On holomorphic two-spheres with constant curvature in the complex Grassmann manifold G(2,n)

TL;DR

This paper investigates holomorphic two-spheres with constant curvature in the complex Grassmannian G(2,n), providing explicit characterizations, curvature distributions, and constructing numerous examples of non-homogeneous cases.

Contribution

It offers a new explicit polynomial characterization of such spheres and constructs many non-homogeneous examples, advancing understanding of their geometric properties.

Findings

Curvature values are explicitly determined.

Explicit polynomial equations characterize the spheres.

Many non-homogeneous examples are constructed.

Abstract

In this paper, the theory of functions of one complex variable is explored to study linearly full unramified holomorphic two-spheres with constant curvature in $G(2,n)$ satisfying that the generated harmonic sequence degenerates at position $2$. Firstly, we determine the value distribution of the curvature and give the explicit characterization of such holomorphic two-spheres in terms of a polynomial equation. Then, applying this characterization, many examples of non-homogeneous constantly curved holomorphic two-spheres are constructed.

Equations

$$\mathcal{K}_{1,n}^{h}=\left\{K=\frac{4}{\texttt{r}}\ \bigg{|}\ \texttt{r}=1,2,\cdots,n-1\right\},~{}\mathcal{K}_{1,n}^{fh}=\left\{\frac{4}{n-1}\right\}.$$

$$\mathcal{K}_{2,4}^{h}=\left\{K=\frac{4}{\texttt{r}}\ \bigg{|}\ \texttt{r}=1,2,3,4\right\},~{}\mathcal{K}_{2,4}^{fh}=\left\{K=\frac{4}{\texttt{r}}\ \bigg{|}\ \texttt{r}=2,3,4\right\}$$

$$\mathcal{K}_{k,n}^{h}=\left\{K=\frac{4}{\texttt{r}}\ \bigg{|}\ \texttt{r}=1,2,\cdots,k(n-k)\right\}.$$

$$\mathcal{K}_{2,n}^{fh}=\left\{K=\frac{4}{\texttt{r}}\ \bigg{|}\ \texttt{r}=n-2,\cdots,2(n-2)\right\},$$

$$|f_0^{(1)}{|}^2|f_0^{(n-3)}{|}^2+|h{|}^2=c(1+z\bar{z}{)}^{\text{d}+1},$$

$$\phi =\text{span}\left\{f_0^{(1)},h{f}_1^{(1)}-c_0{f}_0^{(n-3)}\right\}$$

$$0\stackrel{{\partial }^{\prime }}{⟶}{\phi }_0=\phi \stackrel{{\partial }^{\prime }}{⟶}{\phi }_1\stackrel{{\partial }^{\prime }}{⟶}{\phi }_2\stackrel{{\partial }^{\prime }}{⟶}\cdots \stackrel{{\partial }^{\prime }}{⟶}{\phi }_l\stackrel{{\partial }^{\prime }}{⟶}0,$$

$${\phi }_i=\text{span}\{e_{2i+1},e_{2i+2}\},\text{for}0\le i\le r-1,$$

$${\phi }_i=\text{span}\{e_{r+i+1}\},\text{for}r\le i\le n-r-1,$$

$${\partial }^{\prime }[e_{2r-1}]=0,{\partial }^{\prime }[e_{2r}]={\phi }_r.$$

$$\left(\begin{array}[]{ccccccccc}\Omega_{1}&A_{1}\phi&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\ -A_{1}^{*}\bar{\phi}&\Omega_{2}&A_{2}\phi&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\ {}\hfil&-A_{2}^{*}\bar{\phi}&\Omega_{3}&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\ {}\hfil&{}\hfil&{}\hfil&\ddots&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\ {}\hfil&{}\hfil&{}\hfil&{}\hfil&\Omega_{r}&A_{r}\phi&{}\hfil&{}\hfil&{}\hfil\\ {}\hfil&{}\hfil&{}\hfil&{}\hfil&-A_{r}^{*}\bar{\phi}&\Omega_{r+1}&{}\hfil&{}\hfil&{}\hfil\\ {}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&\ddots&{}\hfil&{}\hfil\\ {}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&\omega_{n-1\,n-1}&a_{n-1\,n}\phi\\ {}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&-\overline{a}_{n-1\,n}\bar{\phi}&\omega_{n\,n}\\ \end{array}\right),$$

$$d\varphi =-\sqrt{-1}\rho \wedge \varphi ,d(\sqrt{-1}\rho )=\frac{K}{2}\varphi \wedge \overline{\varphi },$$

$$\Delta \log |\det A_i|=2K+2(L_{i-1}-2L_i+L_{i+1}),1\le i\le r-1,$$

$$K=4-8|\det A_1{|}^2-\frac{1}{2}S.$$

$$0\stackrel{{\partial }^{\prime \prime }}{⟵}{\psi }_0^{(m)}\stackrel{{\partial }^{\prime }}{⟶}\cdots \stackrel{{\partial }^{\prime }}{⟶}\psi ={\psi }_i^{(m)}\stackrel{{\partial }^{\prime }}{⟶}\cdots \stackrel{{\partial }^{\prime }}{⟶}{\psi }_m^{(m)}\stackrel{{\partial }^{\prime }}{⟶}0,$$

$$f_i^{(m)}=\frac{\partial }{\partial z}{f}_{i-1}^{(m)}-\frac{⟨\frac{\partial }{\partial z}{f}_{i-1}^{(m)},f_{i-1}^{(m)}⟩}{|f_{i-1}^{(m)}{|}^2}{f}_{i-1}^{(m)}$$

$$\frac{\partial }{\partial z}{f}_i^{(m)}=f_{i+1}^{(m)}+\frac{\partial }{\partial z}\log |f_i^{(m)}{|}^2{f}_i^{(m)},i=0,\cdots ,m-1,$$

$$\frac{\partial }{\partial \overline{z}}{f}_i^{(m)}=-l_{i-1}^{(m)}{f}_{i-1}^{(m)},i=1,\cdots ,m,$$

$$\frac{{\partial }^2}{\partial z\partial \overline{z}}\log |f_i^{(m)}{|}^2=l_i^{(m)}-l_{i-1}^{(m)},$$

$$\frac{{\partial }^2}{\partial z\partial \overline{z}}\log l_i^{(m)}=l_{i+1}^{(m)}-2l_i^{(m)}+l_{i-1}^{(m)},i=0,\cdots ,m-1,$$

$$d{e}_i^{(m)}=-\sqrt{l_{i-1}^{(m)}}d\bar{z}{e}_{i-1}^{(m)}+{\theta }_{ii}^{(m)}{e}_i^{(m)}+\sqrt{l_i^{(m)}}dz{e}_{i+1}^{(m)},$$

$$F_p^{(m)}=f_0^{(m)}\wedge \cdots \wedge f_p^{(m)},0\le p\le m.$$

$$\frac{{\partial }^2}{\partial z\partial \overline{z}}\log |F_i^{(m)}{|}^2=l_i^{(m)},0\le p\le m-1.$$

$${\delta }_p^{(m)}:={\int }_{S^2}{l}_p^{(m)}dz\wedge d\bar{z}\cdot \frac{\text{i}}{2\pi }.$$

$${\delta }_{p-1}^{(m)}-2{\delta }_p^{(m)}+{\delta }_{p+1}^{(m)}=-2-r_p^{(m)}.$$

$$f_{i,p}(z)=\frac{i!}{(1+z\bar{z}{)}^i}\sqrt{\left(\genfrac{}{}{0ex}{}{m}{p}\right)}{z}^{p-i}\sum _k(-1{)}^k(\genfrac{}{}{0ex}{}{p}{i-k}\left)\right(\genfrac{}{}{0ex}{}{m-p}{k})(z\bar{z}{)}^k.$$

$$K_i^{(m)}=\frac{4}{m+2i(m-i)},\cos {\alpha }_i^{(m)}=\frac{m-2i}{m+2i(m-i)}.$$

$$0\stackrel{{\partial }^{\prime }}{⟶}{\psi }_0^{(m)}\stackrel{{\partial }^{\prime }}{⟶}\cdots \stackrel{{\partial }^{\prime }}{⟶}{\psi }_{m-n+5}^{(m)}={\phi }_2\stackrel{{\partial }^{\prime }}{⟶}\cdots \stackrel{{\partial }^{\prime }}{⟶}{\psi }_m^{(m)}={\phi }_{n-3}\stackrel{{\partial }^{\prime }}{⟶}0.$$

$${\phi }^{\ast }d{s}^2=\frac{4/K}{(1+z\bar{z}{)}^2}dzd\bar{z},$$

$$\Delta \log |\det A_1|=2L_2+2K-4$$

$$\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\\ ⋮\\ e_n\end{array}\right)=\left(\begin{array}{cccccc}{u}_{11}& u_{12}& u_{13}& 0& \cdots & 0\\ u_{21}& u_{22}& u_{23}& 0& \cdots & 0\\ u_{31}& u_{32}& u_{33}& 0& \cdots & 0\\ 0& 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& \cdots & 1\end{array}\right)\left(\begin{array}{c}{e}_0^{(n-1)}\\ e_1^{(n-1)}\\ e_2^{(n-1)}\\ e_3^{(n-1)}\\ ⋮\\ e_{n-1}^{(n-1)}\end{array}\right),$$

Full text

**On holomorphic two-spheres with constant curvature in the complex Grassmann manifold $G(2,n)$ **

Jie Fei

---

J. Fei

Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou 215123, P. R. China

e-mail: jie.fei@xjtlu.edu.cn

and Ling He ††† L. He (Corresponding author)

Center for Applied Mathematics, Tianjin University, Tianjin 300072, P. R. China

e-mail: heling@tju.edu.cn

Abstract. In this paper, the theory of functions of one complex variable is explored to study linearly full unramified holomorphic two-spheres with constant curvature in $G(2,n)$ satisfying that the generated harmonic sequence degenerates at position $2$. Firstly, we determine the value distribution of the curvature and give the explicit characterization of such holomorphic two-spheres in terms of a polynomial equation. Then, applying this characterization, many examples of non-homogeneous constantly curved holomorphic two-spheres are constructed.

Keywords and Phrases. Harmonic sequence, Holomorphic two-sphere, constant curvature, Functions of one complex variable, Complex Grassmann manifold.

Mathematics Subject Classification (2010). Primary 53C42, 53C55, 32H02.

