论文标题
Minkowski Norm和Hessian等轴测图由单位球上的等轴叶诱导
Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere
论文作者
论文摘要
令$ m_t $为单位球上的等式叶叶$（s^{n-1}（1），g^{\ mathrm {st}}）$，带有$ d $ princional曲率值。使用$ M_T $引起的球形坐标，我们构建了一个Minkowski Norm，其中包括$ f = r \ sqrt {2f（t）} $，它概括了$（α，β）$ - norm和$（α_1，α_2）$ - norm的概念。使用球形本地帧的技术，当$ f = r \ sqrt {2f（t）} $真正定义了Minkowski Norm时，我们给出了一个精确而明确的答案。使用类似的技术，我们研究了由$ M_T $引起的两个Minkowski规范之间的Hessian等轴测$φ$，该规范可以保留方向并确定球形$ξ$ - 坐标。有两种方法可以通过odes系统或以$ m_t $为$ m_t $的任何普通平面来描述这种$φ$，然后将其简化为$ \ mathbb {r}^2 $在满足某些对称和dproperties上的Minkowski Norms之间的Hessian等轴测图。当$ d> 2 $时，我们可以证明可以通过在legendre转换和积极的标量乘法之间粘合积极的标量乘以和组成来获得此$φ$，因此它必须满足任何正交分解$ \ \ \ mathbb {r}^n = r}^n = \ Mathbf { nonZero $ x = x'+x''$和$φ（x）= \ edimelline {x} = \ edimelline {x}'+\+\ edline {x}'$，带有$ x'，\ overline {x}}'\ in \ mathbf in \ mathbf {v}'$和$ x''，\ edline {x}''\ in \ mathbf {v}''$，我们有$ g_x^{f_1}（x''，x''，x）= g _ {\ overline {x}}}}}}^{f_2} {f_2}（f_2}（f_2}）作为副产品，我们证明了以下结果。在Indicatrix $（s_f，g）$上，其中$ f $是$ m_t $和$ g $引起的minkowski norm，是hessian公制，叶子$ n_t = s_f \ cap \ cap \ cap \ mathbb {r} _ {> 0} m_0 $ _0 $ isoparametric。 Laugwitz的猜想对于由$ M_T $引起的Minkowski Norm $ f $有效，即，如果其Hessian Metric $ G $在$ \ Mathbb {r}^n \ backslash \ {0 \ {0 \ {0 \} $上均固定，则使用$ n> 2 $，那么$ f $ is euclidean。
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 $(α,β)$-norm and $(α_1,α_2)$-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 $Φ$ between two Minkowski norms induced by $M_t$, which preserves the orientation and fixes the spherical $ξ$-coordinates. There are two ways to describe this $Φ$, 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 $Φ$ 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 $Φ(x)=\overline{x}=\overline{x}'+\overline{x}''$, 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.
