On the Steinness of strongly convex K$\ddot{a}$hler Finsler manifolds

Abstract

In this paper, we study Steinness of nonpositively or nonnegatively curved strongly convex Kähler Finsler manifolds. In particular, we prove that every strongly convex Kähler Finsler manifold with a pole and nonpositive bisectional curvature must be Stein. In addition, every complete noncompact and strongly convex Kähler Berwald manifold is Stein if it has positive flag curvature everywhere, or it has nonnegative flag curvature and everywhere positive holomorphic bisectional curvature.

Appendix

In this appendix, we shall establish Proposition 5.1 below. From this, we can obtain a series of strongly convex Kähler Berwald metrics with nonpositive or nonnegative holomorphic bisectional (resp. sectional) curvature. First of all, we introduce some notations.

Let $(M_i, \alpha _i)$ be $m_i$-dimensional Hermitian manifolds, where $i=1, 2$. We denote by $(z_i, v_i)$ the local coordinates on $T^{1, 0}M_i$ and (z, v) the local coordinates on $T^{1, 0}(M_1\times M_2)\cong T^{1, 0} M_1\oplus T^{1, 0} M_2$, where

$$\begin{aligned} z= & {} (z_1, z_2)=(z^1, \dots , z^{m_1}, z^{m_1+1}, \cdots , z^{m_1+m_2})\in M_1\times M_2, \\ v= & {} (v_1, v_2)=(v^1, \dots , v^{m_1}, v^{m_1+1}, \dots , v^{m_1+m_2})\in T_z^{1, 0}(M_1\times M_2).\end{aligned}$$

Write

$$\begin{aligned} \alpha _1=\sqrt{a_{i\bar{j}}(z_1)v^i {\bar{v}}^j}, \ \ \ \ \ \alpha _2=\sqrt{a_{\alpha \bar{\beta }}(z_2)v^{\alpha } {\bar{v}}^\beta }, \end{aligned}$$

where $1\le i, j, k, \dots \le m_1$ and $m_1+1\le \alpha , \beta , \gamma , \dots \le m_1+m_2.$ We also use the indices conventions $1\le I, J, \dots \le m_1+m_2$ in the following.

Let $f: [0, \infty )\times [0, \infty )\rightarrow [0, \infty )$ be an arbitrary $C^\infty$ function satisfying

$$\begin{aligned} f(\lambda s, \lambda t)=\lambda f(s, t) \ (\lambda >0), \ \ \text{ and }\ \ f(s, t)\ne 0\ \text{ if }\ (s, t)\ne 0. \end{aligned}$$

(6.1)

Define

$$\begin{aligned} F(z, v): =\sqrt{f([\alpha _1(z_1, v_1)]^2, [\alpha _2(z_2, v_2)]^2)}. \end{aligned}$$

(6.2)

Now we are going to find additional conditions on f such that $\sqrt{-1}\partial \bar{\partial }G$ is positive, where $G(z, v)=F^2(z, v)$, that is, F is a complex Finsler metric. Recall some notations in §2.2. By a direct calculation, we get

$$\begin{aligned} G_{i\bar{j}}=f_1a_{i\bar{j}}+f_{11}v_iv_{\bar{j}}, \ \ \ G_{i\bar{\beta }}=f_{12}v_iv_{\bar{\beta }}, \ \ \ G_{\alpha \bar{j}}=f_{12}v_{\alpha }v_{\bar{j}}, \ \ \ G_{\alpha \bar{\beta }}=f_2a_{\alpha \bar{\beta }}+f_{22}v_{\alpha }v_{\bar{\beta }}, \end{aligned}$$

(6.3)

where $f_1$ and $f_2$ mean the usual partial derivatives of f with respect to $s: =\alpha _1^2$ and $t: =\alpha _2^2$ respectively. Similar meanings are suitable for $f_{ij} (i, j=1, 2)$. By an elementary argument, one can show that $(G_{I\bar{J}})$ is positive if and only if f(s, t) satisfies

$$\begin{aligned} f_1>0, \ \ f_2>0, \ \ f_1+sf_{11}>0, \ \ f_2+tf_{22}>0, \ \ f_1f_2-ff_{12}>0. \end{aligned}$$

