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We discuss $(K,N)$-convexity and gradient flows for $(K,N)$-convex functionals on metric spaces, in the case of real $K$ and negative $N$. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of $(K,N)$-convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness., Comment: 23 pages

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor $\text{ric}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories., Comment: 44 pages. We corrected a mistake in Theorem 5.3, present in the previous version, and added Theorem 1.8

The aim of this paper is to show the existence of a canonical distance $\mathsf d'$ defined on a locally Minkowski metric measure space $(\mathsf X,\mathsf d,\mathfrak m)$ such that: i) $\mathsf d'$ is equivalent to $\mathsf d$, ii) $(\mathsf X, \mathsf d', \mathfrak m)$ is infinitesimally Hilbertian. This new regularity assumption on $(\mathsf X, \mathsf d,\mathfrak m)$ essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the Finsler manifolds, and it is actually strictly larger. The required distance $\mathsf d'$ will be the intrinsic distance $\mathsf d_\mathsf{KS}$ associated to the so-called Korevaar-Schoen energy, which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space $(\mathsf X, \mathsf d_\mathsf{KS}, \mathfrak m)$ is in fact the Korevaar-Schoen energy.

In this paper we investigate two important properties of metric measure spaces satisfying the reduced curvature-dimension condition for negative values of the dimension parameter: the existence of a transport map between two suitable marginals and the so-called local-to-global property.

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we introduce an appropriate setting to introduce the CD(K, N)-condition for $N < 0$, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we introduce the distance $d_{\mathsf{iKRW}}$, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K, N)-condition with $N < 0$ is converging with respect to the distance $d_{\mathsf{iKRW}}$ to some metric measure space, then this limit structure is still a CD(K, N) space.

We study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author. Our main results are 1) The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivatives, 2) a quite complete set of results relating the property $f\in W^{2,2}(\X\times\Y)$ on one side with that of $f(\cdot,y)\in W^{2,2}(\X)$ and $f(x,\cdot)\in W^{2,2}(\Y)$ for a.e.\ $y,x$ respectively on the other. Here $\X,\Y$ are $\RCD$ spaces so that second order Sobolev spaces are well defined. \end{itemize} These results are in turn based upon the study of Sobolev regularity, and of the underlying notion of differential, for a map with values in a Hilbert module: we mainly apply this notion to the map $x\mapsto\d_\sy f(x,\cdot)$ in order to build, under the appropriate regularity requirements, its differential $\d_\sx\d_\sy f$.

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-\'Emery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior., Comment: 68 pages; comments welcome!

We give a quick and direct proof of the strong maximum principle on finite dimensional $RCD$ spaces based on the Laplacian comparison of the squared distance.