First of all, these cables do not provide internet to the entire world– what they do is provide connections across water. The vast majority of cabling is on land, either above ground or buried beneath it. Undersea cables are undersea because not every place is connected by land conveniently, so you just lay a cable underwater instead of underground, since there is no ground.
The cables are owned by various private companies, generally in telecommunications. Even Facebook and Google own some cables though now and are investing more into them. Their owners maintain them (generally through subcontractors).
We are also laying more and new cables all the time, there is a significant demand for this both to increase capacity in existing areas, such as US to Europe, or to add new capacity, such as to places in Africa
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