**13**

votes

**0**answers

308 views

### 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 ...

**7**

votes

**0**answers

224 views

### 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
...

**6**

votes

**0**answers

103 views

### 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 ...

**2**

votes

**0**answers

150 views

### 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 ...

**2**

votes

**0**answers

304 views

### 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 ...

**2**

votes

**0**answers

75 views

### 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 ...

**1**

vote

**0**answers

128 views

### Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...

**1**

vote

**0**answers

160 views

### Contractibility of union of smooth rational curves in famillies

Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...

**1**

vote

**0**answers

180 views

### surfaces dominated by a product of curves

I would like to know for which projective smooth surfaces over a finite field there exists a dominant rational map from a product of curves to the surface

**1**

vote

**0**answers

151 views

### Is -(E,E) greater or equal to 2 for a minimal resolution

I'm quite confused by the terminology minimal resolution and minimal model.
Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.
Let $E$ be an ...

**1**

vote

**0**answers

292 views

### Abel-Jacobi map for regular fibered surfaces.

Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let ...

**0**

votes

**0**answers

151 views

### Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...

**0**

votes

**0**answers

244 views

### Integral of Square of Mean Curvature

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.
When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...