{i}\\}_{i\in R}$ be a set of hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ of the common degree $d$ and let $f$ be a
meromorphic mapping of $\Delta_{s,\infty}$ into
${\mathbb{P}}^{n}({\mathbb{C}})$. Assume that $\bigcap_{i\in R}Q_{i}\cap
V=\varnothing$. Then there exist positive constants $\alpha$ and $\beta$ such
that

$\alpha\|f\|^{d}\leq\max_{i\in R}|Q_{i}(f)|\leq\beta\|f\|^{d}.$

######  Lemma 2.6 (cf. [23, Lemma 5]).

Let $V$ be a projective subvariety of dimension $k$ of
${\mathbb{P}}^{n}({\mathbb{C}})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be a set of $q$
hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$ of the common degree $d$ and
let $M=H_{d}(V)-1$. Then there exist $M-k$ hypersurfaces
$\\{T_{i}\\}_{i=1}^{M-k}$ in ${\mathbb{P}}^{n}({\mathbb{C}})$ such that for
any subset $R\subset\\{1,...,q\\}$ with $\sharp
R=\mathrm{rank}\\{[Q_{i}]\\}_{i\in R}=k+1,$ we get

$\mathrm{rank}\\{\\{[Q_{i}]\\}_{i\in R}\cup\\{[T_{i}]\\}_{i=1}^{M-k}\\}=M+1.$

C. Defect relation for holomorphic curves

######  Theorem 2.7.

Let $V$ be a complex projective subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$
of dimension $k\ (k\leq n)$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces of
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position with respect to
$V$ with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$. Let $d$ be the least common
multiple of $d_{i}$’s, i.e., $d=lcm(d_{1},...,d_{q})$, and let $M=H_{d}(V)-1$.
Let $f$ be a holomorphic mapping of $\Delta_{s,\infty}$ into $V$ such that $f$
is nondegenerate over $I_{d}(V)$. Then, we have

$\biggl{|}\biggl{|}\
\left(q-\dfrac{(2N-k+1)(M+1)}{k+1}\right)T_{f}(r,s)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[M]}_{Q_{i}(f)}(r,s)+o(T_{f}(r,s)).$

###### Proof.

By replacing $Q_{i}$ with $Q_{i}^{d/\deg Q_{i}}\ (i=1,...,q)$ if necessary, we
may assume that all $Q_{i}\ (1\leq i\leq q)$ have the same degree $d$. It is
easy to see that there is a positive constant $C_{1}$ such that $\|f\|^{d}\geq
C_{1}|Q_{i}(f)|$ for every $1\leq i\leq q.$ Set $Q:=\\{1,\cdots,q\\}$. Let
$\\{\omega_{i}\\}_{i=1}^{q}$ and $\tilde{\omega}$ be as in Lemma 2.3 for the
family $\\{Q_{i}\\}_{i=1}^{q}$. Let $\\{T_{i}\\}_{i=1}^{M-k}$ be $(M-k)$
hypersurfaces in ${\mathbb{P}}^{n}({\mathbb{C}})$, which satisfy Lemma 2.6.

Take a ${\mathbb{C}}$-basis $\\{[A_{i}]\\}_{i=0}^{M}$ of $I_{d}(V)$, where
$A_{i}\in H_{d}$. Let $F=(f_{0},\ldots,f_{n})$ be a reduced representation of
$f$. Since $f$ is nondegenerate over $I_{d}(V)$, $\\{A_{i}(F);0\leq i\leq
M\\}$ is linearly independent over ${\mathbb{C}}$. Then

$W\equiv\det\bigl{(}A_{j}(F)^{(i)}\bigl{)}_{0\leq i,j\leq M}\not\equiv 0.$

For each $R^{o}=\\{i_{0},...,i_{k}\\}\subset\\{1,...,q\\}$ with
$\mathrm{rank}\\{Q_{i}\\}_{i\in R^{o}}=\sharp R^{o}=k+1$, set

$W_{R^{o}}\equiv\det\bigl{(}A_{i_{j}}(F)^{(l)}(0\leq j\leq
k),T_{j}(F)^{(l)}(1\leq j\leq M)\bigl{)}_{0\leq l\leq M}.$

Since $\mathrm{rank}\\{Q_{i_{j}}(0\leq j\leq k),T_{j}(1\leq j\leq
M-k)\\}=M+1$, there exists a nonzero constant $C_{R^{o}}$ such that
$W_{R^{o}}=C_{R^{o}}\cdot W$.

We denote by $\mathcal{R}^{o}$ the family of all subsets $R^{o}$ of
$\\{1,...,q\\}$ satisfying

$\mathrm{rank}\\{Q_{i}\\}_{i\in R^{o}}=\sharp R^{o}=k+1.$

Let $z$ be a fixed point. For each $R\subset Q$ with $\sharp R=N+1,$ we choose
$R^{o}\subset R$ such that $R^{o}\in\mathcal{R}^{o}$ and $R^{o}$ satisfies
Lemma 2.3 v) with respect to numbers
$\bigl{\\{}\dfrac{C_{1}\|F(z)\|^{d}}{|Q_{i}(F)(z)|}\bigl{\\}}_{i=1}^{q}$. On
the other hand, there exists $\bar{R}\subset Q$ with $\sharp\bar{R}=N+1$ such
that $|Q_{i}(F)(z)|\leq|Q_{j}(F)(z)|,\forall i\in\bar{R},j\not\in\bar{R}$.
Since $\bigcap_{i\in\bar{R}}Q_{i}=\varnothing$, by Lemma 2.5, there exists a
positive constant $C_{2}$ (chosen common for all $R$) such that

$C_{2}\|F\|^{d}(z)\leq\max_{i\in\bar{R}}|Q_{i}(F)(z)|.$

Then, we get

$\displaystyle\dfrac{\|F(z)\|^{d(\sum_{i=1}^{q}\omega_{i})}|W(z)|}{|Q_{1}(F)(z)|^{\omega_{1}}\cdots|Q_{q}(F)(z)|^{\omega_{q}}}$
$\displaystyle\leq\dfrac{|W(z)|}{C_{2}^{q-N-1}C_{1}^{N+1}}\prod_{i\in\bar{R}}\left(\dfrac{C_{1}\|F(z)\|^{d}}{|Q_{i}(F)(z)|}\right)^{\omega_{i}}$
$\displaystyle\leq
C_{3}\dfrac{|W(z)|\cdot\|F\|^{d(k+1)}(z)}{\prod_{i\in\bar{R}^{o}}|Q_{i}(F)|(z)}$
$\displaystyle\leq
C_{4}\dfrac{|W_{\bar{R}^{o}}(z)|\cdot\|F\|^{d(M+1)}(z)}{\prod_{i\in\bar{R}^{o}}|Q_{i}(F)|(z)\prod_{i=1}^{M-k}|T_{i}(F)|(z)},$

where $C_{3},C_{4}$ are positive constants.

