Partial regularity of heat flows for harmonic maps into spheres

Let M, N be Riemannian manifolds, n = dim N {dollar}>{dollar} 2, {dollar}Msubset Rsp{lcub}k{rcub}{dollar}, and let {dollar}Hsp1{dollar}(N, M) be the space of Sobolev maps from N to M. Consider on {dollar}Hsp1{dollar}(N, M) the energy functional{dollar}{dollar}E(u) = intsb{lcub}N{rcub}vertbigtriangledown u vertsp2.{dollar}{dollar}Weakly harmonic maps are critical points of this functional with respect to target variations of the form {dollar}uspvarepsilon = Pi(u + varepsilonvarphi{dollar}), where {dollar}varphi : Nto Rsp{lcub}k{rcub}{dollar} is a smooth function and {dollar}Pi : Rsp{lcub}k{rcub}to M{dollar} is the nearest point projection. This is equivalent to the condition {dollar}Delta uperp M{dollar}. If the target manifold M is a sphere {dollar}Ssp{lcub}m-1{rcub}{dollar}, we obtain the system{dollar}{dollar}-Delta u =vertbigtriangledown u vertsp2 u.{dollar}{dollar}; The corresponding heat equation is{dollar}{dollar}{lcub}partial uoverpartial t{rcub}-Delta u =vertbigtriangledown u vertsp2 u,{dollar}{dollar}and the weak solutions of this system are called weak heat flows for harmonic maps.; In this work we consider regularity of heat flows for harmonic maps. The results and counterexamples on the partial regularity of harmonic maps show that one can not expect any regularity of heat flows in general. For this reason we should consider a more restrictive class of heat flows. For harmonic maps one such is the class of stationary harmonic maps, i.e., harmonic maps that in addition are critical points for another family of variations, domain variations, defined by {dollar}tilde uspvarepsilon{dollar} = u {dollar}circ{dollar} {dollar}{lcub}cal F{rcub}sbvarepsilon{dollar}, where {dollar}{lcub}cal F{rcub}sbvarepsilon{dollar} is a smooth family of diffemorphisms of N, {dollar}{lcub}cal F{rcub}sb0{dollar} = Id, {dollar}{lcub}cal F{rcub}sb{lcub}varepsilonvertpartial N{rcub} = Id{dollar}.; In this work a class of weak heat flows was defined by a condition that, similarly to the stationarity condition for harmonic maps, involves domain variations. Consider a family of diffeomorphims {dollar}{lcub}cal F{rcub}{dollar}: {dollar}Ntimes Rsb+to Ntimes Rsb+{dollar}, defined as {dollar}{lcub}cal F{rcub}sbvarepsilon (x,t) = (Xsbvarepsilon (x,t), Tsbvarepsilon (x,t)){dollar}, where {dollar}{lcub}cal F{rcub}sb{lcub}varepsilonvert{lcub}t=0{rcub}{rcub} = Id, Xsb{lcub}varepsilonvertpartial N{rcub} = Id, Tsbvarepsilon(x,t)ge t{dollar}. Define the family of domain variations as {dollar}tilde uspvarepsilon{dollar} = u {dollar}circ{dollar} {dollar}{lcub}cal F{rcub}sbvarepsilon{dollar}. Let us say a weak heat flow for harmonic maps satisfies the stability condition, if for each family of domain variations {dollar}tilde uspvarepsilon{dollar} the inequality{dollar}{dollar}intsbsp{lcub}0{rcub}{lcub}infty{rcub} intsb{lcub}N{rcub} {lcub}partial uoverpartial t{rcub}({lcub}partialtilde uspvarepsilonoverpartialvarepsilon{rcub})sb{lcub}vertvarepsilon=0{rcub}dxdt + (partialsbsp{lcub}varepsilon{rcub}{lcub}+{rcub} intsbsp{lcub}0{rcub}{lcub}infty{rcub} E(tilde uspvarepsilon({lcub}cdot{rcub},t))dt)sb{lcub}vertvarepsilon=0{rcub}leq 0{dollar}{dollar}holds. Note that this condition is always satisfied (with equality) for weak heat flows that belong to the Sobolev space {dollar}Hsbsp{lcub}loc{rcub}{lcub}2{rcub}(Ntimes Rsb+ ,M).{dollar} The main result of this work is the following partial regularity assertion: a heat flow into a sphere {dollar}Ssp{lcub}m-1{rcub}{dollar} satisfying the stability condition is smooth everywhere in {dollar}Ntimes Rsb+{dollar} except for a closed subset that has n-dimensional Hausdorff measure zero. (Here we consider the Hausdorff measure with respect to the parabolic metric.); The main ideas of the proof are following. The stability condition implies the monoticity inequalities, obtained for smooth harmonic he...