(6.4)

In this case, we have

$$\begin{aligned} \det (G_{I\bar{J}})=f_1^{m_1-1}f_2^{m_2-1}(f_1f_2-ff_{12})\det (a_{i\bar{j}})\det (a_{\alpha \bar{\beta }}) \end{aligned}$$

and the components of the inverse matrix $(G^{I\bar{J}})$ of $(G_{I\bar{J}})$ given by

$$\begin{aligned} G^{i\bar{j}}=f_1^{-1}a^{i\bar{j}}+k_1v^i\bar{v}^j, \ \ \ G^{i\bar{\beta }}=k_2v^i\bar{v}^\beta , \ \ G^{\alpha \bar{j}}=k_3v^\alpha \bar{v}^j, \ \ G^{\alpha \bar{\beta }}=f_2^{-1}a^{\alpha \bar{\beta }}+k_4v^\alpha \bar{v}^\beta ,\nonumber \\ \end{aligned}$$

(6.5)

where $k_2=k_3=-\frac{1}{\Delta }f_{12}$ and

$$\begin{aligned} k_1=-\frac{1}{\Delta }\Big \{f_1^{-1}[f_2f_{11}+t(f_{11}f_{22}-f_{12}^2)]\Big \}, \ \ k_4=-\frac{1}{\Delta }\Big \{f_2^{-1}[f_1f_{22}+s(f_{11}f_{22}-f_{12}^2)]\Big \}, \end{aligned}$$

here $\Delta :=(f_1+sf_{11})(f_2+tf_{22})-stf_{12}^2$. Thus F is a complex Finsler metric if and only if f satisfies (6.1)–6.4).

Further we shall find the conditions that f satisfies such that F is strongly convex complex Finsler metric. For this, we need to find suitable f such that

$$\begin{aligned} \text{ Re }\left( G_{I\bar{J}}w^I\bar{w}^J+G_{IJ}w^Iw^J\right) >0 \end{aligned}$$

for any nonzero vector $w=(w_1, w_2)=(w^1, w^\alpha )\in T_z^{1, 0}(M_1\times M_2)$ by Proposition 2.6.1 in [1]. By (6.3) and a direct calculation, we have

$$\begin{aligned}&5G_{I\bar{J}}w^I\bar{w}^J= G_{i\bar{j}}w^i\bar{w}^j+G_{i\bar{\beta }}w^i\bar{w}^\beta +G_{\alpha \bar{j}}w^\alpha \bar{w}^j\nonumber \\&+G_{\alpha \bar{\beta }}w^\alpha \bar{w}^j \nonumber \\= & {} f_1\Vert w_1\Vert ^2_{\alpha _1}+f_2\Vert w_2\Vert _{\alpha _2}^2+f_{11}\alpha _1(v_1, w_1)\alpha _1(w_1, v_1)+f_{22}\alpha _2(v_2, w_2)\alpha _2(w_2, v_2) \nonumber \\&+f_{12}\left[ \alpha _1(w_1, v_1)\alpha _2(v_2, w_2)+\alpha _1(v_1, w_1)\alpha _2(w_2, v_2)\right] .\end{aligned}$$

(6.6)

and

$$\begin{aligned} G_{IJ}w^Iw^J= & {} G_{ij}w^i w^j+G_{i\beta }w^iw^\beta +G_{\alpha j}w^\alpha w^j+G_{\alpha \beta }w^\alpha w^j \nonumber \\= & {} f_{11}[\alpha _1(w_1, v_1)]^2+f_{22}[\alpha _2(w_2, v_2)]^2 +2f_{12}\alpha _1(w_1, v_1)\alpha _2(w_2, v_2). \nonumber \\ \end{aligned}$$

(6.7)

