This paper first proposes a new approximate scheme to construct a harmonic heat flow u between Q and SD⊂ RD + 1 with Q= (0 , ∞) × Bd and positive integers d and D: We agree that harmonic heat flow u means a solution of ∂u∂t-▵u-|∇u|2u=0.Its scheme is crucially given by ∂uλ∂t-▵uλ+λ1-κ(|uλ|2-1)uλ=0,where the unknown mapping uλ is from Q to RD + 1 with positive number λ and κ(t) = arctan (t) / π(0 ≤ t). The benefit to introduce a time-dependent parameter λ1 - κ is readily to see ∫Qλ1-κ(|uλ|2-1)2dtdx≤Clogλfor some positive constant C independent of λ. Next, making the best of it, we prove that a passing to the limits λ↗ ∞ (modulo sub-sequence of λ) brings the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled energy, (iii) a reverse Poincaré inequality. These inequalities (i), (ii) and (iii) improve the estimates on its singular set of a harmonic heat flow by Chen and Struwe (Math Z 201(1):83–103, 1989), i.e. I show that a singular set of the new harmonic heat flows into spheres has at most finite (d- ϵ) -dimensional Hausdorff measure with respect to the parabolic metric whereupon ϵ is a small positive number. We finally prove that if the harmonic heat flows is a constant at the boundary, then uλ(t) strongly converges to the constant t↗ ∞ in H1 , 2(Bd; SD + 1). We call this the parabolic constancy theorem. We restrict ourselves a harmonic heat flow from the unit ball into a sphere to avoid confusion of notation. But it is readily seen that our results can be extended to it between compact Riemannian manifolds using a distance function combined with Nash’s imbedding theorem.
ASJC Scopus subject areas
- Geometry and Topology