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EDIT: Let $g$ be a smooth Riemannian metric on $S^2$ which is invariant under the antipodal involution $x\mapsto -x$. Assume it has Gauss curvature at least $-1$ and diameter $D$. Does there exis …

Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\ …

Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov …

Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic spa …

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov compa …

The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the u …

Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$. Let $Y_0$ be a smooth …

Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came …

EDIT: The well known Jordan curve theorem says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both homeomorph …