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

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21
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2answers
708 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
9
votes
1answer
113 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 ...
7
votes
1answer
148 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 ...
7
votes
1answer
356 views

why the K3 surfaces are minimal surfaces

I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read beauville's book on complex algebraic surfaces: ...
7
votes
1answer
298 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 ...
5
votes
1answer
123 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 ...
4
votes
2answers
174 views

How to actually use the Weierstrass-Enneper parameterization to draw a minimal surface?

I'm interested in drawing (with Mathematica for example) the generalized Scherk saddle tower with threefold symmetry, a shape that I find very attractive. In an article (see here) I found the ...
4
votes
3answers
120 views

Given a volumen. Which is the suface, that contains it, that has minimal area?

Defining on $R^3$, $V = \iiint_S dx \, dy \, dz $ as the volume of surface $S$, with $S$ closed, bounded and arc-connected. Which is the $S$ of minimal area, that contains $V$. I know it's a bit ...
3
votes
1answer
115 views

How useful is the Weierstrass representation of minimal surfaces?

Weierstrass representation of minimal surfaces says that if I have a holomorphic function $f: U \rightarrow \mathbb{C}$ and a meromorphic function $g: U \rightarrow \mathbb{C}$ such that $f g^2$ is ...
3
votes
1answer
51 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
3
votes
1answer
240 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 $ ...
2
votes
2answers
72 views

Quadric surface as a $\mathbb{F}_n$ surface

The minimal models for rational projective smooth surfaces are $\mathbb{P}^2$ or the surfaces $\mathbb{F}_n$ for $n\neq 1$, where ...
2
votes
1answer
48 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
2
votes
1answer
71 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
2
votes
2answers
76 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
99 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 ...
2
votes
0answers
24 views

Parametrization admitting conservation of K & H

Which mappings admit conservation of K and H ? ( Gauss and mean curvatures). Apart from helicoid/catenoid isometry, which examples can be given of surface bending and distortion so that mean ...
2
votes
1answer
37 views

Shortest path in the plane under derivative constraint

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ...
2
votes
1answer
55 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
136 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
votes
0answers
132 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 ...
2
votes
0answers
66 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 $C$ ...
2
votes
0answers
106 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 ...
1
vote
1answer
56 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
3answers
685 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!
1
vote
1answer
39 views

Name of the surface with two sides and three boundaries

Once i have seen a 3d visualization of a surface with the following characteristics: it had three circular borders. If you imagine the surface inscribed in the earth globe, one of the borders would ...
1
vote
1answer
49 views

Find out the design of a cylinder

A cylindrical can is made from tin.If it can be contain $1000 m^3$ liquid inside it then what is the parameter of design if we are oblige use the minimum amount of tin. My teacher give me this and say ...
1
vote
1answer
103 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) ...
1
vote
1answer
508 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 ...
1
vote
1answer
84 views

The Weierstrass-Enneper representation, the Gauss map

Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad ...
1
vote
1answer
27 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
1
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1answer
41 views

Minimal surface between two non coaxial rings

I'm currently studying minimal surfaces using the Euler-Lagrange equation. I'm particularly interested in minimal surfaces between two circles. I have already examined the case of two coaxial ...
1
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1answer
62 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
1answer
50 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 ...
1
vote
1answer
317 views

Christoffel symbols in Differential geometry iff proof

I need help in proving that $H = 0$ for a surface iff $g_{11}L_{22} - 2g_{12}L_{12} + g_{22}L_{11} = 0.$ I think that these are the Christoffel symbols exploited in some manner and normally, I'm not ...
1
vote
0answers
14 views

Necessary relation for closed lines on minimal surfaces

Is there a necessary relation between curvature and torsion of a closed non-intersecting curve on a minimal surface? While playing with soap films I noted closed light threads migrating on a soap ...
1
vote
0answers
34 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
1
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0answers
28 views

Minimising surface with given curve as a boundary

I have a problem connected to finding a minimal surface with a given boundary. I know that it is the surface with zero mean curvature but as I have to obain differential equations for such surface I ...
1
vote
1answer
48 views

Minimal surfaces maximum principle

This is homework so no answers please. The problem is The domain is unit disk in $\mathbb{R}^{2}$ Suppose u,w satisfy the minimal-surface equation $div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=0$, ...
1
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0answers
51 views

On Yau's (and Schoen's) proof of the positive mass theorem

I would like to face the proof of the positive mass theorem by Yau and Schoen. I have a Bsc in Mathematics and a Msc in Theoretical Physics and I'm preparing a PhD interview-challenge where I have to ...
1
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0answers
49 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
1
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0answers
82 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 ...
1
vote
2answers
135 views

Total Curvature of 4 pi

What does it mean for a surface to have a total curvature of $4\pi $? I have seen that both the catenoid and Enneper surface are the only minimal surfaces that have this total curvature, but I don't ...
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0answers
44 views

Parameterization of the Schwarz P surface

Is there a closed form parameterization of the Schwarz P minimal surface?
1
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0answers
66 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!
1
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0answers
24 views

Comparation of values in minimal submanifold

Let $\phi: M^m\to H^n(k)$ be minimal immersion. Show that ...
1
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2answers
82 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. ...
0
votes
2answers
199 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
0
votes
1answer
51 views

Prove that there are no complete regular minimal surfaces lying above a paraboloid

Prove that there are no complete regular minimal surfaces lying above a paraboloid contained in $U=\{(x,y,z) \in \mathbb{R}^3 : a(x^2+y^2)<z\}$. Here $a>0$. I've had this problem on my mind ...
0
votes
1answer
29 views

Enneper surface is not injective

I'm having trouble proving the following statement: $x(u, v) = (u − u^ 3/ 3 + uv^2 , v − v^ 3/ 3 + u^ 2 v, u^2 − v^ 2 )$ is a minimal surface and x is not injective Proving that $x(u,v)$, ...