Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation (rho) over bar of a p-adic field K together with a cocharacter mu. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K is totally ramified with n = 3 or mu, is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to (rho) over bar of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.