1 Introduction

In differential geometry, the study of minimal two-spheres with constant curvature in symmetric spaces is a fundamental and important topic. Since the domain is a topological sphere, then any immersed harmonic map from two-sphere is conformal and minimal. Hence we can study minimal two-spheres with constant curvature from the viewpoint of harmonic two-spheres. The complex Grassmann manifold $G(k,n)$ is a prototypical example of symmetric spaces. In particular, when $k=1$, it is just the complex projective space $\mathbb{C}P^{n-1}$. There are many outstanding works about the construction and classification of harmonic maps of two-sphere in $G(k,n)$ (cf. [1], [3], [6], [9], [10], [27], [29], [30] etc.). Holomorphic maps of two-sphere in $G(k,n)$ are naturally harmonic. Here, we focus on the study of the structure of linearly full (i.e. not lying in a proper trivial subbundle $S^{2}\times\mathbb{C}^{m}$ of $S^{2}\times\mathbb{C}^{n}$) holomorphic two-spheres with constant curvature in $G(k,n)$. Explicitly, we study the value distributions of curvature and existence of such holomorphic two-spheres with constant curvature in $G(k,n)$.

Denote the set of the value of curvature of constantly curved holomorphic two-spheres and the linearly full ones in $G(k,n)$ by $\mathcal{K}_{k,n}^{h}$ and $\mathcal{K}_{k,n}^{fh}$ respectively. For the case of $k=1$, it is well known that (cf. [2])

[TABLE]

Moreover, the holomorphic two-sphere with constant curvature $K=\frac{4}{\texttt{r}}$ is the first element of the Veronese sequence in $\mathbb{C}P^{\texttt{r}}\subset\mathbb{C}P^{n-1}$, up to unitary equivalence (cf. [2]). The proof was essentially based on the Calabi’s rigidity theorem of holomorphic curves in $\mathbb{C}P^{n}$ (cf. [4], [15]). However, this rigidity does not hold for generic symmetric spaces. Chi and Zheng firstly observed this phenomenon in [7], where they classified non-congruent holomorphic two-spheres in $G(2,4)$ with constant curvature $2$ into two families, up to unitary equivalence, by using the method of moving frames. Since then, only partial results (e.g. under the condition of homogeneity or lower dimension) about $\mathcal{K}_{k,n}^{h}$ and $\mathcal{K}_{k,n}^{fh}$ were known. Indeed, with the homogeneous assumption, Xu and Peng ([26]) and the first author ([11]) gave a complete classification of linearly full homogeneous holomorphic two-spheres in $G(2,n)$ respectively, and obtained the values of curvature are $\frac{4}{n-2}$ and $\frac{4}{2(n-2)}$. For the case of $k=2$ and $n=4$, Li and Yu proved

[TABLE]

and classified all holomorphic two-spheres with constant curvature in $G(2,4)$ (cf. [24], [25]). For the case of $k=2$ and $n=5$, under the condition of nonsingular, Jiao and Peng classified holomorphic two-spheres in $G(2,5)$ with constant curvatures $K=4/\texttt{r}$ where $\texttt{r}=1,2,3,4,5$ into some classes, up to unitary equivalence, and proved that there do not exist the cases of $\texttt{r}=6,7,8,9$ (cf. [21], [22]). Later, with the totally unramified assumption, the second author, Jiao and Zhou ([18]) found that $K=4/\texttt{r}$, $\texttt{r}=3,6$ for linearly full holomorphic two-spheres with constant curvature in $G(2,5)$, and there exist two families of non-homogeneous holomorphic two-spheres in the case of $\texttt{r}=3$, while there exists only a homogeneous one in the case of $\texttt{r}=6$. So the accurate values in $\mathcal{K}_{2,5}^{h}$ and $\mathcal{K}_{2,5}^{fh}$ are still not clear. We can prove that there don’t exist the cases of $\texttt{r}=7,8,9$ for $G(2,5)$ under the unramified condition.

Theoretical physicists are also interested in this problem. In 2013, Delisle, Hussin and Zakrzewski ([8]) recovered the classification results in $G(2,4)$ and $G(2,5)$ mentioned above by their approach from the viewpoint of Grassmannian sigma models, and proposed two conjectures as follows:

Conjecture 1

[TABLE]

Conjecture 2* For $k$ fixed, holomorphic two-spheres with constant curvature $K=4/\emph{{r}}$ in $G(k,n)$ can be constructed for all integer values of r such that $1\leq\emph{{r}}\leq k(n-k)$.

According to the existing results, we see the above two conjectures are true in the cases of $k=1$ and $k=2,n=4$ respectively. Recently, Hussin et al. explored the constant curvature holomorphic solutions of the supersymmetric Grassmannian sigma model $G(k,n)$ using the gauge invariance of the model (cf. [19], [20]).

In this paper, we mainly study linearly full constantly curved holomorphic two-spheres in $G(2,n)$. Based on the above two conjectures and some known results, we propose the following conjecture:

Conjecture 3

[TABLE]

*and the holomorphic two-spheres with constant curvature $K=4/\texttt{r}$ in $G(2,n)$ can be constructed for all integer values of r satisfying $n-2\leq\texttt{r}\leq 2(n-2)$.

It is clear that Conjecture 3 is true for the case of $G(2,4)$. We will discuss Conjecture 3 for general dimension $n$ in some special cases. If the linearly full constantly curved holomorphic two-sphere generates the harmonic sequence that degenerates at position $1$, then the values of curvature are $\frac{4}{n-2}$ and $\frac{4}{2(n-2)}$, see the beginning of section $3$ for details. If the linearly full constantly curved holomorphic two-sphere generates the harmonic sequence that degenerates at position $2$, which contains the case of non-degenerate holomorphic two-spheres in $G(2,5)$, then we obtain our main theorem:

Theorem 1.1

Let $\varphi:S^{2}\rightarrow G(2,n)~{}(n\geq 5)$ be a linearly full holomorphic curve satisfying that the harmonic sequence generated by $\varphi$ degenerates at position $2$. If $\varphi$ is unramified with constant curvature $K$, then there exist holomorphic curves $\psi^{(1)}_{0}$ and $\psi^{(n-3)}_{0}$ of degrees $1$ in $\mathbb{C}P^{1}$ and $\emph{{d}}~{}(\emph{{d}}\geq n-3)$ in $\mathbb{C}P^{n-3}$ respectively, and a holomorphic polynomial $h$ satisfying

[TABLE]

where $c$ is a positive constant and $f_{0}^{(1)},~{}f_{0}^{(n-3)}$ are nowhere zero holomorphic sections of $\psi^{(1)}_{0},~{}\psi^{(n-3)}_{0}$ respectively, such that

[TABLE]

with $K=\frac{4}{\emph{{d}}+1}$, where $f_{1}^{(1)}=\frac{\partial}{\partial z}f_{0}^{(1)}-\frac{\partial\log|f_{0}^{(1)}|^{2}}{\partial z}\ f_{0}^{(1)}$ and $|f_{0}^{(1)}||f_{1}^{(1)}|=c_{0}$. Furthermore, $\emph{{d}}+1\leq 2(n-2)$.

It follows from Theorem 1.1 that we verify the first part of Conjecture 3 in the special situation. Moreover, Theorem 1.1 gives an explicit characterization of constantly curved holomorphic two-spheres, which can be used to consider the second part of Conjecture 3. Here the unramified condition is necessary for applying some techniques of functions of one complex variable. But we believe this condition is not essential.

For the second part of Conjecture 3, under the same assumption of Theorem 1.1, we get the following result, see also Theorem 4.6.

Theorem 1.2

If there exist solutions to the polynomial equation (1.1) satisfying that the multiplicity of zero eigenvalues of the coefficients matrix of $|f_{0}^{(n-3)}|^{2}$ is q, then the holomorphic two-spheres with $K=\frac{4}{n-2+\emph{{q}}}$ can be constructed explicitly.

Our paper is organized as follows. Preliminaries are given in Section 2. Section 3 is devoted to proving Theorem 1.1. The method of moving frames (cf.[5]), harmonic sequences in $\mathbb{C}P^{m}$ and some techniques of complex analysis are used. We know it is pivotal to find a good adapted frame in the method of moving frames. At first we choose a generic adapted frame by using the harmonic sequences in $\mathbb{C}P^{m}$. Then we can obtain a series of globally defined functions on two-spheres of constant curvature with the unramified assumption. Analyzing the properties of these functions through applying some techniques of functions of one complex variable, we simplify the original frame and obtain a good adapted one. As an application of Theorem 1.1, we completely classify such immersions with constant length of second fundamental form (See Theorem 3.7). In Section 4, we construct some families of non-homogeneous constantly curved holomorphic two-spheres in $G(2,n)$ by solving the polynomial equation (1.1), the similar techniques of solving equations have been used in [17].

2 Preliminaries

2.1 Geometry of holomorphic curves in $G(2,n)$

In this subsection, we introduce the geometry of holomorphic curves in $G(2,n)$ by the method of moving frames. More details can be found in [7] and [12]. Let $\varphi$ be a linearly full holomorphic immersion from $S^{2}$ into $G(2,n)$. Then $\varphi$ generates the following harmonic sequence (cf. [3],[6])

[TABLE]

where $\varphi_{i}:S^{2}\rightarrow G(k_{i},n)$ is a harmonic map with $k_{i}\leq 2$, $\partial^{\prime}\varphi_{i}=\varphi_{i+1}$ for $0\leq i<l$, $\partial^{\prime}\varphi_{l}=0$ and $\sum_{i=0}^{l}k_{i}=n$. If $k_{1}=\cdots=k_{r-1}=2$ and $k_{i}=1$ for $r\leq i\leq l$, we say that $\varphi$ degenerates at position $r$ and $l=n-r-1$. For example, $\varphi$ degenerates at position 2 means $k_{1}=2$, $k_{i}=1$ for all $i=2,\cdots,n-3$.

