 
Summary: THE RADIUS OF VANISHING BUBBLES IN EQUIVARIANT
HARMONIC MAP FLOW FROM D 2 TO S 2
S.B. ANGENENT # , J. HULSHOF + , AND H. MATANO #
Abstract. We derive an upper bound for the radius R(t) of a vanishing bubble in a family of
equivariant maps F t : D 2
# S 2 which evolve by the Harmonic Map Flow. The selfsimilar ``type 1''
radius would be R(t) = C # T  t. We prove that R(t) = o(T  t).
1. Introduction. Let N n
# R k be a smooth submanifold. The Dirichlet integral
or energy of a map F from the unit disc D 2
# R 2 into N is defined to be
D[F ] = 1
2 # D 2
#F (x) 2 .
Extremals of this energy with prescribed boundary values F  #D 2 are called harmonic
maps. Eells and Sampson [4] introduced the gradient flow for D[F ], now called the
harmonic map flow, in which a family of maps F t : D # N evolves according to the
nonlinear heat equation
#F
#t