Put
$S_{\bar{R}}=C_{4}\dfrac{|W_{\bar{R}^{o}}|}{\prod_{i\in\bar{R}^{o}}|Q_{i}(F)|\prod_{i=1}^{M-k}|T_{i}(F)|}$.
By the Lemma on logarithmic derivative, it is easy to see that

$\|\
\int\limits_{0}^{2\pi}\log^{+}S_{\bar{R}}(re^{i\theta})\dfrac{d\theta}{2\pi}=O(\log^{+}T_{f}(r,s))+O(\log
r).$

Therefore, for each $z\in\Delta_{s,\infty}$, we have

$\displaystyle\log\left(\dfrac{\|F(z)\|^{d(\sum_{i=1}^{q}\omega_{i})}|W(z)|}{|Q_{1}(F)(z)|^{\omega_{1}}\cdots|Q_{q}(F)(z)|^{\omega_{q}}}\right)\leq\log\left(\|F(z)\|^{d(M+1)}\right)+\sum_{R\subset
Q,\sharp R=N+1}\log^{+}S_{R}.$

Since $\sum_{i=1}^{q}\omega_{i}=\tilde{\omega}_{i}(q-2N+k-1)+k+1$ and by
integrating both sides of the above inequality over $S(r),$ we have

(2.8) $\displaystyle\begin{split}\|\
d\biggl{(}q-2N+k-1&-\dfrac{M-k}{\tilde{\omega}}\biggl{)}T_{f}(r,s)\leq\sum_{i=1}^{q}\dfrac{\omega_{i}}{\tilde{\omega}}N_{Q_{i}(f)}(r,s)\\\
&-\dfrac{1}{\tilde{\omega}}N_{W}(r,s)+O(\log^{+}T_{f}(r,s))+O(\log
r).\end{split}$

By usual arguments (see [23]), we have

$\sum_{i=1}^{q}\omega_{i}N_{Q_{i}(f)}(r,s)-N_{W}(r,s)\leq\sum_{i=1}^{q}\omega_{i}N^{[M]}_{Q_{i}(f)}(r,s).$

Combining this and (2.8), we obtain

$\displaystyle\|\ $ $\displaystyle
d\left(q-2N+k-1-\dfrac{M-k}{\tilde{\omega}}\right)T_{f}(r,s)$
$\displaystyle\leq\sum_{i=1}^{q}\dfrac{\omega_{i}}{\tilde{\omega}}N^{[M]}_{Q_{i}(f)}(r,s)+O(\log^{+}T_{f}(r,s))+O(\log
r)$
$\displaystyle\leq\sum_{i=1}^{q}N^{[M]}_{Q_{i}(f)}(r,s)+O(\log^{+}T_{f}(r,s))+O(\log
r).$

Since $\tilde{\omega}\geq\dfrac{k+1}{2N-k+1}$, the above inequality implies
that

$\biggl{\|}\quad
d\left(q-\dfrac{(2N-k+1)(M+1)}{k+1}\right)T_{f}(r,s)\leq\sum_{i=1}^{q}N^{[M]}_{Q_{i}(f)}(r,s)+O(\log^{+}T_{f}(r,s))+O(\log
r).$

Hence, the theorem is proved. ∎

For each hypersurface $Q$ of degree $d$, we define the truncated defect of $f$
with respect to $Q$ by

$\delta^{[m]}_{f}(Q)=1-\lim\underset{r\rightarrow\infty}{\rm
sup}\dfrac{N_{Q(f)}^{[m]}(r,s)}{dT_{f}(r,s)}.$

As we known that

$\delta^{H,M}_{f,\Delta_{s,\infty}}(Q)\leq\delta_{f,\Delta_{s,\infty}}^{S,M}(Q)\leq\delta_{f}^{[M]}(Q)\leq
1.$

From Lemma 2.4 and Theorem 2.7, we have the following corollary.

######  Corollary 2.9.

Let $f$ be a holomorphic map from $\Delta_{s,\infty}$ into a projective
subvariety $V$ of ${\mathbb{P}}^{n}({\mathbb{C}})$ of dimension $k$. Let
$Q_{1},\ldots,Q_{q}$ be distinct $q$ hypersurface in $N$-subgeneral position
with respect to $V$. Assume that $f$ has an essential singularity at $\infty$,
then

$\sum_{j=1}^{q}\delta^{H,M}_{f,\Delta_{s,\infty}}(Q_{j})\leq\sum_{j=1}^{q}\delta^{[M]}_{f}(Q_{j})\leq\dfrac{(2N-k+1)H_{V}(d)}{k+1}.$

##  3\. Sum to Product inequality for holomorphic curves and families of
hypersurfaces

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

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

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

  * •

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

  * •

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

  * •

$v_{i}(F)^{(l)}_{z}\ (i=0,\ldots,M)$ 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

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

We have some fundamental properties of the wronskian of holomorphic function
$h_{0},h_{1},\ldots,h_{p}$ as follows:

  * •

$W_{\xi}(h_{0},\ldots,h_{p})=W_{z}(h_{0},\ldots,h_{p})(\xi^{\prime}_{z})^{\frac{p(p+1)}{2}}$,

  * •

$W_{z}(hh_{0},\ldots,hh_{p})=h^{p+1}W_{z}(h_{0},\ldots,h_{p})$,

  * •

$h_{0},\ldots,h_{p}$ are ${\mathbb{C}}$-linearly dependent if and only if
$W_{z}(h_{0},\ldots,h_{p})\equiv 0$.

