I will describe some parts of Troynov’s work on conical surface. For details, check his paper. For simplicity, we consider a closed Riemann surface with a real divisor . A conformal metric on is said to represent the divisor if is smooth Riemannian metric on and near each , there exists a neighborhood of and coordinate function and such that and

where . is called the angle of the conical singularity. We will always assume .

For example equipped with the metric is isometric to an Euclidean cone of total angle .

Suppose now is a closed Riemann surface with conical metric . **Assume that the Gauss curvature extends on as a H\”{o}lder continuous function.**

Suppose is a conformal change of the conical metric. Then necessarily we have . In order to prescribe the Gauss curvature , we need to solve the equation

.

Any reasonable solution of (0) will satisfy

where follows from the Gauss-Bonnet formula for conical surface. We will use variational method to attack it. To that end, define to be Sobolev space of functions with

and functionals

It is easy to verify that the minimizer of the following variation problem will be a weak solution of (0)

As one can see, there are two immediate questions we need to answer,

(a)

(b) minimizer exists

The first question follows from the Trudinger inequality. Namely, define . Then

**Lemma:** For any fixed , there exist constant such that,

for any and and .

**Lemma:** Suppose and . Let . Then for any fixed , there exists constants such that

for any and . Here .

From this key lemma, one can derive that is lower bounded on provided and . To do that, choose and such that , then it follows from the above lemma that

For the second question, we need to prove the embedding is compact(actually it is true for any for any ). Note that this is true if is some smooth metric, however is conical one. Therefore, we want to compare with of some smooth metric.

Suppose is equipped with smooth metric such that . Here is smooth and positive outside the support of and near . If we use the denote the gradient of with respect to , then and

This is only true in two dimension. Now and have inner product

From the above analysis, we need to examine the difference of and . It follows from the singular behavior of that

for any and some . Then for any ,

Then . On the contrary, we need the following inequality

**Lemma: **For conical metric

Then we have . Therefore is compact for . Furthermore, after some effort, one can prove

is also compact for .

Remark:

[1] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc. 324 (1991) 793–821. doi:10.1090/S0002-9947-1991-1005085-9.