Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with $SO(k)\times SO(n-k)$-symmetry

TL;DR

This paper proves longstanding conjectures by analyzing isometries of Hessian metrics derived from Minkowski norms with specific symmetries, advancing understanding in Finsler geometry and Minkowski space structures.

Contribution

It proves Laugwitz's conjecture from 1965 and the Landsberg Unicorn Conjecture for certain symmetric Minkowski norms, providing a comprehensive classification of isometries under $SO(k) imes SO(n-k)$ symmetry.

Findings

Proved Laugwitz conjecture for Minkowski norms with $SO(k) imes SO(n-k)$ symmetry.

Described all isometries between Hessian metrics of such norms.

Established Landsberg Unicorn Conjecture for Finsler manifolds with symmetric Minkowski norms.

Abstract

For a smooth strongly convex Minkowski norm $F:\mathbb{R}^n \to \mathbb{R}_{\geq0}$, we study isometries of the Hessian metric corresponding to the function $E=\tfrac12F^2$. Under the additional assumption that $F$ is invariant with respect to the standard action of $SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Further, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$-symmetry