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 ${\mathbb{P}}^{n}({\mathbb{C}})$
defined by a homogeneous polynomial in $H_{d}$ denoted again by $Q$. Then 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

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

For each $p\ (0\leq p\leq M-1)$, let $M_{p}=\binom{M+1}{p+1}-1$ and $\pi_{p}$
be the canonial 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 pullback 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 3.1 (cf. [13, 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
$\delta>\delta_{0}(\epsilon)$

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

We note that, the original proposition of H. Fujimoto (cf. [13, Proposition
2.5.1]) states for hyperplanes. However, by using Serge embedding
${\mathbb{P}}^{n}({\mathbb{C}})$ into ${\mathbb{P}}^{M}({\mathbb{C}})$, we
automatically deduce the above proposition.

######  Theorem 3.2.

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 in ${\mathbb{P}}^{n}({\mathbb{C}})$
located 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 2.3). 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.

For each $j\ (1\leq j\leq q)$, we write

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

and set $a_{j}=(a_{j0},\ldots,a_{jM})\in{\mathbb{C}}^{M+1}$. We 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 Grassmanian manifold of all $(p+1)$-dimension sublinear spaces of
${\mathbb{C}}^{M+1}$), we take a decomposable $(p+1)$-vector $E$ such that

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

and set

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

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

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

As we known, $E\vee a_{j}=0$ means that $a_{j}$ 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, $R\in\mathcal{R}_{p}$. This yields that $\psi_{p}$ is positive
everywhere on $G(M,p)$. Since $\psi_{p}$ is obviously continuous and $G(M,p)$
is compact, we can take a positive constant $\delta$ such that
$\psi_{p}(P)>\delta$ for each $P\in G(M,p)$.

Take a point $z$ with $F_{\mathcal{V},p}(z)=0$. 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 consider 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. We note
that, there are at most $N$ indexes $j$ satisfying $\langle
F^{(0)}_{\mathcal{V}}(z),a_{j}\rangle=0$ (i.e., $Q_{j}(F)(z)=0$) since the
intersection of $V$ and arbitrary $N+1$ hyperplanes $Q_{j}\ (1\leq j\leq q)$
is empty. 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 3.3.

Let the notations and the assumption be as in Theorem 3.2. Moreover, let
$\tilde{\omega}$ be the Nochka constant for these hypersurfaces (defined in
the Lemma 2.3). 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. Then we have

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

Choose a positive number $\delta_{0}(\epsilon/l)$ with the properties as in
Proposition 3.1, where $l=\sum_{j=1}^{q}\omega_{j}$. For an arbitrarily fixed
$\delta\geq\delta_{0}(\epsilon/l)$, we obtain

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

Then, by Theorem 3.2 we have

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

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. ∎

######  Theorem 3.4 (cf. [13, 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 3.5.

With the notations and the assumption in Theorem 3.3, 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.

Indeed, let $z$ be a zero of some $Q_{i}(F)$. Since $\\{Q_{i}\\}_{i=1}^{q}$ is
in $N$-subgeneral position, $z$ is not zero of more than $N$ functions
$Q_{i}(F)$. Without loss of generality, we may assume 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 2.3 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 3.6 (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

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

######  Lemma 3.7.

Let $V,\mathcal{V},d,M,\\{Q_{i}\\}_{i=1}^{q}$ and $\\{\omega_{i}\\}_{i=1}^{q}$
be as in Theorem 3.2. 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 $\eta_{j}$, divisors
$\nu_{j}$, continuous real-valued bounded functions $k_{j}$ with mild
singularities on $S$, a positive constant $c$ such that $\nu_{j}(z)>c$ for
each $z\in\mathrm{Supp}\,(\nu_{j})\ (1\leq j\leq q)$ and

$[\min\\{m,\nu_{Q_{j}(f)}\\}]+[\nu_{j}]=d\eta_{j}f^{*}\Omega+dd^{c}[\log|k_{j}|^{2}].$

Let $h_{j}=k_{j}\|F\|^{d\eta_{j}}\ (1\leq j\leq q)$. Then for an arbitrarily
given $\epsilon$ satisfying

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

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

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

and $\delta$ is the number satisfying the conclusion of Theorem 3.3, is
continuous and has strictly negative curvature.

###### Proof.

Without loss of generality, we may assume that $k_{j}\leq 1$, and then
$h_{j}\leq\|F\|^{d\eta_{j}}\ (1\leq j\leq q)$. We see that the function $\eta$
is continuous at every point $z$ with $\prod_{j=1}^{q}Q_{j}(F)(z)\neq 0$. Now,
for a point $z_{0}\in S$ such that $\prod_{j=1}^{q}Q_{j}(F)(z_{0})=0$, from
Theorem 3.5, we have

$\displaystyle\nu_{\eta}(z_{0})$
$\displaystyle\geq\frac{1}{\sigma_{M}+\epsilon\tau_{M}}\left(\nu_{F_{\mathcal{V},M}}(z_{0})+\sum_{j=1}^{q}(\omega_{j}+\frac{\epsilon\eta_{j}}{cq})\min(\nu_{Q_{j}(F)}(z_{0}),M)-\sum_{j=1}^{q}\omega_{j}\nu_{Q_{j}(F)}(z_{0})\right)$
$\displaystyle\geq 0.$

This implies that $d\tau^{2}$ is 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 pullback of $\Omega$ by the curve $f$. By Theorem
3.5, we see that

