Rigidity of a family of spherical conical metrics

Zhu, Xuwen

doi:10.48550/arxiv.1902.02024

Cited by 3 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, the set of cone angle data β for which there exists a solution metric with 2 in the spectrum is unbounded in (R + ) k . Furthermore, if this spectral condition holds, then there are explicit examples which exhibit that the local deformation theory is obstructed, see [47]. Our main result is that even if 2 does lie in the spectrum, there is an unobstructed deformation space if we allow for more drastic deformations which permit the individual points p j to 'splinter' into a collection of conic points with smaller cone angles.…”

Section: Introductionmentioning

confidence: 89%

Conical metrics on Riemann surfaces, II: spherical metrics

Mazzeo,

Zhu

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than 2π. Deformations are obstructed precisely when the number 2 lies in the spectrum of the Friedrichs extension of the Laplacian. 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].

show abstract

Section: Introductionmentioning

confidence: 89%

Conical metrics on Riemann surfaces, II: spherical metrics

Mazzeo,

Zhu

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…And the situation is even worse in the spherical case that the uniqueness result does not hold in general ([13, Theorem 1.5]). As a consequence, the spherical case of Question 2.1 is still widely open although many mathematicians had attacked or have been investigating it by using various methods and obtained a good understanding of the question ( [28,6,9,13,15,16,17,18,19,20,21,32,33,34,35,39,44,45,47,48]).…”

Section: Cone Spherical Metricsmentioning

confidence: 99%

Irreducible cone spherical metrics and stable extensions of two line bundles

Li,

Song,

2020

Preprint

View full text Add to dashboard Cite

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U(1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus gX ≥ 1 a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in 2πZ>1, which is generically injective in the algebro-geometric sense as gX ≥ 2. As an application, we prove the following two results about irreducible metrics:• as gX ≥ 2 and d is even and greater than 12gX − 7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension ≥ 2(d + 3 − 3gX ) in Sym d (X); • as gX ≥ 1, for almost every effective divisor D of degree odd and greater than 2gX −2 on X, there exist finitely many cone spherical metrics representing D.

show abstract

“…There is ongoing work of the first author and his collaborators [CLSX] on the case when the genus of Σ is positive. In [Zhu19a] the local rigidity of one family of such metrics was shown by using synthetic geometry which exemplifies the constraints on supp D.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of reducible conical metrics

Xu,

Zhu

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace-Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue, and a complete classification of the eigenspace dimension depending on the monodromy. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.

show abstract

Rigidity of a family of spherical conical metrics

Cited by 3 publications

References 20 publications

Conical metrics on Riemann surfaces, II: spherical metrics

Conical metrics on Riemann surfaces, II: spherical metrics

Irreducible cone spherical metrics and stable extensions of two line bundles

Spectral properties of reducible conical metrics

Contact Info

Product

Resources

About