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The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.

Added: here is a recent non-trivial result about arbitrary open Riemann surfaces: https://arxiv.org/abs/2103.16702.

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911 (Zbl 0196.33801), MR0228671 (Zbl 0152.27401), MR0159935 (Zbl 0112.30801), MR0264064 (Zbl 0199.40603), MR1973182 (Zbl 1162.14310). It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838 (Zbl 1041.47502), and Kean, Moerbeke, MR0397076 (Zbl 0319.34024). Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581 (Zbl 0541.60076).

Added: here is a recent non-trivial result about arbitrary open Riemann surfaces: https://arxiv.org/abs/2103.16702.

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.

Added: here is a recent non-trivial result about arbitrary open Riemann surfaces: https://arxiv.org/abs/2103.16702.

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "two many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.