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

Then by Theorems 3.3 and 3.4 and the definition of $\eta$, we have

(3.8)
$\displaystyle\begin{split}dd^{c}\log\eta&\geq\dfrac{\gamma-\epsilon(\sigma_{M+1}+\sum_{j=1}^{q}\frac{\eta_{j}}{q})}{\sigma_{M}+\epsilon\tau_{M}}d\Omega_{f}+\dfrac{\epsilon}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\left(|F_{\mathcal{V},0}|\cdots|F_{\mathcal{V},M}|\right)\\\
&+\dfrac{1}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\dfrac{\prod_{p=0}^{M-1}|F_{\mathcal{V},p}|^{2\epsilon}}{\prod_{p=0}^{M-1}\log^{4\omega_{j}}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}\\\
&\geq\dfrac{\epsilon\tau_{M}}{2\sigma_{M}(\sigma_{M}+\epsilon\tau_{M})}\left(\dfrac{|F_{\mathcal{V},0}|^{2}\cdots|F_{\mathcal{V},M}|^{2}}{|F_{\mathcal{V},0}|^{2\sigma_{M+1}}}\right)^{\frac{1}{\tau_{M}}}dd^{c}|z|^{2}\\\
&+C_{0}\left(\dfrac{|F_{\mathcal{V},0}|^{2\left(\tilde{\omega}(q-2N+k-1)-M+k\right)}|F_{\mathcal{V},M}|^{2}}{\prod_{j=1}^{q}(|Q_{j}(F)|^{2}\prod_{p=0}^{M-1}\log^{2}(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}}}dd^{c}|z|^{2}\\\
&\geq
C_{1}\left(\dfrac{|F_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}|F_{\mathcal{V},M}|\prod_{p=0}^{M}|F_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(F)|\prod_{p=0}^{M-1}\log(\delta/\varphi_{\mathcal{V},p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{2}{\sigma_{M}+\epsilon\tau_{M}}}dd^{c}|z|^{2}\end{split}$

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

$\displaystyle|F_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}$
$\displaystyle\hskip
80.0pt=|F_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}+\sum_{j=1}^{q}\omega_{j}\eta_{j}}\geq|F_{\mathcal{V},0}|^{\gamma-\epsilon(\sigma_{M+1}+\frac{\eta_{j}}{q})}h_{1}^{\omega_{1}+\epsilon\frac{\eta_{1}}{q}}\cdots
h_{q}^{\omega_{q}+\epsilon\frac{\eta_{q}}{q}}.$

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. ∎

Applying Lemma 3.7, we will prove the following Main Lemma of this paper.

######  Lemma 3.9 (Main Lemma).

Let $V,\mathcal{V},d,M,\\{Q_{i}\\}_{i=1}^{q}$ and $\\{\omega_{i}\\}_{i=1}^{q}$
be as in Theorem 3.2. 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 $\eta_{j}$, divisor
$\nu_{j}$, real-valued bounded functions $k_{j}\ (1\leq j\leq q)$ and a
constant $c$ as in Lemma 3.7. Then for every $\epsilon>0$ satisfying

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

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

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

where $h_{j}=k_{j}\|F\|^{d\eta_{j}}\ (1\leq j\leq q)$.

###### Proof.

As in the proof of Lemma 3.7, we have

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

According to Lemma 3.6, this implies that

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

for some positive constant. Then we have

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

It follows that

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

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

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

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

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

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

######  Lemma 4.1 (cf. [13, 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 isometry, 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 $x=(x_{0},\ldots,x_{n}):S\rightarrow{\mathbb{R}}^{n+1}$ be the immersion
of the minimal surface $S$ into ${\mathbb{R}}^{n+1}$. Let $(x,y)$ be a local
isothermal coordinate of $S$. Then $z=x+iy$ is a local holomorphic coordinate
of $S$. The generalized Gauss map $g$ of $x$ is defined (locally) by

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

Then, we denote by $G_{z}$ the local reduced representation of $g$ defined by

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

Then, with other local holomorphic coordinate $\xi$, we have

$G_{\xi}=G_{z}\cdot\left(\dfrac{dz}{d\xi}\right)$

(this yields that the map $g$ is global well-defined). Also, the metric
$ds^{2}$ on $S$ induced by the canonical metric on ${\mathbb{R}}^{n+1}$
satisfies

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

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

Let $V$ be a $k$-dimension projective subvariety of
${\mathbb{P}}^{n}({\mathbb{C}})$ and $d$ is positive integer. We fix a
${\mathbb{C}}$-ordered basis $\mathcal{V}=([v_{0}],\ldots,[v_{M}])$ of
$I_{d}(V)$ as in the Section 3. Assume that the image of $g$ is included in
$V$ and $g$ is nondegenerate over $I_{d}(V)$.

Let $Q_{1},\ldots,Q_{q}$ be $q$ hypersurfaces of the same degree $d$ of
${\mathbb{P}}^{n}({\mathbb{C}})$ in $N$-subgeneral position with respect to
$V$ and $V\not\subset Q_{i}\ (1\leq i\leq q)$. 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

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

here the first subscript $z$ mean $G_{z}$ is the local reduced representation
of $g$ on a local holomorphic chat $(U,z)$, the second subscript $z$ means the
derivatives are taken w.r.t. $z$. 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

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

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

###### Proof of Theorem 1.2.

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 $d$.

From the assumption of the theorem, we have

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

Then, there exist a compact subset $K\subset S$, numbers $\eta_{j}>0\ (1\leq
j\leq q)$ such that

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

divisor $\nu_{j}\ (1\leq j\leq q)$ and bounded continuous functions $k_{j}$
(we may assume $0\leq k_{j}<1$) with mild singularities on $S\setminus K$ such
that $\nu_{j}\geq c^{\prime}$ on $\mathrm{Supp}\,\nu_{j}$ for a positive
constant $c^{\prime}$ and

$[\min\\{\nu_{Q_{j}(f)},M\\}]+[v_{j}]=d\eta_{j}\Omega_{f}+dd^{c}[\log
k_{j}^{2}]$

in the sense of currents. We set $h_{zj}=k_{j}\|G_{z}\|^{d\eta_{j}}$ for each
local reduced representation $G_{z}$ of $g$. Then we have

$\nu_{h_{zj}}\geq c\text{ on }\mathrm{Supp}\,\nu_{h_{zj}},\text{ where
}c=\min\\{1,c^{\prime}\\}.$

From Theorem 2.3, 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,

(4.2)
$\displaystyle\begin{split}\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1&\geq\tilde{\omega}(q-2N+k-1-\sum_{j=1}^{q}\eta_{j})-M+k\\\
&\geq\dfrac{k+1}{2N-k+1}\left(\sum_{j=1}^{q}(1-\eta_{j})-2N+k-1\right)-M+k\\\
&=\dfrac{k+1}{2N-k+1}\left(\sum_{j=1}^{q}(1-\eta_{j})-\dfrac{(2N+k-1)(M+1)}{k+1}\right)\\\
&>\dfrac{k+1}{2N-k+1}\cdot\dfrac{(2N+k-1)M(M+1)}{2d(k+1)}=\dfrac{\sigma_{M}}{d}.\end{split}$