Since any member of $\varphi_{0},\cdots,\varphi_{l}$ are orthogonal, we can choose a local unitary frame $e=\{e_{1},\cdots,e_{n}\}$ along $\varphi$ so that

[TABLE]

Under such frame, the pull back of (right invariant) Maurer-Cartan forms which are denoted by $\omega=(\omega_{AB})$ are

[TABLE]

where $\phi$ is a local unitary coframe of $(1,0)$ type with respect to the induced metric $\varphi^{*}ds^{2}$ (here $ds^{2}$ is the standard K$\ddot{\textrm{a}}$hler metric on $G(2,n)$), $\Omega_{i},1\leq i\leq r$ are $\mathfrak{u}(2)$-valued 1-forms, A_{i}=\left(\begin{array}[]{cc}a_{2i-1\,2i+1}&a_{2i-1\,2i+2}\\ a_{2i\,2i+1}&a_{2i\,2i+2}\\ \end{array}\right) with rank$A_{i}=2$ for $1\leq i\leq r-1$, A_{r}=\left(\begin{array}[]{cc}0&0\\ a_{2r\,2r+1}&0\\ \end{array}\right). The structure equations of $\varphi^{*}ds^{2}$ can be written as

[TABLE]

where $\rho$ is the real-valued connection form and $K$ is the curvature.

Notice that the unitary frame we choose is determined up to a transformation of the group \underbrace{U(2)\times\cdots\times U(2)}\limits_{r-1}\times\underbrace{U(1)\times\cdots\times U(1)}\limits_{n-2(r-1)}, so $|\det A_{i}|$ $(i=1,2,\cdots,r-1)$ are global invariants of analytic type on $S^{2}$ vanishing only at isolated points, and away from their zeros, they satisfy (cf. [7], [12])

[TABLE]

where $L_{0}=0$, $L_{i}=trA_{i}A_{i}^{*},1\leq i\leq r$ are also globally defined invariants on $S^{2}$, and $\Delta$ is Laplace-Beltrami operator with respect to $\varphi^{*}ds^{2}$.

Let $S$ be the square of the length of the second fundamental form. Then the Gauss equation of $\varphi$ (cf. [11],[13]) is

[TABLE]

2.2 Harmonic sequence of two-spheres in $\mathbb{C}P^{m}$

In this subsection, we introduce the harmonic sequence of two-spheres in the complex projective space $\mathbb{C}P^{m}$, which will be used to choose proper frames of holomorphic two-spheres in $G(2,n)$. Let $\psi:S^{2}\rightarrow\mathbb{C}P^{m}$ be a linearly full harmonic map. Eells and Wood’s result (cf. [10]) shows that the following sequence in $\mathbb{C}P^{m}$ is uniquely determined by $\psi$

[TABLE]

for some $i=0,1,\cdots,m$.

Under a local coordinate $z$, we choose a holomorphic section $f_{0}^{(m)}$ of ${{\psi}}^{(m)}_{0}$ such that $\frac{\partial}{\partial\overline{z}}f_{0}^{(m)}=0$. Let $f_{i}^{(m)}$ be a local section of ${{\psi}}^{(m)}_{i}$ such that

[TABLE]

for $i=1,\cdots,m$. Then we have some formulas as follows (cf. [2]):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $l_{i}^{(m)}=|f_{i+1}^{(m)}|^{2}/|f_{i}^{(m)}|^{2}$ for $i=0,\cdots,n$, and $l_{-1}^{(m)}=l_{m}^{(m)}=0$.

Set $e_{i}^{(m)}=\frac{f_{i}^{(m)}}{|f_{i}^{(m)}|}$. Then from (2.5)-(2.8) and by a straightforward computation, we have

[TABLE]

where $\theta_{ii}^{(m)}=\frac{\partial}{\partial z}\log|f_{i}^{(m)}|dz-\frac{\partial}{\partial\bar{z}}\log|f_{i}^{(m)}|d\bar{z}$ is an imaginary 1-form.

The $p$-th osculating curve $\sigma_{p}:S^{2}\rightarrow\mathbb{C}P^{{m+1\choose p+1}-1}$ of ${{\psi}}^{(m)}_{0}$ is defined as follows: Locally

[TABLE]

Here $F_{p}^{(m)}$ may have isolated zeros. At the singular points, factor out the common divisors, we write $F_{p}^{(m)}=h_{p,m}\tilde{F}_{p}^{(m)}$, where $\tilde{F}_{p}^{(m)}$ is a $\mathbb{C}^{{m+1\choose p+1}}$-valued holomorphic map without zeros. So we can extend the definition of $\sigma_{p}$ at these points by $\tilde{F}_{p}^{(m)}$. Here $\sigma_{p}$ is a holomorphic map. By (2.7), we have

[TABLE]

The degree $\delta_{p}^{(m)}$ of $\sigma_{p}$ is given by

[TABLE]

Let $r^{(m)}_{p}$ be the degree of the singular divisor of $\sigma_{p}^{*}ds^{2}_{FS}=l_{p}^{(m)}dzd\bar{z}$, which is a non-negative integer, then we have the global Pl$\ddot{u}$cker formula

[TABLE]

We recommend readers to Bolton et al.[2] for details.

Particularly, let $\psi^{(m)}_{i}=[f_{i}^{(m)}]=[f_{i,0},\cdots,f_{i,p}\ ,\cdots,f_{i,m}]$, where $f_{i,p}$ is explicitly given by

[TABLE]

Such a map $\psi^{(m)}_{i}:S^{2}\rightarrow\mathbb{C}P^{m}$ is a conformal minimal immersion with constant curvature and constant K$\ddot{\textrm{a}}$hler angle, which are given by

[TABLE]

This harmonic sequence is well known as Veronese sequence in [2], which will be denoted by $V_{0}^{(m)},\cdots,V_{m}^{(m)}:S^{2}\rightarrow\mathbb{C}P^{m}$ correspondingly.

3 Proof of Theorem 1.1

Let $\varphi$ be a linearly full holomorphic immersion from $S^{2}$ into $G(2,n)$ with constant curvature $K$. The second author in her Ph.D. Thesis ([16]) showed that if $\varphi$ degenerates at position 1, then up to $U(n)$, $\varphi=V^{(n-1)}_{0}\oplus V^{(n-1)}_{1}$ with $K=\frac{2}{n-2}$ or $\varphi=V^{(n-2)}_{0}\oplus\mathbb{C}v$ with $K=\frac{4}{n-2}$, where $v$ is a non-zero constant vector. In this section, we focus on the case that the harmonic sequence generated by $\varphi$ degenerates at position 2.

**Proof of Theorem 1.1

**Proof: If $\varphi$ generates the harmonic sequence (2.1) with $r=2$, then $\varphi_{2},\cdots,\varphi_{n-3}$ belong to the following harmonic sequence in $\mathbb{C}P^{m}~{}(m=n-1,n-2,n-3)$

[TABLE]

Since $\varphi$ is unramified (cf.[18], Definition 2.4) with constant curvature $K$. Then we can choose the canonical coordinate on $S^{2}$ such that

[TABLE]

and the well-defined invariant $|\det A_{1}|^{2}\phi^{2}\bar{\phi}^{2}$ has no zeros on $S^{2}$ and satisfies

[TABLE]

for $L_{1}=1$.

In the following we discuss the values of $K$ and the corresponding map $\varphi$ in three cases $m=n-1,n-2,n-3$ respectively.

Case I: $m=n-1$.

In this case, we have a diagram below about the harmonic sequences in $\mathbb{C}P^{n-1}$ (cf. [3]),

[TABLE]

Then we choose a local unitary frame $e=(e_{1},\cdots,e_{n})^{T}$ as follows

[TABLE]

where $U_{1}=\begin{pmatrix}u_{11}&u_{12}&u_{13}\\ u_{21}&u_{22}&u_{23}\\ u_{31}&u_{32}&u_{33}\end{pmatrix}\in U(3)$, such that in the Maurer-Cartan forms $(\omega_{AB})$,

[TABLE]

Taking exterior differentiation of (3.4) and using (2.9), we get

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

It follows from (3.5) and (3.6) that $u_{13}=0$ and

[TABLE]

Then from (3.7) and $U_{1}U_{1}^{*}=I_{3}$, we have

[TABLE]

and

[TABLE]

Now we will prove that there doesn’t exist this case. If we choose another unitary frame $\tilde{e}$ satisfying (3.4), then

[TABLE]

Since

[TABLE]

[TABLE]

then $|a_{13}|^{2}$ and $|a_{24}|^{2}$ are globally defined functions on $S^{2}$. Since $|\det A_{1}|^{2}=|a_{13}|^{2}|a_{24}|^{2}$ has no zeros on $S^{2}$, then $|a_{13}|^{2}$ and $|a_{24}|^{2}$ have no zeros on $S^{2}$. Hence, we can obtain some functions which is globally defined on $S^{2}$, such as $\frac{u_{23}}{a_{24}},~{}\frac{u_{12}\overline{u}_{31}}{a_{13}}$,  $\frac{u_{12}\overline{u}_{32}}{a_{13}}$,  $|u_{ij}|^{2}~{}(i,j=1,2,3)$ and so on.

Let $\phi=\frac{\sqrt{4/K}}{1+z\bar{z}}dz$, then from (3.8),

[TABLE]

Since $\frac{4|a_{24}|^{2}}{K(1+z\bar{z})^{2}}dzd\bar{z}$ has no zeros on $S^{2}$, and $|u_{23}|^{2}$ and $l_{2}^{(n-1)}dzd\bar{z}$ are globally defined on $S^{2}$, then $|u_{23}|^{2}$ and $l_{2}^{(n-1)}dzd\bar{z}$ have no zeros on $S^{2}$. Let $f_{0}^{(n-1)}$ be a nowhere zero holomorphic section of $\psi_{0}^{(n-1)}$ in (3.1) such that $\frac{\partial}{\partial\bar{z}}f_{0}^{(n-1)}=0$. Let $F_{j}^{(n-1)}=f_{0}^{(n-1)}\wedge\frac{\partial}{\partial z}f_{0}^{(n-1)}\wedge\cdots\wedge\frac{\partial^{j}}{\partial z^{j}}f_{0}^{(n-1)}$ for $j=1,\cdots,n-1$. Then $F_{j}^{(n-1)}=h_{j,n-1}\tilde{F}_{j}^{(n-1)}$ is the $j$-th osculating curve of $\psi_{0}^{(n-1)}$, where $\tilde{F}_{j}^{(n-1)}$ is nowhere zero and $h_{j,n-1}$ is a monic polynomial that represents the zeros of $F_{j}^{(n-1)}$. Here if $F_{j}^{(n-1)}$ is nowhere zero, then $h_{j,n-1}=1$. Since $l_{2}^{(n-1)}dzd\bar{z}$ has no zeros on $S^{2}$, then we get

[TABLE]

and $r_{2}^{(n-1)}=0$, by (2.12),

[TABLE]

Thus,

[TABLE]

In this case, since $L_{2}\phi\bar{\phi}=l_{3}^{(n-1)}dzd\bar{z}$, then (3.2) becomes

[TABLE]

Integrating both sides of (3.14), we have

[TABLE]

where in the first equation we use the fact that the globally defined function $\left|\det\!A_{1}\right|$ has no zeros. Recall $\varphi$ is of constant curvature $K$, and in the last equation we use the Gauss-Bonnet Theorem. Hence by (3.15), we have

[TABLE]

Since

[TABLE]

then using (3.2) again, we obtain

[TABLE]

Since $\frac{|\det\!A_{1}|(1+z\overline{z})^{\frac{\delta_{3}^{(n-1)}}{2}}}{|\tilde{F}_{3}^{(n-1)}|}$ is a globally defined nowhere zero function on $S^{2}$, then it follows that

[TABLE]

where $c$ is a positive constant.

