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The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces
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abstract
In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space with non-maximal diameter. The same analysis is performed for functions with a Laplacian in $L^{N+\varepsilon}$. As a consequence we show that on smooth manifolds there is no a priori estimate on the modulus of continuity of the gradient of harmonic functions which depends only on lower bounds of the sectional curvature. In the same way we show that there is no a priori Calder\'on-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.
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Lipschitz continuity of harmonic maps between ${\rm RCD}(K,N)$ spaces and ${\rm CAT}(\kappa)$ spaces
Energy-minimizing harmonic maps from RCD(K,N) domains into small balls in CAT(κ) spaces are locally Lipschitz, completing the singular Bochner–Eells–Sampson picture.
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