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

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

We define the following numbers

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

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

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

where $A=S\setminus K$, $z$ is a local holomorphic coordinate around $a$, and
define a new pseudo-metric on $A_{1}$ as follows

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

where $h_{zj}=k_{j}\|F_{z}\|^{d\eta_{j}}$. Here and throughout this section,
for simplicity we will write $\prod_{j,p}$ for
$\prod_{j=1}^{q}\prod_{p=1}^{M-1}$. We note that the definition of $A_{1}$ and
$d\tau^{2}$ do not depend on the choice of the local holomoprhic coordinate.
Indeed, we have

$\displaystyle G_{z}=(\xi^{\prime}_{z})\cdot G_{\xi},\
Q_{j}(G_{z})=(\xi^{\prime}_{z})^{d}\cdot Q_{j}(G_{\xi}),\
v_{j}(G_{z})=(\xi^{\prime}_{z})^{d}\cdot v_{j}(G_{\xi}),\ h_{zj}$
$\displaystyle=|\xi^{\prime}_{z}|^{d\eta_{j}}\cdot h_{\xi j}$
$\displaystyle((G_{z})_{\mathcal{V},p})_{z}=(\xi^{\prime}_{z})^{d(p+1)}\cdot((G_{\xi})_{\mathcal{V},p})_{z}=(\xi^{\prime}_{z})^{d(p+1)+\sigma_{p}}\cdot((G_{\xi})_{\mathcal{V},p})_{\xi}$
$\displaystyle(\psi(G_{z})_{ip})_{z}=(\xi^{\prime}_{z})^{d(p+1)+\sigma_{p}}\cdot(\psi(G_{z})_{ip})_{\xi}\
(0\leq p\leq M).$

and hence $(\psi(G_{z})_{ip})_{z}(a)=0$ if and only if
$(\psi(G_{\xi})_{ip})_{\xi}(a)=0$ for two local holomorphic coordinates $z$
and $\xi$ around $a$. On the other hand, we have

$\displaystyle\left(\dfrac{\prod_{j=1}^{q}|Q_{j}(G_{z})|^{\omega_{j}}}{|((G_{z})_{\mathcal{V},M})_{z})|^{1+\epsilon}\prod_{j=1}^{q}h_{zj}^{\omega_{j}+\frac{\epsilon}{q}}\prod_{j,p}|(\psi(G_{z})_{jp})_{z}|^{\frac{\epsilon}{q}}}\right)^{2\rho^{*}}|dz|^{2}$
$\displaystyle=$
$\displaystyle\left(\dfrac{\left(\prod_{j=1}^{q}|Q_{j}(G_{\xi})|^{\omega_{j}}\right)|\xi^{\prime}_{z}|^{d\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-d\epsilon\sum_{j=1}^{q}\frac{\eta_{j}}{q}}}{|((G_{\xi})_{\mathcal{V},M})_{\xi})|^{1+\epsilon}\prod_{j=1}^{q}h_{\xi
j}^{\omega_{j}+\frac{\epsilon}{q}}\prod_{j,p}|(\psi(G_{\xi})_{jp})_{\xi}|^{\frac{\epsilon}{q}}|\xi^{\prime}_{z}|^{d(M+1)+\sigma_{M}+\epsilon\sum_{p=0}^{M}(d(p+1)+\sigma_{p})}}\right)^{2\rho^{*}}|dz|^{2}$
$\displaystyle=$
$\displaystyle\left(\dfrac{\left(\prod_{j=1}^{q}|Q_{j}(G_{\xi})|^{\omega_{j}}\right)|\xi^{\prime}_{z}|^{d\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-d\epsilon\sum_{j=1}^{q}\frac{\eta_{j}}{q}}}{\left(|((G_{\xi})_{\mathcal{V},M})_{\xi})|^{1+\epsilon}\prod_{j=1}^{q}h_{\xi
j}^{\omega_{j}+\frac{\epsilon}{q}}\prod_{j,p}|(\psi(G_{\xi})_{jp})_{\xi}|^{\frac{\epsilon}{q}}\right)|\xi^{\prime}_{z}|^{d(M+1)+\sigma_{M}+\epsilon(d\sigma_{M+1}+\tau_{M})}}\right)^{2\rho^{*}}|dz|^{2}$
$\displaystyle=$
$\displaystyle\left(\dfrac{\prod_{j=1}^{q}|Q_{j}(G_{\xi})|^{\omega_{j}}}{|((G_{\xi})_{\mathcal{V},M})_{\xi})|^{1+\epsilon}\prod_{j=1}^{q}h_{\xi
j}^{\omega_{j}+\frac{\epsilon}{q}}\prod_{j,p}|(\psi(G_{\xi})_{jp})_{\xi}|^{\frac{\epsilon}{q}}}\right)^{2\rho^{*}}|d\xi|^{2}$

(since
$|\xi^{\prime}_{z}|^{2\rho^{*}(d(\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1)-\sigma_{M}-\epsilon(d\sigma_{M+1}+\tau_{M}+d\sum_{j=1}^{q}\frac{\eta_{j}}{q}))}|dz^{2}|=|\xi^{\prime}_{z}|^{2}|dz^{2}|=|d\xi^{2}|$).
Then the pseudo-metric $d\tau^{2}$ is global well-defined.

Since $Q_{j}(G_{z}),((G_{z})_{\mathcal{V},M})_{z},\psi(G_{z})_{jp})_{z}$ are
all holomorphic functions and $u_{j}\ (1\leq j\leq q)$ are harmonic functions
on $A_{1}$, $d\tau$ is flat on $A_{1}$. So, $d\tau$ can be smoothly extended
over $K$. Thus, we have a metric, still call it $d\tau$, on

$A_{1}^{\prime}=A_{1}\cup K$

that is flat outside the compact set $K$.

######  Claim 4.3.

$d\tau$ is complete on $A_{1}^{\prime}$

Indeed, suppose contrarily that $A_{1}^{\prime}$ is not complete with $d\tau$,
there is a divergent curve $\gamma:[0,1)\rightarrow A_{1}^{\prime}$ with
finite length. By using only the last segment of $\gamma$, we may assume that
the distance $d(\gamma,K)$ between $\gamma$ and $K$ exceeds a positive number
$d$. Then, as $t\rightarrow 1$ there are only two cases: either $\gamma(t)$
tends to a point $a$ with

$\prod_{j=1}^{q}(h_{zj}\prod_{p=0}^{M}|\psi(G_{z})_{ip}|)(a)=0$

for a local holomorphic coordinate $z$ around $a$, or else $\gamma(t)$ tends
to the boundary of $S$.

