# Tagged Questions

A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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### Non-lattice Veech groups

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been ...
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### Approximating homeomorphisms of 2-disk by diffeomorphisms

Any homeomorphism of a compact surface can be approximated by diffeomorphisms. Is there a parametrized version of this result, where the parameter space is an $n$-disk? In other words, if $S$ is a ...
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### Surfaces with many (but not solely) closed geodesics?

Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random ...
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### Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
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### Fiber at infinity of an arithmetic surface $X$ as an element of $\widehat{\operatorname{Div}(X)}$

Introduction: Let $M$ be a Riemann surface, then a Green function on $M$ is an element $g\in C^\infty(V)$ where $V=M\setminus\{x_1,\ldots,x_r\}$ and around each point $p\in M$ we have: g=a\log\...
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### Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a ...
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### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
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### Elliptic surfaces with different Kodaira symbols

Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre? Can ...