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

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.