Lemma 3.1

In the case $m=n-1$, $|u_{31}|^{2}\equiv 0$ on $S^{2}$.

Proof: Suppose that $|u_{31}|^{2}$ is not identically zero on $S^{2}$, then $|u_{12}|^{2}$ is not identically zero on $S^{2}$ by $U_{1}\in U(3)$. Substituting $u_{11}=-u_{12}\cdot\frac{\overline{u}_{32}}{\overline{u}_{31}}$ into (3.9), we obtain

[TABLE]

and

[TABLE]

Substituting $u_{21}=-u_{22}\cdot\frac{\overline{u}_{32}}{\overline{u}_{31}}-u_{23}\cdot\frac{\overline{u}_{33}}{\overline{u}_{31}}$ into (3.10) and using (3.17), we have

[TABLE]

By (3.17), (3.18) and (3.19), a straightforward computation shows

[TABLE]

Let $\eta_{1}=\frac{\sqrt{4/K}a_{13}}{u_{12}\overline{u}_{31}(1+z\bar{z})}\sqrt{l_{0}^{(n-1)}}$, then $\eta_{1}$ is a holomorphic function and

[TABLE]

Set $h_{1,n-1}=(z-z_{1})^{\alpha_{1}}\cdots(z-z_{k})^{\alpha_{k}}$, where $z_{1},\cdots,z_{k}$ are diverse. Since $\frac{u_{12}\overline{u}_{31}}{a_{13}}$ is globally defined on $S^{2}$, then $z_{j}(j=1,\cdots,k)$ are all possible singularities of $\frac{1}{\eta_{1}}$ on $\mathbb{C}=S^{2}\backslash\{\infty\}$. Because $\lim\limits_{z\rightarrow z_{j}}\frac{(z-z_{j})^{\alpha_{j}+1}}{\eta_{1}}=0$, then $z_{j}$ is a removable singularity of $\frac{(z-z_{j})^{\alpha_{j}}}{\eta_{1}}$, which implies that $\frac{1}{\eta_{1}}$ has a pole of order at most $\alpha_{j}$ at $z=z_{j}$. It follows from (2.12) that

[TABLE]

Let $\alpha=\sum\limits_{\j=1}^{k}\alpha_{j},~{}\rho=2+2\delta_{0}^{(n-1)}-\delta_{1}^{(n-1)}-\alpha$, then $\rho=4+r_{0}^{(n-1)}-\alpha\geq 4$ by (3.22). Similarly, since $\infty$ is a removable singularity of $\frac{1}{\eta_{1}z^{\rho}}$, then $\frac{1}{\eta_{1}}$ has a pole of order at most $\rho$ at $z=\infty$. So $\frac{1}{\eta_{1}}$ is a meromorphic function on $S^{2}$, obviously is a rational function. Then $\eta_{1}$ is also a rational function. Let $\eta_{1}=\frac{P_{1}}{Q_{1}}$, where $P_{1}=(z-z_{1})^{\beta_{1}}\cdots(z-z_{k})^{\beta_{k}}$ and $Q_{1}=(z-z_{k+1})^{\beta_{k+1}}\cdots(z-z_{k+l})^{\beta_{k+l}}$ are co-prime monic polynomials. Here $\beta_{j}\leq\alpha_{j}$ for $j=1,\cdots,k$ and $\sum\limits_{j=1}^{l}\beta_{k+j}-\sum\limits_{j=1}^{k}\beta_{j}\leq\rho$. It follows from (3.21) that

[TABLE]

From (3.17), we have

[TABLE]

Let $\eta=\frac{\overline{u}_{32}}{\overline{u}_{31}}\sqrt{l_{0}^{(n-1)}}+\frac{\partial\log|f_{0}^{(n-1)}|^{2}}{\partial{z}}$, then $\eta$ is a holomorphic function and by (3.23),

[TABLE]

Since $\frac{u_{12}\overline{u}_{32}}{a_{13}}$ is globally defined on $S^{2}$, then from (3.25) we find $\eta$ is also a rational function. Let $\eta=\frac{P}{Q}$, where $Q=(z-z_{k+1})^{\nu_{k+1}}\cdots(z-z_{k+l})^{\nu_{k+l}}$ and $P$ are co-prime monic polynomials. Here $\nu_{k+j}\leq\beta_{k+j}$ for $j=1,\cdots,l$. Then,

[TABLE]

where $g_{0}^{(n-1)}=Pf_{0}^{(n-1)}-Q\frac{\partial f_{0}^{(n-1)}}{\partial z}$. Since $U_{1}\in U(3)$, from (3.26),

[TABLE]

It follows from (3.23) and (3.27) that

[TABLE]

Since $|u_{23}|^{2}=|u_{31}|^{2}+|u_{32}|^{2}$, then by (3.27) and (3.28),

[TABLE]

Let $g_{0}^{(n-1)}=h_{0,n-1}\tilde{g}_{0}^{(n-1)}$, where $\tilde{g}_{0}^{(n-1)}$ is nowhere zero and $h_{0,n-1}$ is a monic polynomial that represents the zeros of $g_{0}^{(n-1)}$. Since $|u_{23}|^{2}$ is a globally defined function without zeros on $S^{2}$, then from (3.29), we get $h_{0,n-1}^{2}Q_{1}=h_{1,n-1}P_{1}Q^{2}$. Here $h_{0,n-1}$ and $Q$ are co-prime, $Q_{1}$ and $h_{1,n-1}P_{1}$ are co-prime, so,

[TABLE]

Since $h_{1,n-1}=\prod\limits_{j=1}^{k}(z-z_{j})^{\alpha_{j}}$, $P_{1}=\prod\limits_{j=1}^{k}(z-z_{j})^{\beta_{j}}$ and $\beta_{j}\leq\alpha_{j}$, then from (3.30), $h_{0,n-1}=\prod\limits_{j=1}^{k}(z-z_{j})^{\gamma_{j}}$, where $\beta_{j}\leq\gamma_{j}\leq\alpha_{j}$ satisfying $2\gamma_{j}=\alpha_{j}+\beta_{j}$. Since $F_{1}^{(n-1)}={f}_{0}^{(n-1)}\wedge\frac{\partial}{\partial z}{f}_{0}^{(n-1)}$, then $\frac{\partial}{\partial z}{f}_{0}^{(n-1)}=h_{1,n-1}g_{1}^{(n-1)}$, where $g_{1}^{(n-1)}$ is nowhere zero. So that $g_{0}^{(n-1)}=P{f}_{0}^{(n-1)}-Qh_{1,n-1}g_{1}^{(n-1)}=h_{0,n-1}\tilde{g}_{0}^{(n-1)}$, which implies $h_{0,n-1}\mid P$ by $h_{0,n-1}\mid h_{1,n-1}$. Let $h_{1,n-1}=h_{0,n-1}\tilde{h}_{1,n-1}$ and $P=h_{0,n-1}\tilde{P}$, where $\tilde{h}_{1,n-1}=\prod\limits_{j=1}^{k}(z-z_{j})^{\alpha_{j}-\gamma_{j}}$ and $\tilde{P}$ is a monic polynomial. Then

[TABLE]

Substituting (3.30) into (3.29), we get

[TABLE]

From (3.18), (3.23), (3.26) and (3.30),

[TABLE]

where $T=\frac{P_{1}+QP^{\prime}-PQ^{\prime}+P^{2}}{Q}$. Set $\mu=TF_{1}^{(n-1)}-g_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}f_{0}^{(n-1)}$, then by (3.33),

[TABLE]

Using (3.28), (3.30) and (3.34), we obtain

[TABLE]

Since $|u_{23}|^{2}+|u_{33}|^{2}=1$ by $U_{1}\in U(3)$, then by (3.32) and (3.35),

[TABLE]

A straightforward calculation shows

[TABLE]

which implies by (3.36),

[TABLE]

Set $\frac{\partial^{2}}{\partial z^{2}}{f}_{0}^{(n-1)}=\tilde{h}_{2,n-1}g_{2}^{(n-1)}$, where $\tilde{h}_{2,n-1}$ is a monic polynomial and $g_{2}^{(n-1)}$ is nowhere zero. Then $F_{2}^{(n-1)}={f}_{0}^{(n-1)}\wedge\frac{\partial}{\partial z}{f}_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}{f}_{0}^{(n-1)}=h_{1,n-1}\tilde{h}_{2,n-1}{f}_{0}^{(n-1)}\wedge g_{1}^{(n-1)}\wedge g_{2}^{(n-1)}$. So that $h_{2,n-1}=h_{1,n-1}\tilde{h}_{2,n-1}$ and $\tilde{F}_{2}^{(n-1)}={f}_{0}^{(n-1)}\wedge g_{1}^{(n-1)}\wedge g_{2}^{(n-1)}$. From (3.38), we have

[TABLE]

where $\tilde{\mu}$ is nowhere zero, and

[TABLE]

Let $\tilde{T}=\frac{Th_{1,n-1}}{\tilde{h}_{2,n-1}h_{0,n-1}}$, then $\tilde{T}$ is a rational function and

[TABLE]

Since $\mu=Th_{1,n-1}\tilde{F}_{1}^{(n-1)}-h_{0,n-1}\tilde{h}_{2,n-1}\tilde{g}_{0}^{(n-1)}\wedge g_{2}^{(n-1)}$, then by (3.31) and (3.39),

[TABLE]

Combining (3.13), (3.32) and (3.40), we get

[TABLE]

It follows from (3.40) and (3.43) that

[TABLE]

Substituting (3.44) into (3.16), we obtain

[TABLE]

where $c_{0}$ is a positive constant, which implies $\tilde{T}$ is a polynomial by (3.42).

