Umbilic-index conjecture for a quartic surface with a<b

Let $a<b$ and let $\epsilon$ satisfy

$$\epsilon>\frac{(5a+b)(b-a)}{18}.$$

Consider the surface

$$ax^2+\epsilon x^4+ay^2+\epsilon y^4+bz^2=0.$$

An umbilic point is a point where the principal curvatures coincide, and its index is the index of the principal direction field around that point. Quartic-surface umbilic-index conjecture. Independent of $a$ and $b$, this surface has exactly two umbilics with index $1$, eight with index $-1/2$, and eight with index $1/2$. Although the parser marks this candidate as open, the supplied evidence says that the authors solved the open question; the database therefore records it as solved.