Adding (6.7) to (6.6) yields

$$\begin{aligned}&\text{ Re }\Big (G_{I\bar{J}}w^I\bar{w}^J+G_{IJ}w^Iw^J\Big )=f_1\Vert w_1\Vert ^2_{\alpha _1}+f_2\Vert w_2\Vert _{\alpha _2}^2\nonumber \\&\quad +\text{ Re }\Big \{f_{11}\left[ \alpha _1(v_1, w_1)+\alpha _1(w_1, v_1)\right] ^2\nonumber \\&\quad +f_{22}\left[ \alpha _2(v_2, w_2)+\alpha _2(w_2, v_2)\right] ^2\nonumber \\&\quad +\left[ \alpha _1(v_1, w_1)+\alpha _1(w_1, v_1)\right] \left[ f_{12}\alpha _2(w_2, v_2)-f_{11}\alpha _1(v_1, w_1)\right] \nonumber \\&\quad +\left[ \alpha _2(v_2, w_2)+\alpha _2(w_2, v_2)\right] \left[ f_{12}\alpha _1(w_1, v_1)-f_{22}\alpha _2(w_2, v_2)\right] \Big \}. \end{aligned}$$

(6.8)

Assume that

$$\begin{aligned} sf_{11}+tf_{12}=0, \ \ \ sf_{12}+tf_{22}=0, \ \ \ f_{12}\le 0.\end{aligned}$$

(6.9)

Remember that $s=[\alpha _1(v_1, v_1)]^2$ and $t=[\alpha _2(v_2, v_2)]^2$. By using $v_i+w_i$ instead of $v_i (i=1, 2)$ in (6.9), we have

$$\begin{aligned} f_{11}\text{ Re }(\alpha _1(v_1, w_1))+f_{12}\text{ Re }(\alpha _2(v_2, w_2))=0, \ \ \ f_{22}\text{ Re }(\alpha _2(v_2, w_w))+f_{12}\text{ Re }(\alpha _1(v_1, w_1))=0. \end{aligned}$$

From this and (6.8)–(6.9), one obtains

$$\begin{aligned}\text{ Re }\Big (G_{I\bar{J}}w^I\bar{w}^J+G_{IJ}w^Iw^J\Big )= & {} f_1\Vert w_1\Vert ^2_{\alpha _1}+f_2\Vert w_2\Vert _{\alpha _2}^2 \nonumber \\&-4f_{12}\Big [\frac{\alpha _2}{\alpha _1}\text{ Re }(\alpha _1(v_1, w_1))- \frac{\alpha _1}{\alpha _2}\text{ Re }(\alpha _2(v_2, w_2))\Big ]^2>0.\end{aligned}$$

Hence F is a strongly convex complex Finsler metric if f further satisfies (6.9).

Next we check that F is a Kähler Berwald metric. For this, we need calculate the connection coefficients of Chern Finsler connection. Observe that

$$\begin{aligned} G_{\bar{j}; k}= & {} f_1v_{\bar{j};k}+f_{11}(\alpha _1^2)_{;k}v_{\bar{j}}, \ \ G_{\bar{\beta }; k}=f_{12}(\alpha _1^2)_{; k}v_{\bar{\beta }}, \end{aligned}$$

(6.10)

$$\begin{aligned} G_{\bar{j}; \alpha }= & {} f_{12}(\alpha _2^2)_{;\alpha }v_{\bar{j}}, \ \ G_{\bar{\beta }, \alpha }=f_2v_{\bar{\beta }; \alpha }+f_{22}(\alpha _2^2)_{;\alpha }v_{\bar{\beta }}. \end{aligned}$$

(6.11)

From these, (6.5) and $N^I_{\ J}=G^{I\bar{K}}G_{\bar{K}; J}$, the nonlinear connection coefficients are given by

$$\begin{aligned} N^i_{\ k}=a^{i\bar{j}}v_{\bar{j};k}, \ \ N^{\alpha }_{\ k}=N^i_{\ \alpha }=0, \ \ N^{\beta }_{\ \alpha }=a^{\beta \bar{\gamma }}v_{\bar{\gamma }; \alpha }.\end{aligned}$$

(6.12)

Thus, the connection coefficients of the Chern Finsler connection are given by

$$\begin{aligned} \Gamma _{j; k}^i=a^{i\bar{l}}(a_{j\bar{l}})_{; k}, \ \ \Gamma _{\alpha ; k}^i=\Gamma _{j; k}^\beta =\Gamma _{\alpha ; k}^\beta =\Gamma _{j; \alpha }^i=\Gamma _{\alpha ; \beta }^i=\Gamma _{j; \alpha }^\beta =0, \ \ \Gamma _{\alpha ; \gamma }^\beta =a^{\beta \bar{\sigma }}(a_{\alpha \bar{\sigma }})_{; \gamma }, \end{aligned}$$

which show that F is a Berwald metric and it is a Kähler Finsler metric if and only if both $\alpha _1$ and $\alpha _2$ are Kählerian.

