8
votes
1answer
62 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
0
votes
0answers
38 views

Coordinate frames along the bounday of a minimal area (soap-film) surface

I would like to calculate coordinate frames along a closed Bezier (Or Catmull-Rom) spline. One axis should be tangential to the curve, and another axis normal to the minimal-area surface (soap-film ...
0
votes
1answer
28 views

Unicity solution in this differential equation

I'm studying Sherk surface which is the unique minimal surface with parametrization given by $\phi(x,y)=(x,y,f(x)+g(y))$. Using the mean curvature formula, is easy to show that this surface is minimal ...
1
vote
1answer
39 views

Weierstraß parametrization of minimal surface

I am currently learning about how to generate minimal surfaces using complex analysis — namely by calculating the Weierstraß parametrization (here's the article on Wikipedia). Now in my lecture notes ...
1
vote
2answers
57 views

Solving a certain differential equation when assuming a surface of revolution is minimal

The problem is the following: Consider the surface of revolution $$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$ If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ ...
2
votes
1answer
72 views

History of the Enneper Surface

I was just wondering whether anyone could tell me more about the Enneper surface and its history (why it is important historically in the development of mathematics), or where to go in order to learn ...
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 ...
2
votes
1answer
38 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
2
votes
0answers
94 views

How is Euler-Lagrange equation used to find optimal solutions in minimizing a function?

How is the Euler-Lagrange equation: $$ L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 $$ used mathematically in finding the optimal solutions of minimising a function? Can someone give me an ...
1
vote
3answers
323 views

There are no compact minimal surfaces

This is one of the exercises of 'Do Carmo' (Section 3.5, 12) How do you prove that there are no compact (i.e., bounded and closed in $\mathbb{R}^3$) minimal surfaces? Thanks!
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 ...
2
votes
0answers
56 views

minimise the total distance to a hyperbolic curve from two fixed points

The point $C$ moves along the hyperbolic curve which is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$. The distances $d_{0}$ and $d_{1}$ in from $A$ to $C$ and $B$ to $C$ ...
3
votes
1answer
168 views

Maximum principle for minimal hypersurfaces

The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces $ M_1 $ and $ M_2 $ of $ R^n $ has a common point $ x_0 \in M_1 \cap M_2 $ ...
1
vote
1answer
82 views

Why this equality must holds for minimal surfaces?

When minimizing a surface area with respect to a fixed volume $V$, I found in some notes that the parametrization $X: U \longrightarrow \mathbb{R}^3$ must satisfy the equality $\iint_U (2H - \lambda) ...
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 ...
1
vote
1answer
382 views

Helicoid and Catenoid

Let $X$ and $Y$ be isothermal parametrizations of minimal surfaces such that their component functions are pairwise harmonic conjugates, then $X$ and $Y$ are called conjugate minimal surfaces. My ...