For the first case, by Theorem 3.5, we have

$\displaystyle\nu_{d\tau}(a)$
$\displaystyle\leq-\biggl{(}\nu_{((G_{z})_{\mathcal{V},M})_{z}}(a)-\sum_{j=1}^{q}\omega_{j}\nu_{G_{z}(Q_{j})}(a)+\sum_{j=1}^{q}\omega_{j}\min\\{\nu_{G_{z}(Q_{j})}(a),M\\}$
$\displaystyle\ \
+\bigl{(}\epsilon\nu_{((G_{z})_{\mathcal{V},M})_{z}(a)}+\dfrac{\epsilon}{q}\sum_{j=1}^{q}\big{(}\nu_{h_{zj}}+\sum_{p=0}^{M-1}\nu_{\psi(G_{z})_{jp}}(a)\big{)}\bigl{)}\biggl{)}\rho^{*}$
$\displaystyle\leq-\epsilon\rho^{*}\nu_{((G_{z})_{\mathcal{V},M})_{z}}(a)-\dfrac{\epsilon\rho^{*}}{q}\sum_{j=1}^{q}\big{(}\nu_{h_{zj}}+\sum_{p=0}^{M-1}\nu_{\psi(G_{z})_{jp}}(a)\big{)}\leq-\dfrac{c\epsilon\rho^{*}}{q}$

(since $\nu_{h_{zj}}>c$ on $\mathrm{Supp}\,\nu_{j}$). 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 $a$. Then we get

$L_{d\tau}(\gamma)=\int_{0}^{1}\|\gamma^{\prime}(t)\|_{\tau}dt=\int_{\gamma}d\tau\geq\int_{\gamma}\dfrac{C}{|z-z(a)|^{\frac{c\epsilon\rho^{*}}{q}}}|dz|=+\infty$

($\gamma(t)$ tends to $a$ as $t\rightarrow 1$). This is a contradiction. Then,
the second case must occur, that is $\gamma(t)$ tends to the boundary of $S$
as $t\rightarrow 1$.

Choose $t_{0}$ close $1$ enough such that $L_{d\tau}(\rho|_{(t_{0},1)})<d/3$.
Take a disk $\Delta$ (in the metric induced by $d\tau$) around
$\gamma(t_{0})$. Since $d\tau$ is flat, by Lemma 4.1, $\Delta$ is isometric to
an ordinary disk in the plane. Let
$\Phi:\Delta(r)=\\{|\omega|<r\\}\rightarrow\Delta$ be this isometric with
$\Phi(0)=\gamma(t_{0})$. 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 $\int_{\gamma|_{[t_{0},1)}}d\tau<d/3$, we have $R\leq
d/3$. Hence, the image of $\Phi$ is bounded away from $K$ by distance at least
$2d/3$. 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 $z_{0}$ of $A^{\prime}_{1}$ with
$\prod_{j=1}^{q}h_{zj}(z_{0})\prod_{p=0}^{M}|\psi(G_{z})_{ip}|(z_{0})=0$,
since we have already shown that $\gamma(t_{0})$ is infinitely far away in the
metric with respect to these points. Then the image of $\Phi$ must go to the
boundary $S$. Hence, by again Lemma 4.1, there exists a point $w_{0}$ with
$|w_{0}|=R$ so that $\Gamma=\Phi(\overline{0,w_{0}})$ is a divergent curve on
$S$.

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

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

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

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

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

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

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

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

Hence, we have

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

We note that
$\frac{h}{d}=\sum_{j=1}^{q}\omega_{j}(1-\eta_{j})-M-1-\epsilon(\sigma_{M+1}+\sum_{j=1}^{q}\frac{\eta_{j}}{q})$.
Then the inequality (4.2) yields that the conditions of Lemma 3.6 are
satisfied. Then, by applying Lemma 3.6 we have

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

for some positive constant $C$. Also, we have that $0<\rho<1$. Then

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

This contradicts the assumption of completeness of $S$ with respect to
$ds^{2}$. Thus, Claim 4.3 is proved.

Then, we note that the metric $d\tau^{2}$ on $A_{1}^{\prime}$ is flat outside
$K$, and $A_{1}^{\prime}$ is complete with this metric by Claim 4.3. Also, as
we known, a theorem of Huber (cf. [5, Theorem 13, p.61]) yields that if
$A_{1}^{\prime}$ has finite total curvature then $A_{1}^{\prime}$ is finitely
connected. This means that there is a compact subregion of $A_{1}^{\prime}$
whose complement is the union of a finite number of doubly-connected regions.
Therefore, the functions $\prod_{p=0}^{M}\prod_{j=1}^{q}|\psi(G_{z})_{jp}|$
must have only a finite number of zeros, and the original surface $S$ is
finitely connected. Due to Osserman (cf. [20, Theorem 2.1]), each annular ends
of $A_{1}^{\prime}$, and hence of $S$, is conformally equivalent to a
punctured disk. Thus, the Riemann surface $S$ must be conformally equivalent
to a compact Riemann surface $\overline{S}$ with a finite number of points
removed. By the assumption, in a neighborhood of each of those points the
Gauss map $g$ satisfies

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

Now by Lemma 2.9, the Gauss map $g$ is not essential at those points.
Therefore $g$ can be extended to a holomorphic map from $\overline{S}$ to
$V\subset{\mathbb{P}}^{n}({\mathbb{C}})$. If the homology class represented by
the image of $g:S\rightarrow V\subset{\mathbb{P}}^{n}({\mathbb{C}})$ is $l$
times the fundamental homology class of ${\mathbb{P}}^{n}({\mathbb{C}})$, then
we have

$\iint K(s)dS=-2\pi l$

as the total curvature of $S$, where $K(s)$ is the Gaussian curvature of $S$.
Hence, Theorem 1.2 is proved. ∎

