Sobolev Mappings Between RCD Spaces and Applications to Harmonic Maps: A Heat Kernel Approach
Sobolev Mappings Between RCD Spaces and Applications to Harmonic Maps: A Heat Kernel Approach
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DOI：
10.1007/s12220-023-01334-6
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发表时间：
2021-05
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通讯作者：
Shouhei Honda;Y. Sire
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作者：
Shouhei Honda;Y. Sire
本文研究了从有限维RCD空间到有限维非塌缩紧RCD空间的Sobolev映射。证明了若映象f（X）在弱意义上光滑（满足此条件的条件是前推测度关于Hausdorff测度绝对连续，或满足此条件的条件是在弱意义上光滑），则黎曼度量f的拉回定义为X上的一张量，最小弱上梯度f f可以用表示，且它与Lipschitz时X中几乎处处点的局部斜率一致.特别是，最后一个陈述给出了度量测度空间上Lipschitz函数的Cheeger可微性定理的非线性模拟。此外，这些结果使我们能够定义能量关闭。还证明了该能量与Korevaar-Schoen能量一致，直到乘以一个维数正常数。为了实现这一点，我们使用了通过热核嵌入的平滑，这是由Ambrosio-Portegies-Tewodrose和第一作者建立的（Ambrosio等人，J Funct Anal 280：108968，2021）。此外，我们改进了的正则性，这对得到上述结果起到了关键作用。作为应用，我们证明了在中的n维标准单位球面的等距映射，f是极小等距浸入当且仅当在中的n维标准单位球面上的n维标准用热核嵌入代替Nash嵌入来研究可能奇异空间之间映射的能量，这种方法即使对于闭黎曼流形也是新的。
In this paper, we investigate a Sobolev mapffrom a finite dimensional RCD spaceto a finite dimensional non-collapsed compact RCD space. It is proved that if the imagef(X) is smooth in a weak sense (which is satisfied if the pushforward measureis absolutely continuous with respect to the Hausdorff measure, or ifis smooth in a weak sense), then the pull-backof the Riemannian metricofis well defined as an-tensor onX, the minimal weak upper gradientoffcan be written by using, and it coincides with the local slopefor-almost everywhere points inXwhenfis Lipschitz. In particular, the last statement gives a nonlinear analogue of Cheeger’s differentiability theorem for Lipschitz functions on metric measure spaces. Moreover,these results allow us to define the energy off. It is also proved that the energy coincides with the Korevaar-Schoen energy up to by multiplying a dimensional positive constant. In order to achieve this, we use a smoothing ofvia the heat kernel embedding, which is established by Ambrosio-Portegies-Tewodrose and the first-named author (Ambrosio et al. in J Funct Anal 280:108968, 2021). Moreover,we improve the regularity of, which plays a key role to get the above results. As an application, we show thatis isometric to theN-dimensional standard unit sphere inandfis a minimal isometric immersion if and only ifis non-collapsed up to a multiplication of a constant to, andfis an eigenmap whose eigenvalues coincide with the essential dimension of, which gives a positive answer to a remaining problem from a previous work by the first-named author. This approach, using the heat kernel embedding instead of using Nash’s one, to the study of energies of maps between possibly singular spaces seems new even for closed Riemannian manifolds.