4 added 192 characters in body; deleted 31 characters in body

Let me give a proof using the deformation theory of holomorphic maps developed by Horikawa [Journal Math. Soc. Japan 25]. This can be seen as a purely analytic proof in the spirit of Kodaira's "deformations of complex structures".

Set $X:=\Sigma_{g_1}$ and $Y:=\Sigma_{g_2}$, and fix a non constant holomorphic mapping $f \colon X \to Y$.
Now

Let us denote by $\textrm{Mor}(X, Y)$ the space of holomorphic maps from $X$ to $Y$. It is an analytic space, whose tangent space at the point $[f]$ coincides with the space of first-order deformations of $f$ is given by f$, namely$H^0(X, f^*T_Y)$. Since$Y$is of genus$g \geq 2$, we have$\deg T_Y <0$, so$H^0(X, f^*T_Y)=0$. This means that the morphism$f$is rigid, in other words there are only finitely many first-order deformations of$f$up to composition with automorphisms of$X$. But it is well known that$|\textrm{Aut}(X)| \leq 84(g(X)-1)$, so there are only finitely many first - order deformations of$f$at all. This shows that every component of the space$\textrm{Mor}(X, Y)$is a point. In general, it can happen that$\textrm{Mor}(X,Y)$has countably many components; in this case, however, it has only finitely many of them, since the possible degrees of$f$are bounded from above by the Riemann-Hurwitz formula. This implies that there are finitely many choices for$f$. If you do not like deformation theory, there exists actually a purely analytic (and completely different) proof of a definitely strong result, the so called Kobayashi - Ochiai Theorem: Theorem. Let$X$be a Moishezon space and$Y$a compact complex spece of general type. Then the set of meromorphic maps from$X$to$Y$is finite. From the proof, that is a combination of techniques and uses in an essential way the Schwarz lemma, I refer you to the original Kobayashi-Ochiai paper [Meromorphic mappings onto compact complex spaces of general type, Inventiones Math. 31 (1975)] 3 added 168 characters in body Let me give a proof using the deformation theory of holomorphic maps developed by Horikawa [Journal Math. Soc. Japan 25]. This can be seen as a purely analytic proof in the spirit of Kodaira Kodaira's "deformations of complex structures". Set$X:=\Sigma_{g_1}$and$Y:=\Sigma_{g_2}$, and fix a non constant holomorphic mapping$f \colon X \to Y$. Now the space of first-order deformations of$f$is given by$H^0(X, f^*T_Y)$. Since$Y$is of genus$g \geq 2$, we have$\deg T_Y <0$, so$H^0(X, f^*T_Y)=0$. This means that the morphism$f$is rigid, in other words there are only finitely many first-order deformations of$f$up to composition with automorphisms of$X$. But it is well known that$|\textrm{Aut}(X)| \leq 84(g(X)-1)$, so there are only finitely many first - order deformations of$f$at all. This shows that every component of the space$\textrm{Mor}(X, Y)$is a point. But In general, it can happen that$\textrm{Mor}(X, Y)$is a quasi-projective variety\textrm{Mor}(X,Y)$ has countably many components; in this case, so however, it has only finitely many componentsof them, since the possible degrees of $f$ are bounded from above by the Riemann-Hurwitz formula. This implies that there are finitely many choices for $f$.

If you do not like deformation theory, there exists actually a purely analytic (and completely different) proof of a definitely strong result, the so called Kobayashi - Ochiai Theorem:

Theorem. Let $X$ be a Moishezon space and $Y$ a compact complex spece of general type. Then the set of meromorphic maps from $X$ to $Y$ is finite.

From the proof, that is a combination of techniques and uses in an essential way the Schwarz lemma, I refer you to the original Kobayashi-Ochiai paper [Meromorphic mappings onto compact complex spaces of general type, Inventiones MeathMath. 31 (1975)]

2 added 14 characters in body

Let me give a proof using the deformation theory of holomorphic maps developed by Horikawa. This can be seen as a purely analytic proof in the spirit of Kodaira "deformations of complex structures".

Set $X:=\Sigma_{g_1}$ and $Y:=\Sigma_{g_2}$, and fix a non constant holomorphic mapping $f \colon X \to Y$.
Now the space of first-order deformations of $f$ is given by $H^0(X, f^*T_Y)$. Since $Y$ is of degree genus $g \geq 2$, we have $\deg T_Y <0$, so $H^0(X, f^*T_Y)=0$.

This means that the morphism $f$ is rigid, in other words there are only finitely many first-order deformations of $f$ up to composition with automorphisms of $X$.

But it is well known that $|\textrm{Aut}(X)| \leq 84(g(X)-1)|$84(g(X)-1)$, so there are only finitely many first - order deformations of$f$at all. This shows that every component of the space$\textrm{Mor}(X, Y)$is a point. But$Mor(X, \textrm{Mor}(X, Y)$is a quasi-projective variety, so it has only finitely many components. This implies that there are finitely many choices for$f$. If you do not like deformation theory, there exists actually a purely analytic (and completely different) proof of a definitely strong result, the so called Kobayashi - Ochiai Theorem: Theorem Let$X$be a Moishezon space and$Y$a compact complex spece of general type. Then the set of meromorphic maps from$X$to$Y\$ is finite.

From the proof, that is a combination of techniques and uses in an essential way the Schwarz lemma, I refer you to the original Kobayashi-Ochiai paper [Meromorphic mappings onto compact complex spaces of general type, Inventiones Meath. 31 (1975)]

1