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12
votes
0answers
142 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
0answers
201 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 ...
3
votes
0answers
221 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
0answers
66 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
0answers
151 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
0answers
171 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
0answers
149 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
0answers
284 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
0answers
118 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
0answers
142 views

Integral of Square of Mean Curvature

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus. (1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$. (2)When ...