Mass transport generated by a flow of Gauss maps
| dc.creator | Bogachev, Vladimir I. | |
| dc.creator | Kolesnikov, Alexander V. | |
| dc.date | 2008-03-10 | |
| dc.date | 2008-05-12 | |
| dc.date.accessioned | 2026-07-07T09:37:57Z | |
| dc.date.available | 2026-07-07T09:37:57Z | |
| dc.description | Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ν= μ\circ T^{-1}$ and $T = ϕ\cdot {\rm n}$, where $ϕ\colon A \to [0,r]$ is a continuous potential with convex sub-level sets and ${\rm n}$ is the Gauss map of the corresponding level sets of $ϕ$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $ϕ$ the level sets of $ϕ$ are driven by the Gauss curvature flow $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x) \cdot {\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface. | |
| dc.description | 15 pages; minor changes | |
| dc.identifier | https://arxiv.org/abs/0803.1436 | |
| dc.identifier | http://arxiv.org/abs/0803.1436 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160636 | |
| dc.subject | Differential Geometry | |
| dc.subject | 49Q20; 35J60 | |
| dc.title | Mass transport generated by a flow of Gauss maps | |
| dc.type | text |