Unusual superconducting state of underdoped cuprates

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.