###### Proof of Theorem 1.4.

By Replacing $Q_{i}$ by $Q_{i}^{d/\deg Q_{j}}$ if necessary, we may assume
that $\deg Q_{j}=d\ (1\leq j\leq q)$. We fix an ordered basis $\mathcal{V}$
for $I_{d}(V)$. We take a global reduced representation
$G=(g_{0},\ldots,g_{n})$ of $g$. Suppose contrarily that

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

Then for each $j\ (1\leq j\leq q)$, there is a constant $\eta_{j}$ and a
continuous subharmonic functions $u_{j}$ on $S$ satisfying conditions
$(D_{1}),(D_{2})$ with respect to the number $\eta_{j}$, the reduced
representation $G$ and the hypersurface $Q_{j}$ such that

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

Now, for a (local) holomorphic reduced representation $G_{z}$ of $g$ on a
holomorphic chat $(U,z)$, there is a non-vanishing holomorphic function
$\theta_{z}$ such that $G_{z}=\theta_{z}\cdot G$. We define

$h_{zj}=|\theta_{z}|^{d\eta_{j}}e^{u_{j}}\leq|\theta_{z}|^{d\eta_{j}}\cdot\|G\|^{d\eta_{j}}=\|G_{z}\|^{d\eta_{j}}.$

It easily see that if $\xi$ is another local holomorphic coordinate then
$\theta_{z}=\theta_{\xi}\cdot\xi^{\prime}_{z}$, and hence
$h_{zj}=|\xi^{\prime}_{z}|^{d\eta_{j}}h_{\xi j}$.

From (4.2), we have

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

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

$\gamma-\sigma_{M}=\epsilon(\sigma_{M+1}+\tau_{M}).$

We set

$\lambda_{z}=\left(\dfrac{|(G_{z})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}}\prod_{j=1}^{q}h_{zj}^{\omega_{j}}|(G_{z})_{\mathcal{V},M}|\prod_{p=0}^{M}|(G_{z})_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(G_{z})|\prod_{p=0}^{M-1}\log(\delta/\varphi_{z,p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}},$

where
$\varphi_{z,p}(Q_{j})=\dfrac{|((G_{z})_{\mathcal{V},p})_{z}(Q)|}{|((G_{z})_{\mathcal{V},p})_{z}|}$
and $\delta$ is the number satisfying the conclusion of Theorem 3.3 with
$G_{z},\varphi_{z,p}(Q_{j})$ in place of $F,\varphi_{\mathcal{V},p}(Q_{j})$.

We define a pseudo-metric $d\tau^{2}:=\lambda_{z}^{2}|dz|^{2}$ on a
holomorphic chat $(U,z)$ of $S$. We will show that $d\tau^{2}$ is well-
defined, i.e., not depend the choice of the local coordinate. Indeed, if we
have another local coordinate $\xi$, we have

$\displaystyle\lambda_{z}$
$\displaystyle=\left(\dfrac{|(G_{z})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}}\prod_{j=1}^{q}h_{zj}^{\omega_{j}}|(G_{z})_{\mathcal{V},M}|\prod_{p=0}^{M}|(G_{z})_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(G_{z})|\prod_{p=0}^{M-1}\log(\delta/\varphi_{z,p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}$
$\displaystyle=\left(\dfrac{|(G_{\xi})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}}\prod_{j=1}^{q}h_{\xi
j}^{\omega_{j}}|(G_{\xi})_{\mathcal{V},M}||\xi^{\prime}_{z}|^{d\sigma_{M}}\prod_{p=0}^{M}|(G_{\xi})_{\mathcal{V},p}|^{\epsilon}|\xi^{\prime}_{z}|^{d\epsilon\sum_{p=0}^{M}\sigma_{p})}}{\prod_{j=1}^{q}(|Q_{j}(G_{\xi})|\prod_{p=0}^{M-1}\log(\delta/\varphi_{\xi,p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}$
$\displaystyle=\left(\dfrac{|(G_{\xi})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}}\prod_{j=1}^{q}h_{\xi
j}^{\omega_{j}}|(G_{\xi})_{\mathcal{V},M}|\prod_{p=0}^{M}|(G_{\xi})_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(G_{\xi})|\prod_{p=0}^{M-1}\log(\delta/\varphi_{\xi,p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{1}{\sigma_{M}+\epsilon\tau_{M}}}|\xi^{\prime}_{z}|.$

Hence
$\lambda_{z}^{2}|dz^{2}|=\lambda_{\xi}^{2}|\xi^{\prime}_{z}|^{2}|dz^{2}|=\lambda_{\xi}^{2}|d\xi^{2}|$.
Then, $d\tau^{2}$ is well-defined.

It is obvious that $d\tau^{2}$ is continuous on
$S\setminus\bigcup_{j=1}^{q}g^{-1}(Q_{j})$. Take a point $a$ such that
$\prod_{j=1}^{q}Q_{j}(G_{z})=0$ (for a loal holomorphic coordinate $z$ around
$a$). We have

$\lim_{t\rightarrow
a}(h_{zj}(t)\cdot|z(t)-z(a)|^{-\min\\{M,\nu_{Q_{j}(G_{z})}(a)\\}})<\infty.$

Combining with Theorem 3.5, this implies that

$\displaystyle d\tau_{z}(a)$
$\displaystyle\geq\dfrac{1}{\sigma_{M}+\epsilon\tau_{M}}\biggl{(}\nu_{(G_{z})_{\mathcal{V},M}}+\sum_{j=1}^{q}\omega_{j}(\min\\{M,\nu_{Q_{j}(G_{z})}(a)\\}-\nu_{Q_{j}(G_{z})}(a))\biggl{)}$
$\displaystyle\geq 0.$

Therefore $d\tau$ is continuous at $a$. This yields that $d\tau$ is a
continuous pseudo-metric on $S$.

