Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

TL;DR

This paper explores the construction and characterization of Minkowski norms induced by isoparametric foliations on spheres, analyzing their Hessian isometries, and proving related geometric properties and conjectures.

Contribution

It introduces a new class of Minkowski norms based on isoparametric foliations, provides explicit conditions for their validity, and characterizes their Hessian isometries, extending understanding of geometric structures.

Findings

Minkowski norms can be explicitly constructed from isoparametric foliations.

Hessian isometries between such norms are characterized by specific ODEs and symmetry properties.

Laugwitz Conjecture holds for Minkowski norms with flat Hessian metrics in dimensions greater than two.

Abstract

Let $M_t$ be an isoparametric foliation on the unit sphere $(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvature values. Using the spherical coordinates induced by $M_t$, we construct a Minkowski norm with the presentation $F=r\sqrt{2f(t)}$, which generalizes the notions of $(\alpha,\beta)$-norm and $(\alpha_1,\alpha_2)$-norm. Using the technique of spherical local frame, we give an exact and explicit answer for the question when $F=r\sqrt{2f(t)}$ really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry $\Phi$ between two Minkowski norms induced by $M_t$, which preserves the orientation and fixes the spherical $\xi$-coordinates. There are two ways to describe this $\Phi$, either by a system of ODEs, or by its restriction to any normal plane for $M_t$, which is then reduced to a Hessian isometry between Minkowski norms on $\mathbb{R}^2$ satisfying certain symmetry and d-properties. When $d>2$, we prove this $\Phi$ can be obtained by gluing positive scalar multiplications and compositions between the Legendre transformation and positive scalar multiplications, so it must satisfy the (d)-property for any orthogonal decomposition $\mathbb{R}^n=\mathbf{V}'+\mathbf{V}''$, i.e., for any nonzero $x=x'+x''$ and $\Phi(x)=\overline{x}=\overline{x}'+\overline{x}''$, with $x',\overline{x}'\in\mathbf{V}'$ and $x'',\overline{x}''\in\mathbf{V}''$, we have $g_x^{F_1}(x'',x)=g_{\overline{x}}^{F_2}(\overline{x}'',\overline{x}) $. As byproducts, we prove the following results. On the indicatrix $(S_F,g)$, where $F$ is a Minkowski norm induced by $M_t$ and $g$ is the Hessian metric, the foliation $N_t=S_F\cap \mathbb{R}_{>0}M_0$ is isoparametric. Laugwitz Conjecture is valid for a Minkowski norm $F$ induced by $M_t$, i.e, if its Hessian metric $g$ is flat on $\mathbb{R}^n\backslash\{0\}$ with $n>2$, then $F$ is Euclidean.