1
vote
1answer
48 views

On Constant mean curvature surfaces.

I have two involved questions, firstly, I know that the gauss map sends a surface to the unit sphere, so for a surface $\Sigma\subset\Bbb R^3$, parametrised by $u:U\subset\Bbb R^2\to \Bbb R^3$. Would ...
1
vote
0answers
51 views

Proof check for critical point definition with mean curvature

I'm currently trying to prove: "Definition: We say that a surface $S \in R^3$ is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ...
5
votes
1answer
103 views

Maths undergrad dissertation - minimal surfaces

I'm a third year maths undergrad writing a dissertation on minimal surfaces, and their application in space. Would anyone be willing to read through it (so far) and give me any feedback? positive or ...
0
votes
1answer
227 views

compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...
0
votes
2answers
152 views

Suppose $S$ is a minimal surface, show that the gaussian curvative is negative on all interior points

I am trying to improve my proving skills, started learning by myself, can anybody help me with this? thanks you
7
votes
1answer
262 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...