Many people believe that greatly increased population levels are the inevitable result of increased human longevity. Separately, many people believe that overpopulation is presently happening, and will lead to catastrophe in the near future. I and many others have noted in the past that overpopulation doesn't exist today, that many multiples of today's population could exist on this one planet with a high standard of living using no more than today's technology, and that population models show that even radical life extension doesn't greatly increase the population size. Supporting any level of population is simply another engineering challenge, one well within our capabilities today for any plausible near future population size, never mind using the improved technologies of tomorrow.
What people point to today as overpopulation is more accurately labeled as poverty caused and maintained by bad governance: resources squandered, persistent war, kleptocracies, regulation, serfdom, and so on. It is not a matter of counting heads, but of greed and inhumanity. So little of the light and noise put out on the topic of overpopulation seems particularly rational to me, but here is a generally sensible piece from a pro-longevity author:
By far the most predominant criticism made against indefinite longevity is overpopulation. It is the first "potential problem" that comes to mind. But fortunately it seems that halting the global mortality rate would not cause an immediate drastic increase in global population; in fact, if the mortality rate dropped to zero tomorrow then the doubling rate for the global population would only be increased by a factor of 1.75, which is smaller than the population growth rate during the post-WWII baby-boom.
Finding innovative solutions to new and old problems is what humanity does. Thus while overpopulation is the most prominent and most credible criticism against continually-increasing lifespans, and the one that needs to be planned-for the most (because it will eventually happen, but it will lead to sustainability, resource and living space problems only if we do nothing about it), it is in no way insoluble, nor particularly pressing in terms of the time available to plan and implement solutions to shrinking living-space and resource-space (i.e. the space occupied by resources such as food, energy-production, workplaces, etc.). We have a host of potential solutions today, ones we can use to increase available living space without regulating the global birthrate, and decades following the achievement of indefinite lifespans to consider the advantages and disadvantages of the various possible solutions, to develop them and to implement them.
So then: wherefore from here? Overpopulation is still the most prominent criticism raised against indefinite longevity, and if combated, it could lead to an increase in public support for the longevity movement. You might think that the widespread concern with overpopulation due to increasing longevity won't really matter, if they turn out to be wrong, and overpopulation isn't so insoluble a problem as one is inclined to first presume. But this misses a crucial point: that the time it takes to achieve longevity is determined by and large by how widespread and strongly society and the members constituting it desire and demand it. If we can convince people today that overpopulation isn't an insoluble problem, then continually-increasing longevity might happen much sooner than otherwise. At the cost of 100,000 deaths due to age-correlated causes per day, I think hastening the arrival of indefinite longevity therapies by even a modest amount is somewhat imperative. Hastening its arrival by one month will save 3 million lives, and achieving it one year sooner than otherwise will save an astounding 36.5 million real, human lives.