Every point is critical

With respect to whom are you critical?

“…In [1] the authors proved together with Imre Bárány, in the framework of Alexandrov surfaces, the following three results. (See [6] for a variational proof of the first one, valid for finite dimensional Riemannian manifolds.)…”

“…A finite number of cycles were defined in [1] to prove Lemma 1, by joining points in T y corresponding to the vertices in C cp 3 (y) by line segments or arcs in T y . Next we indicate a geometrical interpretation (i.e., an equivalent definition) for some of those cycles, useful for our purpose.…”

“…Let C 1 , ..., C n be all these cycles. If 0 ∈ ∪ n j=1 C j then, for some x ∈ C cp 3 (y), there are two segments of diametrally opposite tangent directions in T y , see [1].…”

“…If 0 ∈ ∪ n j=1 C j consider, as in [1], the winding number w(C j ) = w(0, C j ) of every cycle C j with respect to 0. We have…”

“…We proved in [1], together with Imre Bárány, that the set Q −1 y of all points with respect to which y is critical is never empty. It is also shown in [1] that Q −1 y is single-valued for all y ∈ S if and only if the genus of S is 0. We continue this study in the following.…”