From (2.12),

[TABLE]

Combining (3.12), (3.22) and (3.46), we get

[TABLE]

In the following we consider the last three terms with maximum degrees of $\tilde{\mu}$. Let

[TABLE]

where $a_{n-3},a_{n-2},a_{n-1}$ are non-zero complex numbers, and $k_{1},k_{2}$ are positive integers. Then we have

[TABLE]

[TABLE]

[TABLE]

Since $f_{0}^{(n-1)}\wedge\frac{\partial f_{0}^{(n-1)}}{\partial z}=h_{1,n-1}\tilde{F}_{1}^{(n-1)}$, then from (3.49),

[TABLE]

Combining (3.22) and (3.52), we get

[TABLE]

Since $f_{0}^{(n-1)}\wedge\frac{\partial f_{0}^{(n-1)}}{\partial z}\wedge\frac{\partial^{2}f_{0}^{(n-1)}}{\partial z^{2}}=h_{2,n-1}\tilde{F}_{2}^{(n-1)}$, then

[TABLE]

Combining (3.22), (3.46) and (3.54), we obtain

[TABLE]

Using (3.53), (3.55) and $h_{2,n-1}=h_{1,n-1}\tilde{h}_{2,n-1}$, we have

[TABLE]

Applying (3.53), (3.22), (3.47) and (3.56) successively, we get

[TABLE]

By (3.57), we have

[TABLE]

We claim $T$ and $P$ don’t vanish at the same time. Otherwise if $T=P=0$, then $P_{1}=0$. It is a contradiction. Hence $\tilde{T}$ and $\tilde{P}$ don’t vanish at the same time. Observing (3.42) and using (3.49)-(3.50), (3.57)-(3.58), we conclude the degree of everyone of the last three terms in $\tilde{\mu}$ is bigger than $\frac{\delta_{3}^{(n-1)}}{2}+2$. But (3.45) implies the degree of $\tilde{\mu}$ is $\frac{\delta_{3}^{(n-1)}}{2}+2$, so that the last three terms of $\tilde{\mu}$ must vanish. Then the last three terms of ${\mu}$ also vanish. Denote the last three terms of $F_{1}^{(n-1)},~{}f_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}f_{0}^{(n-1)},~{}\frac{\partial}{\partial z}f_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}f_{0}^{(n-1)}$ by $v_{1},v_{2},v_{3}$ respectively. Since $\mu=TF_{1}^{(n-1)}-Pf_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}f_{0}^{(n-1)}+Q\frac{\partial}{\partial z}f_{0}^{(n-1)}\wedge\frac{\partial^{2}}{\partial z^{2}}f_{0}^{(n-1)}$, then

[TABLE]

From (3.49)-(3.51), we see that $v_{1},v_{2},v_{3}$ are linearly independent at any point $z\in\mathbb{C}=S^{2}\backslash\{\infty\}$, then by (3.59), $T=P=Q=0$. It’s a contradiction. Thus we finish the proof of this lemma.   $\Box$

From Lemma 3.1, we have $u_{31}=0$, then $u_{21}\overline{u}_{23}=0$ by $U_{1}^{*}U_{1}=I_{3}$. Since $u_{23}\neq 0$, we have $u_{21}=0$, which implies $u_{12}=0$. Now we can choose a unitary frame $e$ such that

[TABLE]

From (3.5)-(3.7), we get

[TABLE]

and

[TABLE]

where $\Theta_{1},\Theta_{2}$ is given by (3.7). It follows from (3.10) that

[TABLE]

which implies that $|{u}_{22}|^{2}$ is not identically zero on $S^{2}$, and

[TABLE]

Since $|a_{13}|^{2}\phi\bar{\phi}=|{u}_{23}|^{2}l_{0}^{(n-1)}dzd\bar{z}$ has no zeros on $S^{2}$, then $l_{0}^{(n-1)}dzd\bar{z}$ has no zeros on $S^{2}$, which implies $h_{1,n-1}=1,~{}r_{0}^{(n-1)}=0$ and

[TABLE]

Now $|\det\!A_{1}|=|{u}_{23}|^{2}\cdot\frac{K(1+z\bar{z})^{2}|{F}_{1}^{(n-1)}|^{2}|\tilde{F}_{3}^{(n-1)}|}{4|{f}_{0}^{(n-1)}|^{2}|\tilde{F}_{2}^{(n-1)}|^{2}}$, then from (3.16), we get

[TABLE]

Proposition 3.2

If $\varphi:S^{2}\rightarrow G(2,n)$ is a linearly full unramified holomorphic curve of constant curvature and generates the harmonic sequence (2.1) with $r=2$, then $m\neq n-1$.

Proof: From (3.61), we have

[TABLE]

Let $\eta=\frac{{u}_{22}}{{u}_{23}}\sqrt{l_{1}^{(n-1)}}-\frac{\partial\log|F_{1}^{(n-1)}|^{2}}{\partial{z}}$, then $\eta$ is a holomorphic function by (3.64) and

[TABLE]

Here $\frac{{u}_{22}}{{u}_{23}}$ is globally defined on $S^{2}$, then from (3.65) we find $\eta$ has no singularity on $\mathbb{C}=S^{2}/\{\infty\}$, hence is an entire function. Since $2\delta_{1}^{(n-1)}-\delta_{2}^{(n-1)}-\delta_{0}^{(n-1)}=2+r_{1}^{(n-1)}>0$, then $\lim\limits_{z\rightarrow\infty}\frac{|F_{2}^{(n-1)}||f_{0}^{(n-1)}|}{|F_{1}^{(n-1)}|^{2}}=0$. And $\lim\limits_{z\rightarrow\infty}\frac{\partial\log|F_{1}^{(n-1)}|^{2}}{\partial{z}}=0$, then $\lim\limits_{z\rightarrow\infty}\eta=0$. So that $\eta$ is a bounded entire function, which is a constant by Liouville theorem. Thus, $\eta=0$. Then it follows from (3.65) that

[TABLE]

Since $|u_{22}|^{2}+|u_{23}|^{2}=1$ by $U_{1}\in U(3)$, then a straightforward calculation shows

[TABLE]

Because $h_{1,n-1}=1$, then $\frac{\partial^{2}}{\partial z^{2}}{f}_{0}^{(n-1)}={h}_{2,n-1}g_{2}^{(n-1)}$, where $g_{2}^{(n-1)}$ is nowhere zero. Now (3.67) becomes

[TABLE]

Combining (3.63) and (3.68), we obtain

[TABLE]

where $c_{0}$ is a positive constant.

Let $f_{0}^{(n-1)}$ be the same as (3.48). Since $r_{0}^{(n-1)}=0,~{}h_{1,n-1}=1$, then (3.47) and (3.53) become

[TABLE]

and

[TABLE]

respectively. Substituting $r_{0}^{(n-1)}=0$ and (3.71) into (3.55), we have

[TABLE]

which implies by (3.62) and (3.70),

[TABLE]

It follows from (3.50) and (3.73) that the degree of $f_{0}^{(n-1)}\wedge g_{2}^{(n-1)}$ is bigger than $\frac{\delta_{3}^{(n-1)}}{2}+2$. It contradicts (3.69). So there doesn’t exist the case of $m=n-1$, i.e. $m\neq n-1$.   $\Box$

Case II: $m=n-2$.

In this case, similarly we choose a local unitary frame $e$ as follows

[TABLE]

where $v_{0}=(0,\cdots,0,1)$, so that $A_{1},\ A_{2}$ have the same form as in Case I, here the trivial bundle $S^{2}\times\mathbb{C}^{n}$ over $S^{2}$ has a corresponding decomposition $S^{2}\times\mathbb{C}^{n}=S^{2}\times\mathbb{C}^{n-1}\oplus S^{2}\times\mathbb{C}$. In this case we get the similar equations with (3.5) and (3.6), which implies $u_{13}=0$ and

[TABLE]

In the equation (3.7), the matrix $\Theta_{2}$ becomes

[TABLE]

Then from (3.7) we have

[TABLE]

[TABLE]

and

[TABLE]

We choose another unitary frame $\tilde{e}$ satisfying (3.74), then by similar argument as in Case I, we obtain some globally defined functions on $S^{2}$, such as $\frac{u_{23}}{a_{24}},~{}\frac{u_{12}\overline{u}_{31}}{a_{13}}$,  $\frac{u_{12}\overline{u}_{32}}{a_{13}}$,  $|u_{ij}|^{2}~{}(i,j=1,2,3)$ and so on. Here $|a_{13}|^{2}$ and $|a_{24}|^{2}$ have no zeros on $S^{2}$.

Let $\phi=\frac{\sqrt{4/K}}{1+z\bar{z}}dz$, then from (3.75),

[TABLE]

Since $\frac{4|a_{24}|^{2}}{K(1+z\bar{z})^{2}}dzd\bar{z}$ has no zeros on $S^{2}$, then $|u_{23}|^{2}$ and $l_{1}^{(n-2)}dzd\bar{z}$ have no zeros on $S^{2}$.

Lemma 3.3

In the case $m=n-2$, $|u_{31}|^{2}$ and $|u_{12}|^{2}$ are not identically zero on $S^{2}$.

Proof: Suppose $|u_{31}|^{2}\equiv 0$ on $S^{2}$, then $|u_{21}|^{2}\equiv 0$ by $U_{1}^{*}U_{1}=I_{3}$ and the fact that $|u_{23}|^{2}$ has no zeros on $S^{2}$. It means $|u_{11}|^{2}\equiv 1$ and $|u_{12}|^{2}\equiv 0$. From (3.78) we find $a_{13}=0$. It’s a contradiction. Similarly, if $|u_{12}|^{2}\equiv 0$, then $|u_{11}|^{2}\equiv 1$ and $|u_{21}|^{2}=|u_{31}|^{2}\equiv 0$ by $U_{1}^{*}U_{1}=I_{3}$. It contradicts the above argument. So the lemma is proved.   $\Box$

Substituting $u_{11}=-u_{12}\cdot\frac{\overline{u}_{32}}{\overline{u}_{31}}$ into (3.78), we obtain

[TABLE]

and

[TABLE]

Substituting $u_{21}=-u_{22}\cdot\frac{\overline{u}_{32}}{\overline{u}_{31}}-u_{23}\cdot\frac{\overline{u}_{33}}{\overline{u}_{31}}$ into (3.79) and using (3.80), we have

[TABLE]

In the following we can also prove that there doesn’t exist this case $m=n-2$.

