http://www.gsmarena.com/specs_of_samsung_galaxy_note_iii_leak_57inch_display_confirmed-news-6464.php
As additional rumors seem to be settling at 5.7", I have to say I went from initially being mildly disappointing it wasn't slightly larger (say 5.9"), but I'm coming back around.
As someone who loves the Note 2 and think it's the best phone I've ever used, I'm not going to get a phone that's any heavier. It's totally fine for normal use, but holding it up for a while does get a little tiring, much less when you're lying in bed and holding up over your face. So for enough benefit, I'd be OK with a minor weight increase, but I'd really prefer something a bit lighter.
Secondly, I still think I would have liked a bit more screen increase as you get used to the 5.5" screen so quickly, even when using DPI changing programs to shrink program headers (like the browser header/address bar) to have even more usable screen real estate. So I would have preferred a 5.9" phone with smaller bezels and a slightly larger overall size. Slightly be the key word as the Note 2 is close to the limit IMO of what a guy can wear in pants comfortably (and it still took some getting used to). So 5.7" with smaller bezels and the same (if not even slightly smaller) overall size is still pretty good.
I have to say I love the look of the Sony Xperia Z Ultra, but I think they went too far in terms of a mainstream phablet and ended up on the tablet size of the equation.