"define A as ...''

When I use define as a verb intransitive, are the following four usages all correct?

1. I shall define a certain object as to satisfy the following conditions.

2. I shall define a certain object as satisfying the following conditions.

3. I shall define a certain object as satisfies the following conditions.

4. I shall define a certain object to satisfy the following conditions.

There are two possible things you could mean here, and I don't know which one you want.

1. I define a certain object to be anything that satisfies the following conditions.
2. I propose the following definition for a certain object, and I have chosen the definition in order for the object to satisfy the following conditions.

For (1), in a mathematical paper I would just use the wording above—"to be anything that satisfies ...". There are lots of other ways you could say that, but none of your four suggestions work (and notice that in this case, there is no reason to use shall or will; you're defining it right now and not in the future).

For (2), I would use

We will define a certain object so as to satisfy the following conditions.

Your fourth choice above is grammatical, but mathematicians generally use so as to rather than just to; using just to would sound strange in a mathematical paper.

Finally, in scientific journal papers, you generally use "we" rather than "I", even for single-author papers.