Proposition 3.4

If $\varphi:S^{2}\rightarrow G(2,n),\ n\geq 5$ is a linearly full unramified holomorphic curve of constant curvature and generates the harmonic sequence (2.1) with $r=2$, then $m\neq n-2$.

Proof: By (3.80), (3.81) and (3.82), a straightforward computation shows

[TABLE]

Let $\eta_{1}=\frac{\sqrt{4/K}a_{13}|f_{0}^{(n-2)}|}{u_{12}\overline{u}_{31}(1+z\bar{z})}$, then $\eta_{1}$ is a holomorphic function and

[TABLE]

Since $\frac{u_{12}\overline{u}_{31}}{a_{13}}$ is globally defined on $S^{2}$, then $\frac{1}{\eta_{1}}$ has no singularities on $\mathbb{C}=S^{2}\backslash\{\infty\}$, which implies that it is an entire function. If $\delta_{0}^{(n-2)}>2$, then $\lim\limits_{z\rightarrow\infty}\frac{1+z\bar{z}}{|f_{0}^{(n-2)}|}=0$. So that $\lim\limits_{z\rightarrow\infty}\frac{1}{\eta_{1}}=0$, which shows that $\frac{1}{\eta_{1}}$ is a bounded entire function. It is a constant by Liouville theorem. Thus, $\frac{1}{\eta_{1}}=0$. From (3.84), we get $|u_{12}|^{2}|u_{31}|^{2}\equiv 0$ on $S^{2}$, which contradicts to Lemma 3.3. Hence $\delta_{0}^{(n-2)}\leq 2$. Since $\delta_{0}^{(n-2)}\geq n-2$, then $n\leq 4$. It contradicts that $n\geq 5$. Thus there doesn’t exist the case of $m=n-2$ if $n\geq 5$.   $\Box$

Remark 3.5

From above Proposition, we have $n=4$ in this case. The unramified holomorphic two-spheres of constant curvature in $G(2,4)$ was classified by Chi and Zheng ([7], see also [14]).

**Case III: $m=n-3$.

**In this case, the corresponding diagram (cf. [3]) is as follows,

[TABLE]

Here, the trivial bundle $S^{2}\times\mathbb{C}^{n}$ over $S^{2}$ has a corresponding decomposition $S^{2}\times\mathbb{C}^{n}=S^{2}\times\mathbb{C}^{n-2}\oplus S^{2}\times\mathbb{C}^{2}$. Let $G$ be a smooth section of $S^{2}\times\mathbb{C}^{2}$. By the above harmonic sequence, there exists a local section $V=G+x_{0}e_{0}^{(n-3)}$ such that $\varphi_{1}=\text{span}\left\{V,e_{1}^{(n-3)}\right\}$. Since $\partial^{\prime}\varphi_{1}=\varphi_{2}=\text{span}\left\{e_{2}^{(n-3)}\right\}$, then a straightforward computation shows that $\frac{\partial}{\partial z}G=\frac{\langle\frac{\partial}{\partial z}V,V\rangle}{|V|^{2}}G$, which implies that $\text{span}\{G\}$ is an anti-holomorphic line bundle of $S^{2}\times\mathbb{C}^{2}$. So it belongs to the harmonic sequence in $\mathbb{C}P^{1}$ as follows

[TABLE]

Let $f_{0}^{(1)}$ be a nowhere zero holomorphic section of $\underline{Im}\psi^{(1)}_{0}$. Without loss of generality we assume that $\partial f_{0}^{(1)}/\partial\overline{z}=0$, then by (2.5) we obtain $f_{1}^{(1)}$, which is a local section of $\underline{Im}\psi^{(1)}_{1}$. Then we can choose a local unitary frame $e$ as follows

[TABLE]

where$\begin{pmatrix}u_{22}&u_{23}\\ -\bar{u}_{23}&\bar{u}_{22}\\ \end{pmatrix}\in U(2)$. Here $A_{1},\ A_{2}$ have the same form as in Case I.

In this case, we get the similar equations with (3.5) and (3.6), which implies

[TABLE]

In the equation (3.7), the matrix $\Theta_{2}$ becomes

[TABLE]

Then from (3.7) we have

[TABLE]

and

[TABLE]

In this case, since

[TABLE]

has no zeros on $S^{2}$, then we obtain that $|u_{23}|^{2},~{}l_{0}^{(1)}dzd\bar{z},~{}l_{0}^{(n-3)}dzd\bar{z}$ all have no zeros on $S^{2}$. Because $l_{0}^{(1)}dzd\bar{z}$ has no zeros, it follows from section $3$ of [2] that

[TABLE]

then from (2.7),

[TABLE]

which implies that the degree of the holomorphic curve $\psi_{0}^{(1)}$ is $1$. Without loss of generality we assume that

[TABLE]

for some complex numbers $\alpha,\beta,\delta$ with $\alpha,\delta$ real, and $\alpha\delta-\beta\overline{\beta}>0$. A straightforward computation shows that $|f_{0}^{(1)}|^{2}|f_{1}^{(1)}|^{2}=|f_{0}^{(1)}|^{4}\cdot\frac{\partial^{2}}{\partial z\partial\overline{z}}\log|f_{0}^{(1)}|^{2}=\alpha\delta-\beta\overline{\beta}$ is a positive constant denoted by $c_{0}^{2}$, which implies $\theta_{11}^{(1)}=-\frac{\partial}{\partial z}\log|f_{0}^{(1)}|dz+\frac{\partial}{\partial\overline{z}}\log|f_{0}^{(1)}|d\overline{z}$. From (3.89), we obtain

[TABLE]

and

[TABLE]

It follows from (3.91) that $\frac{\partial}{\partial\overline{z}}(\frac{{u}_{22}}{{u}_{23}}\cdot|f_{0}^{(1)}||f_{0}^{(n-3)}|)=0$, which implies that the function $\frac{{u}_{22}}{{u}_{23}}\cdot|f_{0}^{(1)}||f_{0}^{(n-3)}|$ is a holomorphic function denoted by $h(z)$. Since $\frac{{u}_{22}}{{u}_{23}}$ is globally defined on $S^{2}$, then $h(z)$ is a holomorphic polynomial of degree at most $\delta_{0}^{(n-3)}+1$. Thus we have

[TABLE]

Assume the degree of the holomorphic curve $\psi_{0}^{(n-3)}$ is a positive integer $\texttt{d}~{}(\texttt{d}\geq n-3)$. Since $l_{0}^{(n-3)}dzd\bar{z}$ has no zeros, it follows from section $3$ of [2] that

[TABLE]

In this case since $L_{2}\phi\bar{\phi}=l_{1}^{(n-3)}dzd\bar{z}$, then (3.2) becomes

[TABLE]

Integrating both sides of (3.94), we have by the same reasons for (3.15) that

[TABLE]

which gives

[TABLE]

Then the induced metric $\varphi^{*}ds^{2}=\phi\overline{\phi}$ is given by

[TABLE]

So we have

[TABLE]

[TABLE]

Using (3.2) again, we obtain

[TABLE]

Since $\frac{|u_{23}|^{2}(1+z\overline{z})^{\texttt{d}+1}}{|f_{0}^{(1)}|^{2}|f_{0}^{(n-3)}|^{2}}$ is a globally defined nowhere zero function on $S^{2}$, then it follows that

[TABLE]

where $c$ is a positive constant. So we get

[TABLE]

Combining (3.93) and (3.97), we have

[TABLE]

Then by (3.87) and (3.93), the corresponding holomorphic map $\varphi$ in this case is given by

[TABLE]

From (3.92), (3.93), (3.97) and (3.98), a straightforward calculation shows $(|a_{13}|^{2}+|a_{24}|^{2}+|a_{23}|^{2})\phi\bar{\phi}=\frac{\texttt{d}+1}{(1+z\bar{z})^{2}}dzd\bar{z}$, which verifies that the holomorphic curve defined by (3.99) is of constant curvature $K=\frac{4}{\texttt{d}+1}$. Since $f_{0}^{(1)}=(**\underbrace{0\cdots 0}_{n-2})^{T}$, using Pl$\ddot{\text{u}}$cker embedding $\Phi_{n}:G(2,n)\rightarrow\mathbb{C}P^{\frac{n(n-1)}{2}-1}$, we find that $\Phi_{n}\circ\varphi$ is a holomorphic curve of constant curvature in $\mathbb{C}P^{2n-4}$, then $\texttt{d}+1\leq 2(n-2)$.

Thus, we have completed the discussions of the three cases and finished the proof of Theorem 1.1.   $\Box$

In the following, we make some remarks regarding Theorem 1.1.

Remark 3.6

(i)* In Theorem 1.1, if the condition of $\varphi$ being unramified is improved to being totally unramified, that is, every element of the harmonic sequence generated by $\varphi$ is unramified (cf.[18], Definition 2.4), then $\emph{{d}}=n-3$ and $K=\frac{4}{n-2}$. This result was firstly obtained by Jiao and Yu in [23].

(ii)* Through finding the solutions of the polynomial equation (1.1) (i.e. (3.98)), we can get many examples of holomorphic two-spheres of constant curvature in $G(2,n)$. If $h=0$, by using Calabi’s Rigidity Theorem, the holomorphic map $\varphi=V_{0}^{(1)}\oplus V_{0}^{(n-3)}$ with $K=\frac{4}{n-2}$, up to $U(n)$, which is homogeneous. And for non-trivial solutions of $h$, we can check that the corresponding holomorphic curve are not homogeneous. In fact, in [18] the second author and her co-authors have given the symmetric solutions of the polynomial equation (1.1) in the case $n=5$, then determine all totally unramified holomorphic two-spheres of constant curvature in $G(2,5)$, which include two families of non-homogeneous holomorphic two-spheres.*

(iii)* From analysis of Case III, we know that if $\varphi$ is unramified with constant curvature then the equation (1.1) holds. But the solutions of (1.1) do not guarantee that the corresponding holomorphic curve $\varphi$ is unramified (cf.[18]). So by solving the polynomial equation (1.1), we can also obtain ramified holomorphic two-spheres of constant curvature in $G(2,n)$.*

To end this section, we give an application of our main Theorem 1.1. It is well known that the rigidity of holomorphic curves in $G(k,n)\ (k\geq 2)$ does not hold if we only assume they have the same first fundamental form. However, if we further assume that $\varphi$ has constant length of second fundamental form, then we have

Theorem 3.7

Let $\varphi:S^{2}\rightarrow G(2,n)~{}(n\geq 5)$ be a linearly full unramified holomorphic curve of constant curvature satisfying that the harmonic sequence generated by $\varphi$ degenerates at position $2$. If the holomorphic curve $\varphi$ has constant length of second fundamental form, then $\varphi=V_{0}^{(1)}\oplus V_{0}^{(n-3)}$ with $K=\frac{4}{n-2}$, up to $U(n)$.

