Naoya Ando

An Isolated Umbilical Point of the Graph of a Homogeneous Polynomial

Suppose that the origin o of R3 is an isolated umbilical point of the graph of a homogeneous polynomial in two real variables of degree k≥3. Then we see that the index of o is an element of the set 1−k/2+i[k/2]i=0. Moreover, we see that each element of 1−k/2+i[k/2]i=0 may be the index of o on the graph of a suitable homogeneous polynomial of degree k.

C1-umbilics with arbitrarily high indices

In this paper, the existence of C^1-umbilics with arbitrarily high indices is shown. This implies that more than C^1-regularity is required to prove Loewners conjecture.