Metrics with fixed conformal structure and diameter

Question

I have three questions.

1. I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possible that the diameter of $h_n$ tends to infinity?

2. Consider such a sequence of metrics along which the diameter is bounded but whose conformal class is allowed to vary. Is it possible that the associated sequence of conformal classes diverges? (the moduli space of conformal classes on the torus is $\mathbb{H}^2/ \mathrm{SL}_2(\mathbb{Z}))$ and by diverging I mean escaping from every compact set of this moduli space).

3. Fix a conformal structure on a torus. Can I holomorphically embed cylinders of arbitrarily large modulus in this torus in a $\pi_1$-injective way?

Thanks for your attention.

Selim

Answer 1

Question 2: Yes, there are conformal metrics on a divergent sequence of tori with Area=1 and bounded diameter:

Cutting the torus open along an embedded essential curve, you obtain a cylinder, conformally equivalent to $[0,R] \times S^1$. The fact that the sequence of conformal structures diverges, means that we can choose the cutting curve such that $R_n\rightarrow \infty$. However this is not really relevant here and we just directly construct such a conformal metric for any such cylinder (which gives rise to a conformal metric on the torus).

Note that any conformal metric on such a standard cylinder has the form $h|dz|^2$. If $R\leq 5$ choose just any function $1\leq h\leq 2$ which glues nicely back to a metric on the torus; up to rescaling we obtain conformal metric of area 1 and bounded diameter. If $R\geq 5$, choose $h\leq 2$ everywhere and $h\geq 1$ on a neighborhood $[0,1]\times S^1 \cup [R-1,R]\times S^1$ of the boundary (and such that it glues nicely) and arbitrary small away from the boundary, say $h=1/R^2$ on $[2,R-2]\times S^1$. Up to rescaling by a factor in $[1,10]$, we can ensure that the area is 1 and clearly the diameter is bounded.

Question 3: The argument of Thomas Richard shows you can embed any cylinder of the form $[0,R)\times S^1$ for $R\in (0,\infty]$ into any torus. The cylinder $\mathbb{R}\times S^1$ cannot be embedded into any compact Riemann surface $S\neq S^2$, since it would extend to an embedding of $S^2$. On the other hand, covering maps give immersions from $\mathbb{R}\times S^1$ into any torus (but no such immersions exist into surfaces of hyperbolic type).

If the embedding is supposed to be $\pi_1$-injective, then the maximal $R$ such that $(0,R)\times S^1$ embeds into a complex torus $(T^2,j)$ is determined by the length $l_j$ of the systole of the torus in the conformal class determined by $j$. Indeed one can see that the maximal $R$ satisfies $R=1/l_j^2$.

Answer 2

For question 1: Write $g_0$ for your base unit volume flat metric on $\mathbb{T}^2$. Fix a point $p\in\mathbb{T}^2$. Let $u_i$ be a sequence of functions on $\mathbb{T}^2$ such that :

• $\int_{\mathbb{T}^2}u_i^2dv_{g_0}=1$
• $u_i$ is constant equal to $1/i$ outside a ball (for $g_0$) of radius $\tfrac{1}{i}\to 0$ around $p$
• $u_i(p)$ goes to $\infty$ as $i\to\infty$.
• $u_i$ depends only on $d_{g_0}(p,\cdot)$

Now $g_i=u_i^2g_0$ has a long "finger" at $p_0$ attached to a tiny flat torus. And its diameter is going to $+\infty$ (it takes about $u_i(p)^2$ to go from the tip of the finger to the flat part).

For question 2 : This was wrong, I did not understand the question properly.

For question 3 : modulus is invariant under scaling so any holomorphic disk contains annuli of arbitrary finite modulus.