riemann surfaces – Recovering a family of rational functions from branch points

Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:Xrightarrow Y$ of degree $d$ with branch points in $B$ are in a correspondence with homomorphisms $rho:pi_1(Y-B)rightarrow S_d$ with transitive image modulo conjugation by elements of the permutation group $S_d$. Writing a formula for $f$ from the knowledge of $Bsubset Y$ and $rho$ is often hard, e.g. the task of recovering a Belyi map from its dessin where $|B|=3$. I am interested in the case of $X=Y=Bbb{CP}^1$, and some points from $B$ moving in the Riemann sphere. Here is an example:

  • Consider rational functions $f:Bbb{CP}^1rightarrowBbb{CP}^1$ of degree $3$ with four simple critical points that have $1,omega,bar{omega}$ among their critical values $left(omega={rm{e}}^{frac{2pi{rm{i}}}{3}}right)$, thus $B={1,omega,bar{omega},beta}$ with $beta$ varying in a punctured sphere. To fix an element in the isomorphism class, we can pre-compose $f$ with a suitable Möbius transformation so that $1$, $bar{omega}$ and $omega$ are the critical points lying above $1$, $omega$ and $bar{omega}$ respectively: $f(1)=1, f(bar{omega})=omega, f(omega)=bar{omega}$. A normal form for such functions is
    $$
    left{f_alpha(z):=frac{alpha z^3+3z^2+2alpha}{2z^3+3alpha z+1}right}_alpha.
    $$

    A simple computation shows that the fourth critical point is $alpha^2$, and hence $beta=beta_alpha=:f_alpha(alpha^2)=frac{alpha^4+2alpha}{2alpha^3+1}$. Here is my question:

Why $beta$ is not a degree one function of $alpha?$ Shouldn’t the knowledge of the branch locus and the monodromy determine $f_alpha(z)$ in the normalized form above? I presume the monodromy does not change because there are only finitely many possibilities for it and this is a continuous family.

To monodromy of $f_alpha$ is a homomorphism
$$
rho_alpha:pi_1left(Bbb{CP}^1-{1,omega,bar{omega},beta_alpha}right)rightarrow S_3
$$

where small loops around $1,omega,bar{omega},beta_alpha$ generate the fundamental group, and are mapped to transpositions in $S_3$ whose product is identity and are not all distinct. So I guess my question is how can such a discrete object vary with $alpha$; and if it doesn’t, why the assignment $alphamapstobeta(alpha)$ is not injective. The degree of this assignment is four, and there are also four conjugacy classes of homomorphisms
$rho:langlesigma_1,sigma_2,sigma_3,sigma_4midsigma_1sigma_2sigma_3sigma_4=mathbf{1}ranglerightarrow S_3$ with ${rm{Im}}(rho)$ being a transitive subgroup of $S_3$ generated by transpositions $rho(sigma_i)$:
$$
sigma_1mapsto (1,2),sigma_2mapsto (1,2),sigma_3mapsto (1,3), sigma_4mapsto (1,3);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,2), sigma_4mapsto (2,3);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,3), sigma_4mapsto (1,2);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (2,3), sigma_4mapsto (1,3).
$$

Is it accidental that the degree of $alphamapstobeta(alpha)$ is the same as the number of possibilities for the monodromy representations compatible with our ramification structure?

real analysis – Main utility of the monotonicity formula for generalized surfaces

A basic answer is that “the monotonicity formula places constraints on the shape of a minimal surface” e.g., you cannot have a lot of area concentrated in a ball if then later there is a (relatively) small amount of area. This, along with the convex hull property, already tells you a lot about the possible shape of a minimal surface.

However, I think you are looking for a concrete application. Here is one sample application that is somewhat different from what I have described above:

Suppose that $Sigma_i,Sigma subset B_2(0) subset mathbb{R}^3$ are smooth embedded minimal surfaces and for any $f in C^0_c(B_2(0))$ it holds that
$$
tag{*} int_{Sigma_i} f|_{Sigma_i} to int_Sigma f|_Sigma
$$

as $itoinfty$ (this is a very weak notion of convergence of surfaces). Then, if $x_i in Sigma_icap B_1(0)$ has $x_ito x$ then $xinSigma$.

Proof: Assume $xnot in Sigma$. Then, there is $B_{3epsilon}(x)$ disjoint from $Sigma$. Choose a bump function $f=1$ on $B_{2epsilon}(x)$ and $f$ vanishing outside of $B_{3epsilon}(x)$. Now, by (*) we have that
$$
int_{Sigma_i} f|_{Sigma_i}to 0.
$$

Thus, for $i$ large, $B_{epsilon}(x_i)subset B_{2epsilon}(x)$, so we have arranged that
$$
|Sigma_icap B_epsilon(x_i)|leq int_{Sigma_i} f|_{Sigma_i} to 0.
$$

On the other hand the monotonicity formula implies that
$$
frac{|Sigma_icap B_epsilon(x_i)|}{pi epsilon^2} geq lim_{rto0}frac{|Sigma_icap B_r(x_i)|}{pi r^2} = 1
$$

since $Sigma_i$ is smooth (and thus nearly flat on small scales). This is a contradiction.


Note that if the $Sigma_i$ are not minimal (i.e., if they don’t satisfy the monotonicity formula), it is easy to find a counterexample to the above result by taking a flat disk $Sigma$ and forming $Sigma_i$ by gluing on a lot of “tentacles” which all have small area (and thus small contribution to the integral as in (*)). Thus, this is a nontrivial result.

