Abstract
The p-curve of maximal slopes (p-CMS) for a function defined on a metric space plays a similar role to the gradient flow in the Euclidean setting, decreasing the function along it in the steepest descent direction. Minimizing-movement (MM) schemes is one of the methods for constructing p-CMS. In the case p=2, Mayer proved that MM is (i) unique and (ii) p-CMS on nonpositively curved metric spaces. Ambrosio-Gigli-Savar’e studied the case p=2 and showed that the above result holds for a wider class of metric spaces including positively curved spaces. However, the case of p \neq 2 is still widely open. In this talk, after an introduction to related notions, we provide the first example satisfying (i) for a fixed p \gt 2 and then discuss (ii) for the example.
Motivation
For an initial point find a condition such that
- the p-MM starting from exists uniquely
- It is p-CMS
The main theorem is: Unter einer Bedingung gilt 1. und der p-MM ist trivial.
Betrachte eine Funktion auf einem metrischen Raum. Betrachte den metrischen Gradienten der Funktion. (Die größte Steigung bei Annäherung an einen punkt )
Geodesics can be defined metrically. Is there a well-defined derivative you get by following a geodesic?