# Questions tagged [surfaces]

For questions about two-dimensional manifolds.

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### Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
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### How can I modify a surface to satisfy two distance conditions?

I have two variables, $\phi$ and $\theta$, and I'm trying to create a smooth surface such that the following rules are met for the distance between on the surface, $D$ \begin{align*} 1)& \: D[(\...
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### Genus of a graph consisting of two faces homeomorphic to open disks

Suppose the graph $G$ is embedded in a surface $Q$ such that there are two faces $F_1,F_2$ of the embedding, each homeomorphic to the open disk, such that each node of $G$ lies on $F_1$ or $F_2$. Is ...
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### To prove that $k_1 + \dots + k_m = mH/2$, where H - average curvature: $H = \lambda_1 + \lambda_2$
Faced with such a task: Let $k_1, \dots , k_m$ be the normal curvatures of the surface in the directions dividing the plane into angles $\frac{\pi}{m}$ To prove that $k_1 + \dots + k_m = mH/2$, where ...
### Determine a tangent plane on the surface $\psi(u,v)=(3u^2-2v^2,u-v,u+v)$ at the point $\psi(1,2)$.
So in this question basically I'm getting stuck after calculating the Jacobian of the function $\psi$, somehow I'm trying to find the values of a vector that is on this plane, but I'm not so sure of ...