A common concern about life extension is that we would rapidly overrun the planet, and without management, this could happen. However, there a few factors in our favour.
In many countries, fertility rates are already dropping below that required to maintain a population, and without immigration the number of people living there would be falling.
Even with biological immortality, if people were limited to one child each, then there would be a finite maximum population to deal with. And biological immortality does not mean fictional immortality - people will still die in accidents, conflict or by choice, meaning the global population would eventually start falling again.
We will have a lot of time to come up with a solution - the population won't double overnight, it might take a 100 years or so, and it's impossible to imagine what technologies may have been developed by then to help.
According to David Wood of London Futurists if death stopped tomorrow it
would take a hundred years to double the planet's
See also: Space Colonization
Overpopulation Blog Posts
Falling birth rates - a matter of concern in Russia
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Crude birth rate of England and Wales hits a new low
BBC - 01-Aug-2019
Total fertility rate is lower than many other years except 1977 and 1999 to 2002Read more...
China child restrictions leading to fall in population
Futurism - 18-Jan-2019
More countries are having to plan for increasingly aged populationsRead more...