An umbilical point on a non-real-analytic surface
The index of an isolated umbilical point on a surface in three-dimensional Euclidean space is studied. The main attention is paid to the following index conjecture: the index of an isolated umbilical point is not more than one. This conjecture is motivated by studying constant mean curvature surfaces whose isolated umbilical points are always of negative index. Moreover, it is proved by the author in [\textit{N. Ando}, Tohoku Math. J., II. Ser. 54, No. 2, 163--177 (2002; Zbl 1022.53006)] that if the point (0,0,0) is an isolated umbilical point on the graph \(z=g(x,y)\) of a homogeneous polynomial of degree \(k\) then its index is an element of \(\{ 1-k/2, 2-k/2,\dots, 1+[k/2]-k/2 \}\). Similar analysis is produced for the graph of an arbitrary analytic function. The present paper is devoted to the study of smooth non-zero functions with zero coefficients of all orders in Taylor's expansion like \(\exp(-1/(x^2+y^2))\). The author starts from a smooth function \(F(x,y)\) defined in a neighborhood of \((0,0)\) in such a way that all coefficients of order less than \(l\) in Taylor's expansion vanish at \((0,0)\). It is also supposed that \(F(x,y)\) is positive in a punctured neighborhood of \((0,0)\) and that for each \(c>0\) there exists a punctured neighborhood of \((0,0)\) on which \(| \operatorname{grad} F | / F > c\) holds. Then it is proven that \((0,0)\) is an isolated umbilical point with index 1 on the graph of \(E_n (aF(x,y))\) for any natural \(n\) and positive real \(a\). Here \(E_0(b)=b\), \(E_1(b)= \exp(-1/b)\), and \(E_k(b)=E_1(E_{k-1}(b))\) is defined for each \(k\) inductively. Some particular examples are analyzed, a subtler version of the formulated statement is also proven.
scientific article; zbMATH DE number 1991027