We now prove that $d\tau^{2}$ has strictly negative curvature on $S$. By
Theorem 3.5, we see that

$\displaystyle
dd^{c}\left[\log\dfrac{|(G_{z})_{\mathcal{V},M}|}{\prod_{j=1}^{q}|Q_{j}(G_{z})|^{\omega_{j}}}\right]$
$\displaystyle+\sum_{j=1}^{q}\omega_{j}dd^{c}[\log h_{zj}]$
$\displaystyle\geq\sum_{i=1}^{q}(-[\min\\{M,\nu_{Q_{j}(f)}\\}]+dd^{c}[\log
h_{zj}])\geq 0.$

Then by Theorems 3.3 and 3.4, similarly as (3.8) we have

$\displaystyle dd^{c}\log\lambda_{z}$
$\displaystyle\geq\dfrac{\gamma-\epsilon\sigma_{M+1}}{\sigma_{M}+\epsilon\tau_{M}}d\Omega_{g}+\dfrac{\epsilon}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\left(|(G_{z})_{\mathcal{V},0}|\cdots|(G_{z})_{\mathcal{V},M}|\right)$
$\displaystyle+\dfrac{1}{2(\sigma_{M}+\epsilon\tau_{M})}dd^{c}\log\dfrac{\prod_{p=0}^{M-1}|(G_{z})_{\mathcal{V},p}|^{2\epsilon}}{\prod_{p=0}^{M-1}\log^{4\omega_{j}}(\delta/\varphi_{\mathcal{V},p}(Q_{j}))}$
$\displaystyle\geq
C\left(\dfrac{|(G_{z})_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}|(G_{z})_{\mathcal{V},M}|\prod_{p=0}^{M}|(G_{z})_{\mathcal{V},p}|^{\epsilon}}{\prod_{j=1}^{q}(|Q_{j}(F)|\prod_{p=0}^{M-1}\log(\delta/\varphi_{z,p}(Q_{j})))^{\omega_{j}}}\right)^{\frac{2}{\sigma_{M}+\epsilon\tau_{M}}}dd^{c}|z|^{2}$

for some positive constant $C$. On the other hand, we have
$|h_{zj}|\leq\|G_{z}\|^{d\eta_{j}}$,

$\displaystyle|(G_{z})_{\mathcal{V},0}|^{\tilde{\omega}(q-2N+k-1)-M+k-\epsilon\sigma_{M+1}}$
$\displaystyle=|(G_{z})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}+\sum_{j=1}^{q}\omega_{j}\eta_{j}}$
$\displaystyle\geq|(G_{z})_{\mathcal{V},0}|^{\gamma-\epsilon\sigma_{M+1}}h_{1}^{\omega_{1}}\cdots
h_{q}^{\omega_{q}}.$

This implies that $\Delta\log\eta^{2}\geq C\eta^{2}$. Therefore, $d\tau^{2}$
has strictly negative curvature.

As we known that the universal covering surface of $S$ is biholomorphic to
${\mathbb{C}}$ or a disk in ${\mathbb{C}}$. If the universal covering of $S$
is holomorphic to ${\mathbb{C}}$ then from [23, Theorem 2] we have

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

which contradicts the supposition. Now, we consider the case where the
universal covering surface $\tilde{S}$ of $S$ is biholomorphic to the unit
disc $\Delta$. Without loss of generality, we may assume that
$\tilde{S}=\Delta$ and have the covering map $\Phi:\Delta\rightarrow S$. From
Lemma 3.6 we have

$\Phi^{*}d\tau^{2}\leq d\sigma_{\Delta}^{2},$

where $d\sigma_{\Delta}=\dfrac{2}{1-|z|^{2}}|dz|^{2}$ with the complex
coordinate $z$ on $\Delta$.

Since $S$ has total finite curvature, $S$ is is conformally equivalent to a
compact surface $\overline{S}$ punctured at a finite number of points
$P_{1},\ldots,P_{r}$. Then, there are disjoint neighborhoods $U_{i}$ of
$P_{i}\ (1\leq i\leq r)$ in $\overline{S}$ and biholomorphic maps
$\phi_{i}:U_{i}\rightarrow\Delta$ with $\phi_{i}(P_{i})=0$. Note that, the
Poincare-metric on $\Delta^{*}=\Delta\setminus\\{0\\}$ is given by
$d\sigma_{\Delta^{*}}^{2}=\dfrac{4|dz|^{2}}{|z|^{2}\log^{2}|z|^{2}}$. Hence,
by the decreasing distance property, we have

$\Phi^{*}d\tau^{2}\leq d\sigma_{\Delta}^{2}\leq
C\cdot(\Phi\circ\phi_{i}^{-1})^{*}d\sigma_{\Delta^{*}}^{2}\ (1\leq i\leq r)$

for some positive constant $C$. This implies that

$\int_{U_{i}}\Omega_{d\tau^{2}}\leq\int_{\Phi^{-1}(U_{i})}\Phi^{*}\Omega_{\sigma_{\Delta}^{2}}\leq
lC\int_{\Delta^{*}}\Omega_{d\sigma_{\Delta^{*}}^{2}}<\infty.$

where $l$ is the number of the sheets of the covering $\Phi$. Then, it yields
that

$\int_{S}\Omega_{d\tau^{2}}\leq\int_{S\setminus\bigcup_{i=1}^{r}U_{i}}\Omega_{d\tau^{2}}+lCr\int_{\Delta^{*}}\Omega_{d\sigma_{\Delta^{*}}^{2}}<\infty.$

Now, denote by $ds^{2}$ the original metric on $S$. Similar as (3.8), we have

$dd^{c}\log\lambda_{z}\geq\dfrac{\gamma-\epsilon\sigma_{M+1}}{\sigma_{M}+\epsilon\tau_{M}}d\Omega_{g}.$

Then there is a subharmonic function $v_{z}$ such that

$\displaystyle\lambda_{z}^{2}|dz|^{2}$
$\displaystyle=e^{v_{z}}\|G_{z}\|^{2\frac{\gamma-\epsilon\sigma_{M+1}}{\sigma_{M}+\epsilon\tau_{M}}}|dz|^{2}$
$\displaystyle=e^{v_{z}}\|G_{z}\|^{2\frac{\gamma-\sigma_{M}-\epsilon\tau_{M+1}}{\sigma_{M}+\epsilon\tau_{M}}}\|G_{z}\|^{2}|dz|^{2}$
$\displaystyle=e^{w}ds^{2}$

for a subharmonic function $w$ on $S$. Since $S$ is complete with respect to
$ds^{2}$, applying a result of S. T. Yau [28] and L. Karp [14] we have

$\int_{S}\Omega_{d\tau^{2}}=\int_{S}e^{w}\Omega_{ds^{2}}=+\infty.$

This is a contradiction.

Then we must have

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

The theorem is proved. ∎

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