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My native language is German, where we have a phrase "Im Fol­gen­den" to start a sentence.

In scientific writing (e.g., in a paper) can I start a sentence with

In the following, we refer to three examples that are ...

Is it better to use

In this section, we refer to three examples that are...

In other words, is it true that "In the following" cannot be used by itself, there must be another word, like "In the following chapter..."

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It is fine to use "the following" by itself. It takes the role of a noun in such usage. If you say "the following section", then its role is an adjective. Both are fine.

Here is a reference.

Definition:
the following,
   that which comes immediately after, as pages, lines, etc.:
  See the following for a list of exceptions.

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