  There is increasing experimental evidence that the superconducting energy gap
$\Delta_0$ in the underdoped cuprates is independent of doping concentration
$x$ while the superfluid density is linear in $x$. We show that under these
conditions, thermal excitation of the quasiparticles is very effective in
destroying the superconducting state, so that $T_c$ is proportional to
$x\Delta_0$ and part of the gap structure remains in the normal state. We then
estimate $H_{c2}$ and predict it to be proportional to $x^2$. We also discuss
to what extent the assumptions that go into the quasiparticle description can
be derived in the U(1) and SU(2) formulations of the t-J model.