Proof: From Gauss equation (2.3), we know $|\det A_{1}|$ is also constant. In Case III, using (3.2), $L_{2}=2-K=\frac{2(\texttt{d}-1)}{\texttt{d}+1}$, which implies $l_{1}^{(n-3)}dzd\overline{z}=\frac{2(\texttt{d}-1)}{(1+z\overline{z})^{2}}dzd\overline{z}$. Since $l_{1}^{(n-3)}dzd\overline{z}$ is the induced metric on $S^{2}$ by the 1-th osculating curve $\sigma_{1}$ of $\psi_{0}^{(n-3)}$, it follows that $\sigma_{1}$ has constant curvature. By using the rigidity theorems for the Veronese sequences ([2],[28]), we know that $\psi_{0}^{(n-3)}$ is the Veronese map $V_{0}^{(n-3)}$, up to $U(n-3)$, and $|f_{0}^{(n-3)}|^{2}=(1+z\overline{z})^{n-3}$. It follows from (1.1) that $1+z\overline{z}$ is a factor of $|h|^{2}$, which yields $h=0$ since $h$ is holomorphic function and $1+z\overline{z}$ is an irreducible polynomial. Thus from Remark (3.6) (ii), up to $U(n)$, $\varphi=V_{0}^{(1)}\oplus V_{0}^{(n-3)}$ with $K=\frac{4}{n-2}$.   $\Box$

4 Construction of non-homogeneous holomorphic two-spheres of constant curvature in

$G(2,n)$

In this section we will construct examples of non-homogeneous holomorphic two-spheres of constant curvature in $G(2,n)$, by finding the solutions of the polynomial equation (1.1).

At first, we give a family of solutions of the polynomial equation (1.1) as follows.

Proposition 4.1

Let $\psi^{(1)}_{0}:S^{2}\rightarrow\mathbb{C}P^{1}$ and $\psi^{(n-3)}_{0}:S^{2}\rightarrow\mathbb{C}P^{n-3}$ be holomorphic curves of degree $1$ and $\emph{{d}}~{}(\emph{{d}}\geq n-3)$ respectively. Let $f_{0}^{(1)}$ and $f_{0}^{(n-3)}$ be nowhere zero holomorphic sections of $\psi^{(1)}_{0}$ and $\psi^{(n-3)}_{0}$ respectively, let $h$ be a holomorphic polynomial, satisfying

[TABLE]

where $c_{ij}=c_{ji}=c_{\emph{{d}}-i,\emph{{d}}-j},~{}\alpha_{i},~{}\texttt{t}$ are real coefficients, and $c_{ij}~{}(i\geq j),~{}\alpha_{i}$ are given by

[TABLE]

[TABLE]

Then (4.1) gives a family of solutions of the polynomial equation (1.1), taking t as a parameter in $(-1,1)$ such that $|f_{0}^{(n-3)}|^{2}>0$.

Proof: Assume $|f_{0}^{(1)}|^{2},~{}|f_{0}^{(n-3)}|^{2},~{}h$ are given by (4.1). In order to prove they are solutions of the polynomial equation (1.1) it’s enough to prove that for $i=0,\cdots,\texttt{d}+1$,

[TABLE]

holds and for $i>j$,

[TABLE]

holds, where $c_{-1,j}=c_{\texttt{d}+1,j}=0$ and $c=1+\alpha_{0}^{2}$.

Substituting (4.1) into the left right of (4.2), we get that (4.2) holds if and only if the following equations

[TABLE]

hold. In the following we prove (4) holds by showing the coefficient of the term $\texttt{t}^{p}$ is zero for $p=0,\cdots,\texttt{d}+1$.

For $p=0$, the coefficient of the term $\texttt{t}^{0}$ is given by

[TABLE]

which is obviously zero.

For $p=1$, the coefficient of the term $\texttt{t}^{1}$ is given by

[TABLE]

which is also obviously zero.

For $p=2s~{}(s\geq 1,p\leq\texttt{d})$, the coefficient of the term $\texttt{t}^{2s}$ is given by

[TABLE]

which is zero.

For $p=2s+1~{}(s\geq 1,p\leq\texttt{d})$, the coefficient of the term $\texttt{t}^{2s+1}$ is given by

[TABLE]

which is zero.

For $p=\texttt{d}+1$, the coefficient of the term $\texttt{t}^{\texttt{d}+1}$ is given by

[TABLE]

which is zero.

From the above cases, we know that (4) holds. It verifies (4.2).

Similarly, substituting (4.1) into the left right of (4.3), we get that (4.3) holds if and only if the following equations

[TABLE]

hold. In the following we prove (4) holds by showing the coefficient of the term $\texttt{t}^{p}$ is zero for $p=i-j,\cdots,\texttt{d}+1$.

For $p=i-j$, the coefficient of the term $\texttt{t}^{i-j}$ is given by

[TABLE]

which is obviously zero.

For $p=i-j+1$, the coefficient of the term $\texttt{t}^{i-j+1}$ is given by

[TABLE]

which is also obviously zero.

For $p=i-j+2s~{}(s\geq 1,p\leq\texttt{d})$, the coefficient of the term $\texttt{t}^{i-j+2s}$ is given by

[TABLE]

which is zero.

For $p=i-j+2s+1~{}(s\geq 1,p\leq\texttt{d})$, the coefficient of the term $\texttt{t}^{i-j+2s+1}$ is given by

[TABLE]

which is zero.

For $p=\texttt{d}+1$, the coefficient of the term $\texttt{t}^{\texttt{d}+1}$ is given by

[TABLE]

which is zero.

From the above cases, we know that (4) holds. It verifies (4.3).   $\Box$

Next, using this family of solutions given by Proposition 4.1, we give the explicit expressions of $f_{0}^{(1)}$ and $f_{0}^{(n-3)}$, then characterize the corresponding holomorphic curve $\varphi$ explicitly by Theorem 1.1.

I For $\texttt{d}+1=3$. Since $\texttt{d}+1\geq n-2$, then $n\leq 5$, so $n=5$.

Set

[TABLE]

Set ${f}_{0}^{(2)}={A}_{0}^{(2)}{V}_{0}^{(2)}$, where ${A}_{0}^{(2)}\in M(3;\mathbb{C})$, then from (4.1), we have

[TABLE]

where

[TABLE]

By calculating the eigenvalues and corresponding eigenvectors of matrix $\left({A}_{0}^{(2)}\right)^{*}{A}_{0}^{(2)}$, we get

[TABLE]

where

[TABLE]

[TABLE]

Set ${U}_{0}^{(2)}={A}_{0}^{(2)}{W}_{0}^{(2)}\left({D}_{0}^{(2)}\right)^{-1}$, then ${U}_{0}^{(2)}\in U(3)$, and ${A}_{0}^{(2)}={U}_{0}^{(2)}{D}_{0}^{(2)}\left({W}_{0}^{(2)}\right)^{T}$, thus we have

[TABLE]

Similarly, set

[TABLE]

then we get

[TABLE]

where ${U}_{0}^{(1)}\in U(2)$,

[TABLE]

In this case, from (4.1) we have

[TABLE]

From Theorem 1.1 we know

[TABLE]

Using (4.6) and (4.7), we obtain a family of linearly full holomorphic curves in $G(2,5)$ with $K=\frac{4}{3}$, up to $U(5)$, as follows,

[TABLE]

where $-1<\texttt{t}<\frac{1}{2}$.

From the above discussions, we have the following proposition.

Proposition 4.2

For $\emph{{d}}+1=3$, the solutions (4.1) give a family of linearly full non-homogeneous holomorphic curves of constant curvature $K=\frac{4}{3}$ in $G(2,5)$, that is (4.8), which is also given in [18].

II For $\texttt{d}+1=4$. Since $\texttt{d}+1\geq n-2$, then $n\leq 6$, so $n=5,6$.

II(1) For $\texttt{d}+1=4,~{}n=6$. Set

[TABLE]

Set ${f}_{0}^{(3)}={A}_{0}^{(3)}{V}_{0}^{(3)}$, where ${A}_{0}^{(3)}\in M(4;\mathbb{C})$, then from (4.1), we have

[TABLE]

where

[TABLE]

[TABLE]

By calculating the eigenvalues and corresponding eigenvectors of matrix $\left({A}_{0}^{(3)}\right)^{*}{A}_{0}^{(3)}$, we get

[TABLE]

where

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

Set ${U}_{0}^{(3)}={A}_{0}^{(3)}{W}_{0}^{(3)}\left({D}_{0}^{(3)}\right)^{-1}$, then ${U}_{0}^{(3)}\in U(3)$, and ${A}_{0}^{(3)}={U}_{0}^{(3)}{D}_{0}^{(3)}\left({W}_{0}^{(3)}\right)^{T}$, thus we have

[TABLE]

In this case, from (4.1) we have

[TABLE]

Using (4.10) and (4.7), from Theorem 1.1 we obtain a family of linearly full holomorphic curves in $G(2,6)$ with $K=1$, up to $U(6)$, as follows,

[TABLE]

where $-1<\texttt{t}<\frac{1}{3}$.

II(2) For $\texttt{d}+1=4,~{}n=5$. Set ${f}_{0}^{(2)}={A}_{0}^{(3)}{V}_{0}^{(3)}$, where ${A}_{0}^{(3)}\in M(3\times 4;\mathbb{C})$, then from (4.1), we know that $\left({A}_{0}^{(3)}\right)^{*}{A}_{0}^{(3)}$ is the same with (4.9). But in this case, ${A}_{0}^{(3)}$ is a $(3\times 4)$-matrix, then we conclude that the matrix ${D}_{0}^{(3)}$ is singular. So we get $\texttt{t}=\frac{1}{3}$ in this case. Substituting $\texttt{t}=\frac{1}{3}$ into (4.11) we obtain a holomorphic curve in $G(2,5)$ with $K=1$, up to $U(5)$, as follows,

[TABLE]

Hence we have the following conclusion.

