Falling birth rates are a huge & growing problem.
… for the rich.
Fertility rate is important for some things, but not total world population, i.e., "too many people".
My Country Has (F) Fertility Rate (%) How many more I must feed next year
1000 2.1 21
2000 1.8 36
WRT total population projections, the two largest countries have started levelling out in population, which has a large effect.
link
But world population is still growing. We'll still hit 11B, barring a major change, just later than they thought in the decade where they thought we would have flying cars 20 years ago. (And we actually do have them, we just don't allow their use. It's not a tech problem, it's a "people already suck driving in two dimensions problem".)
WRT immortality, one of the major causes of aging is thought to be improperly recombined DNA. Not just damaging cells through various means that are avoidable to certain degrees, but having the replacement cells either not happen (fail to reproduce) or reproduce in the diseased/damaged state. Theoretically, we have the technology to fix this. Practically, nobody is rich enough to have this done for themselves. Increasing the availability of these technologies works toward immortality, but the thesis of this part was the cellular health problem is not the only cause, so fixing it doesn't fix every cause of aging.
Second, there is only a finite total of matter-energy in the universe. In order to keep a human body in the complex form we are talking about (above the heat-death of the universe), we have to expend energy, thus moving things more toward the total heat-death state. Basically, against infinite time, with finite energy available, we would run out of gas.
That second one is a way far-off limit beyond the idea of pushing human age from around 80-90 to 800-900. To bring that idea closer to home, the human brain is also finite. There is only a certain amount of memories/process/identity that it can hold. We already lose some things and gain new ones. And there are second- and higher-order historyless state effects that can maintain continuity (for example, we can have no memories of an early childhood event that affected us, but because it did, we changed our behaviour, so our new brain patterns are influenced by the changed behaviour, but not the actual event), but even that has a limit. We can, of course, futz with the size of our brain (theoretically), but you still just end up with a larger finite limit. So WRT infinite time, even if we don't physically die, our identity "fades out" and we become someone else.
Slow it down? Sure. Stop it? Nope.
begins to think about terrain for Team Yankee.
