Non-Contraction of Heat Flow on Minkowski Spaces

By Shin-ichi Ohta, Karl‐Theodor Sturm. 2010

67 citations recorded by the source corpus.

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/84fd075bb3577c07e1a630c51c22ac3cdff4e07e

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “5) is specific of Riemannian geometry, because OhtaSturm proved in [28] that no exponential contraction holds in (Rd, ‖ · ‖,Ld) if the norm ‖ · ‖ does not come from a scalar product.”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “Even if φ is geodesically λ-convex (the natural extension of convexity in metric spaces), however, the evolution does not enjoy all the nice semigroup properties of the Hilbertian case: it can be easily checked even in finite-dimensional spaces, endowedwith a non-quadratic norm [74].”