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M. Berger, Geometrie, vol. V. MR0536874 Edit. Let me sketch a proof for the spherical triangle. Let the sphere have area $4\pi$. First you derive the area of digon. It is $2\alpha$, where $\alpha$ is …

Without the deep theorem of Nash-Kuiper, it is easy to construct and visualize a bi-Lipschitz embedding: think of those Dutch collars of the XVII century like this: https://en.wikipedia.org/wiki/File: …

Let me add to this nice answer of Ian Agol, that the isometry group is always finite, and contains at most $84(g-1)$ elements, if we count orientation-preserving (conformal) isometries, where $g\geq 2 …

There are indeed very few pairs (except spheres with 3 or 4 singularities, or tori, and what can be obtained from them by finite coverings, where correspondence 2)-3) is completely explicit. See: H. P …

W. Thurston, Three-dimensional geometry and topology.

MR0725161 Ahlfors, Lars V. Möbius transformations in several dimensions, Minneapolis, MN, 1981.

For the equivalence of definitions of quasiconformal maps the reference is J. Heinonen, Lectures on analysis on metric spaces, Springer 2001. Notice that the $K$ in the definiton you cite is not the s …

Yes, this is a theorem of Siegel, and a reference with simple proof is Theorem XI.12 in M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo 1959 (there is an AMS Chelsea reprint, 1975 …

Let me rephrase the proof of @Ian Agol as I understand it. Let $\Gamma(2)$ be the congruence subgroup of level $2$ (it consists of all $2\times 2$ integer matrices $A$ with ${\mathrm{det}\, }A=1$ and …

This paper is probably relevant for your question: MR0528966 Wolpert, Scott The length spectra as moduli for compact Riemann surfaces. Ann. of Math. (2) 109 (1979), no. 2, 323–351. And here is a more …

Yes, the answer is complicated, it is related to holomorphic dynamics, and the question was much studied, see for example: MR0869581 Lyubich, M. Yu.; Suvorov, V. V. Free subgroups of SL2(C) with two …

For statement 2, you have to specify whether $f$ is allowed to have zeros. (If yes, this is a metric with isolated singularities, but can be complete. If $f$ is free of zeros, it is always incomlete b …

This is a classical theorem: hyperbolic Riemann surfaces of finite hyperbolic area are compact surfaces with finitely many punctures. Tsuji (Theorem XI.12) credits this to Siegel (1945). The proof is …

There is a theory of conformal map for circular polygons (polygons bounded by arcs of circles). But in this case, instead of an integral in the Schwarz-Christoffel formula, you obtain a linear differe …

Alexandr Solynin told me that he solved this problem (even the more general one, for hyperbolic n-gons with all zero angles) in 1993. A. Solynin, Some extremal problems for circular polygons, (Russian …