Co-axial monodromy

Conical metrics on Riemann surfaces, II: spherical metrics

“…This bubbling set is in fact the same as the set of critical angles appearing in the variational approach in [30]. Furthermore, when M = S 2 , this bubbling set strictly contains the set of cone angle vectors associated to spherical cone metrics with coaxial monodromy, see [14].…”

“…Their main result, a tour-de-force in classical geometry, is that any point in the interior of MP k is the vector of cone angles for at least one spherical cone metric on S 2 ; they are not able, however, to specify the locations of the conic points p j , and do not address whether solutions exist when β is on the boundary of this region. After partial results of Dey [12] and Kapovich [25], a complete answer was obtained by Eremenko [14] on which β ∈ ∂MP k are possible.…”

Rigidity of a family of spherical conical metrics

“…and showed the existence when the strict inequality holds; the boundary cases have been considered in [10,16,12]. The same two authors [23] also showed that when M = S 2 , the condition χ(M, β) > 0 is sufficient for existence.…”

“…One approach is through complex analysis, see [13,14,15,28]. For metrics with special monodromy which is of particular interest of this note, see the works of Xu and collaborators [8,24,25] and Eremenko [12]. We also mention here the variational approach by Malchiodi and collaborators [1,2,3,4] and the Leray-Schauder degree counting method by Chen and Lin [5,6].…”

Cone spherical metrics and stable vector bundles

“…However, this natural necessary condition of deg D > 2g X − 2 is not sufficient for the existence of cone spherical metrics ( [33]). In this case Question 1.1 has been open over 20 years although many mathematicians had attacked or have been investigating it by using various methods and obtained a good understanding of the question ( [34,35,11,5,13,14,15,16,12,7,25,26,10,30,23]). Then we list some of the known results which are relevant to this manuscript.…”