Question on minimal surfaces, or surfaces that have zero mean curvature.

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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 $a,b$ ...
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2answers
74 views

Minimize the distance in the Euclidean space

The objective is to minimise the distance $d_{0}+d_{1}$. The points $c_{0}$ and $c_{1}$ are given. I need to locate the point $c$ which minimises the distance $d_{0}+d_{1}$. I have worked like this. ...
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1answer
102 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 ...
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1answer
37 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 ...
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0answers
68 views

Do K3 surfaces with an Enriques involution have a polarization of bounded degree

Does there exists a real number $C$ with the following property. For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 ...
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0answers
113 views

Castelnuovo's rationality criterion following Beauville

Following Beauville's book "Complex algebraic Surfaces", in order to prove Castelnuovo's rationality criterion i need to prove one lemma and one proposition. There is one point of proof of lemma V.8 ...
2
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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 ...
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0answers
55 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 $C$ ...
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74 views

Minimal surface representation from a 3D contour

I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
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0answers
48 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 ...
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0answers
24 views

Parameterization of the Schwarz P surface

Is there a closed form parameterization of the Schwarz P minimal surface?
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0answers
20 views

What is the function describing the minimal surface of this object?

What function describes the minimal surface of this object? The object consists of four circular arcs glued together. One arc is parallell to another arc, and a third arc is parallell to a fourth ...
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0answers
47 views

Help with Plateau's Laws

Can someone please explain mathematically what is meant by the term 'smooth' in Plateau's First Law: "Soap films are made of entire smooth surfaces" Thank you in advance!
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0answers
21 views

Comparation of values in minimal submanifold

Let $\phi: M^m\to H^n(k)$ be minimal immersion. Show that ...
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0answers
28 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 ...
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0answers
3 views

Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
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0answers
26 views

How to find the infintesimal generator and conserved current of the symmetries of the minimal surface problem

For the Lagrangian $L(x,y,z,z_x,z_y)=\sqrt{1+z^2_x+z^2_y}$ how do you find infinitesimal generator and conserved current of the six symmetries (3 translations and 3 rotations)? I was using Noether's ...
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24 views

zero mean curvature and a variational problem

given the lagrangian $$ J(u)= \int_{V} \sqrt{1+|gra(u)|^{2})} $$ with the constraint $ \int_{V}udx =1 $ why is the volumen constraint there ? for teh case of R^{3} i know this must satisfy the ...