Since the Kähler metrics $\alpha _i (i=1, 2)$ are strongly convex, the associated real Riemannian metrics $\alpha _i^\circ (x_i, u_i)$ are given by $\alpha ^\circ (x_i, u_i)=\alpha (z_i, v_i)$. We identify $(z_i, v_i)=(z_i^1, z_i^2, \cdots , z_i^{m_i}, v_i^1, v_i^2,$ $\cdots v_i^{m_i})$ with $(x_i, u_i)=(x_i^1, x_i^2, \cdots , x_i^{2m_1+2m_2};$ $u_i^1, u_i^2, \dots , u_i^{2m_1+2m_2})$ in the sense that

$$\begin{aligned}&z_1^{j_1}=x_1^{j_1}+\sqrt{-1}x_1^{m_1+j_1}, \ \ \ \ z_2^{j_2}=x_2^{2m_1+j_2}+\sqrt{-1}x_2^{2m_1+m_2+j_2}, \\&v_1^{j_1}=u_1^{j_1}+\sqrt{-1}u_1^{m_1+j_1}, \ \ \ \ v_2^{j_2}=u_2^{2m_1+j_2}+\sqrt{-1}u_2^{2m_1+m_2+j_2}\end{aligned}$$

for $1\le j_i\le m_i$ and $i=1, 2$. It is easy to check that $v_i=v_i^{j_i}\frac{\partial }{\partial z_i^{j_i}}=\frac{1}{2}(u_i-\sqrt{-1}Ju_i)=(u_i)_{\circ }$. Thus, if F is a strongly convex Kähler Finsler metric, then the associated real Finsler metric $F^\circ$ is given by

$$\begin{aligned} F^\circ (x, u)=\sqrt{f([\alpha ^\circ _1(x_1, u_1)]^2, [\alpha ^\circ _2(x_2, u_2)]^2)}, \end{aligned}$$

where $x=(x_1, x_2)\in M_1\times M_2$ and $u=(u_1, u_2)\in T_x(M_1\times M_2)$, and $F^\circ$ is a real Berwald metric on $M_1\times M_2$ by Theorem 1.2 in [28]. By the same arguments as in Example 6.3.4 in [5], F (equivalently, $F^\circ$) has nonnegative or nonpositive flag curvature if both $\alpha _1$ and $\alpha _2$ have nonnegative or nonpositive sectional curvature.

Finally we calculate the holomorphic bisectional curvature of F by (2.21). From (6.10)–(6.12), one obtains

$$\begin{aligned} \delta _{\bar{i}}(G_{; j})= & {} \frac{\partial }{\partial \bar{z}^i}(G_{; j})-\overline{N^k_{\ i}}G_{\bar{k}; j}-\overline{ N^{\beta }_{\ i}}G_{\bar{\beta }; j} \nonumber \\= & {} f_{11}(\alpha _1^2)_{; \bar{i}}(\alpha _1^2)_{;j}+f_1(\alpha _1^2)_{; \bar{i} j}-a^{l\bar{k}}v_{l;\bar{i}}\left[ f_1v_{\bar{k}; j}+f_{11}(\alpha _1^2)_{; j}v_{\bar{k}}\right] \nonumber \\= & {} f_1(\alpha _1^2)_{; \bar{i} j}-f_1a^{l\bar{k}}v_{l;\bar{i}}v_{\bar{k}; j}=f_1\left[ \frac{\partial ^2a_{p\bar{q}}}{\partial \bar{z}^i\partial z^j}-a^{l\bar{k}}\frac{\partial a_{p\bar{k}}}{\partial z^j}\frac{\partial a_{l\bar{q}}}{\partial \bar{z}^i}\right] v^p\bar{v}^q. \end{aligned}$$

(6.13)