Proposition 4.3

For $\emph{{d}}+1=4$, the solutions (4.1) give a family of linearly full non-homogeneous holomorphic curves of constant curvature $K=1$ in $G(2,6)$, that is (4.11), and a linearly full non-homogeneous holomorphic curve of constant curvature $K=1$ in $G(2,5)$, that is (4.12).

III For $\texttt{d}+1=5$. Then $5\leq n\leq 7$.

III(1) For $\texttt{d}+1=5,~{}n=7$. Set

[TABLE]

Set ${f}_{0}^{(4)}={A}_{0}^{(4)}{V}_{0}^{(4)}$, where ${A}_{0}^{(4)}\in M(5;\mathbb{C})$, then from (4.1), we have

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By calculating the eigenvalues and corresponding eigenvectors of matrix $\left({A}_{0}^{(4)}\right)^{*}{A}_{0}^{(4)}$, we get

[TABLE]

where

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

Set ${U}_{0}^{(4)}={A}_{0}^{(4)}{W}_{0}^{(4)}\left({D}_{0}^{(4)}\right)^{-1}$, then ${U}_{0}^{(4)}\in U(5)$, and ${A}_{0}^{(4)}={U}_{0}^{(4)}{D}_{0}^{(4)}\left({W}_{0}^{(4)}\right)^{T}$, thus we have

[TABLE]

In this case, from (4.1) we have

[TABLE]

Using (4.14) and (4.7), from Theorem 1.1 we obtain a family of linearly full holomorphic curves in $G(2,7)$ with $K=\frac{4}{5}$, up to $U(7)$, as follows,

[TABLE]

where $-1<\texttt{t}<\texttt{t}_{0}$ with $\texttt{t}_{0}$ being the unique zero of $1-6\texttt{t}+16\texttt{t}^{2}-26\texttt{t}^{3}+31\texttt{t}^{4}$ in the interval $(0,\frac{2}{3})$.

III(2) For $\texttt{d}+1=5,~{}n=6$. Set ${f}_{0}^{(3)}={A}_{0}^{(4)}{V}_{0}^{(4)}$, where ${A}_{0}^{(4)}\in M(4\times 5;\mathbb{C})$, then from (4.1), we know that $\left({A}_{0}^{(4)}\right)^{*}{A}_{0}^{(4)}$ is the same with (4.13). But in this case, ${A}_{0}^{(4)}$ is a $(4\times 5)$-matrix, then we concludes that the matrix ${D}_{0}^{(4)}$ is singular and it’s multiplicity of zero eigenvalues is 1. So we get $\texttt{t}=\texttt{t}_{0}$ in this case. Substituting $\texttt{t}=\texttt{t}_{0}$ into (4.15) we obtain a holomorphic curve in $G(2,6)$ with $K=\frac{4}{5}$, up to $U(6)$, as follows,

[TABLE]

III(3) For $\texttt{d}+1=5,~{}n=5$. Set ${f}_{0}^{(2)}={A}_{0}^{(4)}{V}_{0}^{(4)}$, where ${A}_{0}^{(4)}\in M(3\times 5;\mathbb{C})$, then from (4.1), we know that $\left({A}_{0}^{(4)}\right)^{*}{A}_{0}^{(4)}$ is also the same with (4.13). But in this case, ${A}_{0}^{(4)}$ is a $(3\times 5)$-matrix, then we concludes that the matrix ${D}_{0}^{(4)}$ is singular and it’s multiplicity of zero eigenvalues is $2$. But by analyzing eigenvalues of the matrix ${D}_{0}^{(4)}$ we find there doesn’t exist such t that it’s multiplicity of zero eigenvalues is $2$. Hence this case doesn’t happen.

So we get the following proposition.

Proposition 4.4

For $\emph{{d}}+1=5$, the solutions (4.1) give a family of linearly full non-homogeneous holomorphic curves of constant curvature $K=\frac{4}{5}$ in $G(2,7)$, that is (4.15), and a linearly full non-homogeneous holomorphic curve of constant curvature $K=\frac{4}{5}$ in $G(2,6)$, that is (4.16).

In summary, we present the table below to list all examples of non-homogeneous holomorphic two-spheres we obtained in $G(2,5)$, $G(2,6)$ and $G(2,7)$.

Table 1 Non-homogeneous examples in $G(2,5)$, $G(2,6)$ and $G(2,7)$.

[TABLE]

Generally, we have the following conclusion.

Proposition 4.5

For any integer $\emph{{d}}+1\geq 3$, the solutions (4.1) can give a family of non-homogeneous holomorphic curves (may be not linearly full) of constant curvature $K=\frac{4}{\emph{{d}}+1}$ in $G(2,\emph{{d}}+3)$.

Proof: Let $\psi^{(1)}_{0}:S^{2}\rightarrow\mathbb{C}P^{1}$ and $\psi^{(n-3)}_{0}:S^{2}\rightarrow\mathbb{C}P^{n-3}$ be holomorphic curves of degree $1$ and $\texttt{d}~{}(2n-5\geq\texttt{d}\geq n-3)$ respectively. Let $f_{0}^{(1)}$ and $f_{0}^{(n-3)}$ be nowhere zero holomorphic sections of $\psi^{(1)}_{0}$ and $\psi^{(n-3)}_{0}$ respectively, let $h$ be a holomorphic polynomial, satisfying (4.1). Then by Theorem 1.1 we know

[TABLE]

is a holomorphic curve of constant curvature $K=\frac{4}{\texttt{d}+1},~{}(2n-4\geq\texttt{d}+1\geq n-2)$ in $G(2,n)$. In the following we prove that the solutions (4.1) can give the holomorphic curves (may be not linearly full) of constant curvature $K=\frac{4}{\texttt{d}+1}$ in $G(2,\texttt{d}+3)$.

For $\texttt{d}+1=n-2$. Set

[TABLE]

Set ${f}_{0}^{(n-3)}={A}_{0}^{(n-3)}{V}_{0}^{(n-3)}$, where ${A}_{0}^{(n-3)}\in M(n-2;\mathbb{C})$, then from (4.1), we have

[TABLE]

where

[TABLE]

Since ${{A}_{0}^{(n-3)}}^{*}{A}_{0}^{(n-3)}$ is a Hermitian matrix, then there exist the eigenvalues and corresponding eigenvectors. Set

[TABLE]

where ${W}_{0}^{(n-3)}\in U(n-2)$ and

[TABLE]

where $\lambda_{j}^{2}(\texttt{t})$ is a non-negative rational function of parameter t. Assume for any $j$ the value of $\lambda_{j}(\texttt{t})$ is non-negative. Define a set as follows,

[TABLE]

We claim the set $\Delta$ is nonempty. In fact, since $g(\texttt{t})=\lambda_{0}(\texttt{t})\lambda_{1}(\texttt{t})\cdots\lambda_{n-3}(\texttt{t})$ is a continuous function except finite points in $(-1,1)$, and $g(0)=1>0$, then there exists an enough small neighborhood $(-\epsilon,\epsilon)$ such that for any $\texttt{t}\in(-\epsilon,\epsilon)$, $g(\texttt{t})>0$. Then for $\texttt{t}\in\Delta$, the matrix ${D}_{0}^{(n-3)}$ is invertible. Set ${U}_{0}^{(n-3)}={A}_{0}^{(n-3)}{W}_{0}^{(n-3)}\left({D}_{0}^{(n-3)}\right)^{-1}$, then ${U}_{0}^{(n-3)}\in U(n-2)$, and ${A}_{0}^{(n-3)}={U}_{0}^{(n-3)}{D}_{0}^{(n-3)}\left({W}_{0}^{(n-3)}\right)^{T}$, thus we have

[TABLE]

From (4.1) we have

[TABLE]

Then using (4.17) and (4.7), we obtain a family of linearly full holomorphic curves in $G(2,\texttt{d}+3)$ with $K=\frac{4}{\texttt{d}+1}$, up to unitary equivalence.

Let $\texttt{t}_{0}$ be a zero of $g(\texttt{t})$ in $(-1,1)$ ($\texttt{t}_{0}$ must exist). Then the matrix ${D}_{0}^{(n-3)}$ is singular at $\texttt{t}=\texttt{t}_{0}$. Assume that it’s multiplicity of zero eigenvalues is q (the value of q can be determined by the given number $n$). Then we can obtain a linearly full holomorphic curve in $G(2,\texttt{d}+3-\texttt{q})$ with $K=\frac{4}{\texttt{d}+1}$, up to unitary equivalence, which is not linearly full in $G(2,\texttt{d}+3)$.   $\Box$

Finally, combining Theorem 1.1 and the above discussions, we get the following theorem.

Theorem 4.6

Let $\varphi:S^{2}\rightarrow G(2,n)~{}(n\geq 5)$ be a linearly full holomorphic curve, satisfying that the harmonic sequence generated by $\varphi$ degenerates at position $2$. If the holomorphic curve $\varphi$ is unramified with constant curvature $K=4/\emph{{r}}$, then r is an integer and satisfies $n-2\leq\emph{{r}}\leq 2n-4$. Moreover, if there exist solutions to the polynomial equation (1.1) satisfying that the multiplicity of zero eigenvalues of the coefficients matrix of $|f_{0}^{(n-3)}|^{2}$ is q, then the holomorphic two-spheres with $K=\frac{4}{n-2+\emph{{q}}}$ can be constructed explicitly.

Remark 4.7

The above theorem verifies the first part of Conjecture 3 completely and the second part partly for linearly full non-degenerated unramified holomorphic two-spheres in $G(2,n)$, generating the harmonic sequence that degenerates at position $2$. In fact, in order to prove the second part completely in this special case, we need to construct solutions of the polynomial equation (1.1) such that q can take $0,1,\cdots,n-2$ respectively. By Proposition 4.5, the solutions (4.1) can give a family of examples for the case of $\emph{{q}}=0$. We guess that the solutions (4.1) can also give one example for the case of $\emph{{q}}=1$, and we checked this by Mathematica for the cases of low dimensions such as $G(2,5),G(2,6),G(2,7),G(2,8)$ and so on.

Acknowledgement. The first author was supported by the NSFC (No.11401481), the Research Enhancement Fund and Continuous Support Fund of Xi’an Jiaotong-Liverpool University (REF-18-01-03, RDF-SP-43). The second author was supported by the NSFC (No.11501548).