Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components

Conical metrics on Riemann surfaces, II: spherical metrics

“…Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37,38].…”

“…Forthcoming work of Bin Xu and the second author shows that coaxial metrics always have eigenvalue 2. On the other hand, the work of Mondello and Panov [38], and Eremenko, Gabrielov and Tarasov [21], indicate that there also exist non-coaxial metrics with eigenvalue 2.…”

“…For cone angles lying in (0, 2π), Troyanov [42] discovered an auxiliary set of linear inequalities on the β j which are necessary and sufficient for existence; later, Luo and Tian [28] proved uniqueness of the solution in this angle regime. When M = S 2 , existence was recently proved by Mondello and Panov [38] for any β with χ(M, β) > 0, at the expense of not being able to specify the conformal class on M , see also [1]. When M = S 2 , the same two authors [37] gave necessary conditions on β for existence, again in the form of a set of linear inequalities, and proved existence in the interior of this region.…”

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. In either cases, one is unable to specify the marked conformal class, i.e., the location of the points p.…”

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.…”