从RCD空间到CAT(0)空间的调和映射热流的Lipschitz正则性
Lipschitz regularity of harmonic map heat flows from $\mathrm{RCD}$ spaces into $\mathrm{CAT}(0)$ spaces
AI总结：
该研究证明从有限维RCD(K,N)空间到完备CAT(0)目标的Korevaar-Schoen能量的EVI梯度流在正时间的局部Lipschitz正则性，结合度量Hamilton-Jacobi论证与椭圆接触估计完成证明。
AI中文摘要：
到完备CAT(0)空间的调和映射热流即使目标空间奇异也可采用变分构造，但当源空间也非光滑时其正则性认知不足。我们证明：在初始数据满足有界像假设下，从有限维RCD(K,N)空间到完备CAT(0)目标的Korevaar-Schoen能量的EVI梯度流，在正时间具有局部Lipschitz正则性，所得代表在空间和时间上联合局部Lipschitz。证明结合了度量Hamilton-Jacobi论证与应用于RCD源合适时间切片的椭圆接触估计。
英文摘要：
We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells--Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space--time contact selection required by the Hamilton--Jacobi argument.
