论文标题
Hessian等级1的分类1在r^{n+1}中相关的Hypersurfaces H^n在尺寸n = 2、3、4中
Classification of Hessian Rank 1 Affinely Homogeneous Hypersurfaces H^n in R^{n+1} in Dimensions n = 2, 3, 4
论文作者
论文摘要
在上一本回忆录2202.03030中，我们表明，在每个维度$ n \ geq 5 $中，都存在（出乎意料的）同质性高hyperface $ h^n \ subset \ subset \ mathbb {r}^{r}^{n+1} $具有常数级别1的Hessian and and contance an \ geqslant 1} $）。目前的工作致力于确定所有非产物常数Hessian等级1均匀均匀的超曲面$ h^n \ subset \ subset \ mathbb {r}^{n+1} $ in Dimensions $ n = 2，3，4 $，$ n = 1，2 $。有了完整的详细信息，在情况下$ n = 2 $，我们说明了可以称为“等价的功率系列方法”的主要功能。要点是仅在原点捕获不变性，创建分支并无限地拟合计算。在尺寸$ n = 3 $中，我们找到了一个同质模型：\ [u \，= \，\ frac {1} {3 \，z^2} \ big \ {\ big \ big（1-2 \，y+y+y^2-2-2-2-2-2 \，x z z z \ big）^big） \ big）\ big \}，\]奇异性$ \ frac {1} {3z^2} $是虚幻的。在尺寸$ n = 4 $中，没有达到封闭表格，我们找到了两个简单的均匀型号，不同的是$ \ pm $ sign。
In a previous memoir 2202.03030, we showed that in every dimension $n \geq 5$, there exists (unexpectedly) no affinely homogeneous hypersurface $H^n \subset \mathbb{R}^{n+1}$ having Hessian of constant rank 1 (and not being affinely equivalent to a product with $\mathbb{R}^{m \geqslant 1}$). The present work is devoted to determine all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimensions $n = 2, 3, 4$, the cases $n = 1, 2$ being known. With complete details in the case $n = 2$, we illustrate the main features of what can be termed the "Power Series Method of Equivalence". The gist is to capture invariants at the origin only, to create branches, and to infinitesimalize calculations. In dimension $n = 3$, we find a single homogeneous model: \[ u \,=\, \frac{1}{3\,z^2} \Big\{ \big( 1-2\,y+y^2-2\,xz \big)^{3/2} - (1-y)\, \big( 1-2\,y+y^2-3\,xz \big) \Big\}, \] the singularity $\frac{1}{3z^2}$ being illusory. In dimension $n = 4$, without reaching closed forms, we find two simply homogeneous models, differing by some $\pm$ sign.
