Connected sums of unstabilized Heegaard splittings are unstabilized
  Let M_1 and M_2 be closed, orientable 3-manifolds. Let H_i denote a Heegaard surface in M_i. We prove that if H_1 # H_2 comes from stabilizing a lower genus splitting of M_1 # M_2 then either H_1 or H_2 comes from stabilizing a lower genus splitting. If H_i and G_i are non-isotopic Heegaard surfaces in M_i, and H_i is unstabilized, then we show H_1 # H_2 is not isotopic to G_1 # G_2 in M_1 # M_2. The former result answers a question of C. Gordon (Problem 3.91 from Kirby's problem list). 
David Bachman