(Note that I have not attempted to prove the most general version of this result, if you want you might see Simon’s GMT book or other sources on minimal surfaces.)

There are many related applications of the monotonicity formula. A simple yet powerful one consists of White’s proof of the Allard regularity theorem. See Section 1.1 here https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n3-p07.pdf or Theorem 7.8 here http://web.stanford.edu/~ochodosh/MinSurfNotes.pdf).

Contraction of loops on algebraic surfaces.

Suppose ${Bbb C}$ be a complex number field and $S$ be a projective smooth surface over ${Bbb C}$. We consider a finite etale Galois covering $pi colon T to S$ of degree $d$. We consider the curve $D$ on $S$ such that

  1. $D = underset{i = 1,cdots,n}{cup} C_i$, where each $C_i$ is a smooth irreducible curve and $C_i cdot C_j = 1$ for $i not= j$.

  2. $D$ forms a closed loop, i.e. the dual graph of $D$ is a $n$-gon.

  3. Each irreducible component $C_i$ of $D$ splits into $d$ irreducible components in $T$.

  4. $D$ can be contracted to a point. I.e., $exists f colon S to U$ such that $U$ is a proper algebraic surface over ${Bbb C}$ and $f(D) = {mathrm{a~point}}$.

Q. For a given etale covering $pi$, is it true that sufficiently many $D$‘s satisfying above 4 conditions exist in quantity? Or is there any measure to size up the number of above $D$‘s?

complex analysis – Question about the definition of holomorphic maps between riemann surfaces

The definition of holomorphic maps between Riemann surfaces confuses me a little. When we say a (continuous) map between two Riemann surfaces $f: X to Y$ is holomorphic, we mean that for any pair of charts $varphi: U_1subseteq X to V_1$ and $psi: U_2 subseteq Y to V_2$, the composition $psi circ fcirc varphi^{-1}: varphi(U_1 cap f^{-1}(U_2))to psi(f(U_1)cap U_2)$ is differentiable at every point in its domain.

Now the word ‘charts’ here confuses me. Do we require the charts to be in the complex atlases of $X$ and $Y$? Or do we require the charts to be in the maximal atlases on $X$ and $Y$? Or do we just take arbitrary local homomorphisms $varphi, psi$ regardless of the atlases on $X$ and $Y$?

In the first case, will the definition of holomorphic maps depends on the specific atlases (not necessarily maximal) we impose on $X$ and $Y$?

It seems to me that the second case is somehow equivalent to the first one as we can simply write, for any charts $varphi’,psi$ in the maximal atlases,
$$psi’circ fcircvarphi’^{-1}=(psi’circpsi^{-1})circ(psicirc fcircvarphi^{-1})circ(varphicircvarphi’^{-1})$$
given some charts $varphi, psi$ in the given atlases. But I am not quite sure because even the same author may refer to different things when they mention the word ‘chart’ in different places. Thanks in advance.

complex analysis – classification of Riemann surfaces for higher genus?

It is well known that by uniformization theorem the simply connected Riemann surface can be classified into three equivalent classes by conformal mapping: $C, C^infty$, and $D$. This may be seen as a generalization of the Riemann mapping theorem. And also as a general fact, the annuli in
$C$ can be classified into equivalent classes of one parameter (they are all conformally equivalent to the unit disk removing a hole of radius $s$). So I’m wondering if there are any similar theorems for Riemann surfaces of the higher genus that generalizes the latter result.

Subvarieties in cotangent bundles of Riemann surfaces

Let $X$ be a Riemann surface and $pi colon T^*Xrightarrow X$ be its holomorphic cotangent bundle. Other than (1) cotangent fibers and (2) ramified coverings in $T^*X$ of $X$ minus some divisor, what are examples of irreducible analytic subvarieties of $T^*X$? Give me examples.

riemann surfaces – Dessin d’enfant of Dynkin diagrams?

Dessin d’enfant have a nice particular case of shabbat trees, where we take a tree, bicolor it, and get a polynomial map.

A very famous set of trees is the Dynkin diagrams, I wonder what are the special properties of the polynomial the Dessin d’enfant with them produces.

As an example, An becomes the chebychev polynomials.

multivariable calculus – Why are level surfaces of positive definite functions closed near critical points

Consider a positive definite scalar function $textit{f}:mathbb{R^n} rightarrow mathbb{R}$, $f$ is $C^1$.

Let $M={xin mathbb{R^n}: f(x)<c}$, $c$ is some positive constante.

As long as $f$ is positive definite, its level surfaces $f(x)=c$ that are close to the origin are garanteed to be closed.
Further away, the level surfaces are not necessarily closed if $f$ is not radially unbounded.

Why is that?

Any explanation please?

multivariable calculus – Find the volume bounded by the following surfaces

I need to find the volume bounded by the following surfaces:

$$x^2+y^2=2z$$ $$x^2+y^2=3-z$$

I don’t know how to proceed in solving this exercise, all I could think of was trying to make a system of these 2 and get $z=1$. I just wanna understand how am I supposed to proceed in finding the boundaries, no need to evaluate the integral.

mg.metric geometry – Rigidity for convex surfaces in elliptic/hyperbolic space

From Alexandrov’s work we know that any metric on the sphere with lower curvature bound $kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact convex domain) in the $3$-space form of constant curvature $kappa$.

For $kappa=0$, the surface is unique up to isometry of the surrounding space $mathbb{R}^3$ due to Pogorelov. Is the same true for the elliptic and hyperbolic case? In other words, are two closed convex surfaces in these spaces congruent when they are isometric with respect to their inner metrics?

I asked this question here on Math SE, but did not get an answer.