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

Part of: Global differential geometry Local differential geometry General convexity

Published online by Cambridge University Press:  03 June 2021

Ming Xu  and
Vladimir S. Matveev
Ming Xu
Affiliation:
School of Mathematical Sciences, Capital Normal University, Beijing100048, P.R. China e-mail: mgmgmgxu@163.com
Vladimir S. Matveev*
Affiliation:
Institut für Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany
*
e-mail: vladimir.matveev@uni-jena.de
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Abstract

For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$ . 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. Furthermore, 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.

Keywords

Minkowski normHessian isometryLaugwitz ConjectureLandsberg Unicorn ConjectureLegendre transformation

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C30: Homogeneous manifolds 53B40: Finsler spaces and generalizations (areal metrics)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 5 , October 2022 , pp. 1486 - 1516
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author is supported by Beijing Natural Science Foundation (No. Z180004), NSFC (No. 11771331 and No. 11821101), and Capacity Building for Sci-Tech Innovation—Fundamental Scientific Research Funds (No. KM201910028021). The second author thanks DFG for partial support via projects MA 2565/4 and MA 2565/6.

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