On the Finite Time Blow-up of Biharmonic Map Flow in Dimension Four

Abstract

Abstract. In this paper, we show that for certain initial values, the (extrinsic) biharmonic map flow in dimension four must blow up in finite time.

Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

“…is an approximate Diracharmonic map with boundary-data 20) which satisfies the assumptions of Theorem 1.2. In fact, h(φ n , ψ n ) = 0 and…”

“…The idea is from Qing-Tian's paper [22], which used a special case of the three circle theorem due to Simon [25] to show that the tangential energy of the sequence in the neck region decays exponentially. The second author in cooperation with H.Yin has extended this idea to some fourth order equations, see [19,20]. Let us first state the three circle theorem for harmonic functions (see [18,22,25]).…”

Finite time blowup of the n-harmonic flow on n-manifolds

“…The proof here is to use similar ideas in [20]. Due to that there are several modifications for the case of n-harmonic maps, we give a proof for completeness here.…”

“…Recently, Chen-Li [3] verified the Qing-Tian program by constructing a special target manifold N with a proper topology to show that the harmonic map flow blows up at finite time for n = 2. Later, Liu and Yin [20] successfully applied this idea to construct a proper manifold N to show that the bi-harmonic maps flow on 4-manifolds blows up at finite time.…”

“…Liu-Yin in [21] established the no-neck result of a sequence of biharmonic maps. Later, Liu-Yin [20] generalized the no-neck result to a sequence of approximate biharmonic maps. By combining the no-neck result with a construction of a proper target manifold, they introduced a concept of width of bi-harmonic maps in the covering space to show that the bi-harmonic map flow blows up in finite time.…”