  We consider closed immersed surfaces in R^3 evolving by the geometric
triharmonic heat flow. Using local energy estimates, we prove interior
estimates and a positive absolute lower bound on the lifespan of solutions
depending solely on the local concentration of curvature of the initial
immersion in L^2. We further use an {\epsilon}-regularity type result to prove
a gap lemma for stationary solutions. Using a monotonicity argument, we then
prove that a blowup of the flow approaching a singular time is asymptotic to a
non-umbilic embedded stationary surface. This allows us to conclude that any
solution with initial L^2-norm of the tracefree curvature tensor smaller than
an absolute positive constant converges exponentially fast to a round sphere
with radius equal to the cube root of 3V_0/4{\pi}, where V_0 denotes the signed
enclosed volume of the initial data.
