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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 $R\leq 10$ 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 $R\geq 10$, choose $1\leq h\leq 2$ 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 $h=1/R$ on $[1,R-1]\times S^1$. Up to rescaling by a factor in $[1,5]$, 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 $R=1/l_j^2$.

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 $R=1/l_j^2$.

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 $R\leq 10$ 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 $R\geq 10$, choose $1\leq h\leq 2$ 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 $h=1/R$ on $[1,R-1]\times S^1$. Up to rescaling by a factor in $[1,5]$, we can ensure that the area is 1 and clearly the diameter is bounded.

Question 3: The argument of Richard Thomas 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 $R=1/l_j^2$.

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 $R\leq 10$ 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 $R\geq 10$, choose $1\leq h\leq 2$ 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 $h=1/R$ on $[1,R-1]\times S^1$. Up to rescaling by a factor in $[1,5]$, 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 $R=1/l_j^2$.

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 $R\leq 10$ 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 $R\geq 10$, choose $1\leq h\leq 2$ 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 $h=1/R$ on $[1,R-1]\times S^1$. Up to rescaling by a factor in $[1,5]$, we can ensure that the area is 1 and clearly the diameter is bounded.

Question 3: The argument of Richard Thomas 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.

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 $R=1/l_j^2$.

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 $R\leq 10$ 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 $R\geq 10$, choose $1\leq h\leq 2$ 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 $h=1/R$ on $[1,R-1]\times S^1$. Up to rescaling by a factor in $[1,5]$, we can ensure that the area is 1 and clearly the diameter is bounded.

Question 3: The argument of Richard Thomas 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 $R=1/l_j^2$.