In the same way, we have

$$\begin{aligned} \delta _{\bar{i}}(G_{; \beta })=\delta _{\bar{\beta }}(G_{; i})=0, \ \ \ \delta _{\bar{\beta }}(G_{; \alpha })=f_2\left[ \frac{\partial ^2a_{\gamma \bar{\sigma }}}{\partial \bar{z}^\alpha \partial z^\beta }-a^{\delta \bar{\tau }}\frac{\partial a_{\delta \bar{\sigma }}}{\partial z^\beta }\frac{\partial a_{\gamma \bar{\tau }}}{\partial \bar{z}^\alpha }\right] v^\gamma \bar{v}^\sigma . \end{aligned}$$

(6.14)

Plugging (6.13)–(6.14) in (2.21) yields

$$\begin{aligned} K_F(v, w)=\frac{1}{G(v)\Vert w^{\mathcal H}\Vert ^2_{h_v}} \left\{ f_1 K_{\alpha _1}(v_1, w_1)\Vert v_1\Vert ^2_{\alpha _1}\Vert w_1\Vert ^2_{\alpha _1}+f_2K_{\alpha _2}(v_2, w_2)\Vert v_2\Vert ^2_{\alpha _2}\Vert w_2\Vert ^2_{\alpha _2}\right\} ,\nonumber \\ \end{aligned}$$

(6.15)

where $K_{\alpha _1}(v_1, w_1)$ and $K_{\alpha _2}(v_2, w_2)$ are respectively the bisectional curvatures of the Kähler metrics $\alpha _1$ and $\alpha _2$. In particular, the holomorphic sectional curvature $K_F(v)$ of F is

$$\begin{aligned} K_F(v)=\frac{1}{F^4(v)} \left\{ f_1 K_{\alpha _1}(v_1)\Vert v_1\Vert ^4_{\alpha _1}+f_2K_{\alpha _2}(v_2)\Vert v_2\Vert ^4_{\alpha _2}\right\} .\end{aligned}$$

(6.16)

Thus, F has nonpositive or nonnegative holomorphic bisectional (resp. sectional) curvature if both $\alpha _1$ and $\alpha _2$ have nonpositive or nonnegative holomorphic bisectional (resp. sectional) curvatures respectively.

Summing up, we have proved the following

Proposition 6.1

Let $(M_i, \alpha _i)$ be $m_i$-dimensional Hermitian manifolds, where $i=1, 2$, and F be defined by (6.2), in which $f: [0, \infty )\times [0, \infty )\rightarrow [0, \infty )$ is a smooth function satisfying (6.1). If f further satisfies (6.4) and (6.9), then F is a strongly convex complex Finsler metric on $M_1\times M_2$. Further,

• (1) F is a Berwald metric;

• (2) F is a Kähler Finsler metric if and only if both $\alpha _1$ and $\alpha _2$ are Kähler metrics;

• (3) F has nonnegative or nonpositive (resp. positive or negative) flag curvature if both $\alpha _1$ and $\alpha _2$ have nonnegative or nonpositive (resp. positive or negative) sectional curvature;

• (4) The holomorphic bisectional (resp. sectional) curvature $K_F$ is given by (6.15) (resp. (6.16)). In particular, F has nonpositive or nonnegative holomorphic bisectional (resp. sectional) curvature if both $\alpha _1$ and $\alpha _2$ have nonpositive or nonnegative bisectional (resp. sectional) curvatures respectively.

A typical example of f is given by

$$\begin{aligned} f_{\lambda , k}(s, t)=s+t+\lambda \root k \of {s^k+t^k}, \end{aligned}$$

where $\lambda$ is a positive number and $k>1$. It is easy to check that $f_{\lambda , k}$ satisfy (6.1), (6.4) and (6.9). The corresponding complex Finsler metric is exactly (1.1). In [25], Xia-Zhong studied strongly convexity and Kähler property of this special class of complex Finsler metrics. The associated real Finsler metrics are given by $F=\sqrt{f_{\lambda , k}(\alpha _1, \alpha _2)}$, which were introduced and studied by Z. I. Szabó ([22]), where $\alpha _i$ are Riemannian metrics on $M_i (i=1, 2)$. For more general extensions of the Szabó’s metric, we refer to Examples 4.3.1 and 6.3.4 in [5].