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

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53 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
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1answer
84 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 ...
2
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1answer
45 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 ...
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50 views

Area of the minimal surface of a non-planar quadrilateral in 3d [migrated]

Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
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1answer
20 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 ...
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1answer
39 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 ...
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25 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 ...
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39 views

To minimize surface area of integer cuboid of ​​the known volume

There is a cuboid (a * b * c), (a, b, c ∈ N). S (Surface area of a cuboid) = 2 * (ab + bc + ca). V (Volume of a cuboid) = a * b * c = n. I need to minimize S, provided that I specified the volume ...
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31 views

A minimal surface

http://en.wikipedia.org/wiki/Minimal_surface Ref: The first figure with soap film at right. What is surface parametrization or references? How is it connected to the helicoid/catenoid ? Thanks.
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15 views

Minimal surface containing tetrahedron

Find the minimal smooth surface passing through four vertices of a regular tetrahedron bounded by three geodesic arcs connecting any two vertices pairwise. I remember a similar discussion in Courant ...
8
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1answer
79 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 ...
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44 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 ...
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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 ...
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0answers
5 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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1answer
32 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 ...
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1answer
53 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 ...
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1answer
31 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 ...
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1answer
15 views

Applications of Derivatives invloving surface area

A piece of wire 16 cm long is cut into two pieces, one piece is bent to form a square and the other is bent to form a circle. What is the exact length of the sides of the square, (let each side of the ...
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1answer
46 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 ...
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2answers
60 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 $a,b$ ...
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1answer
76 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 ...
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66 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
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1answer
42 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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2answers
104 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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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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29 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 ...
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0answers
28 views

Parameterization of the Schwarz P surface

Is there a closed form parameterization of the Schwarz P minimal surface?
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1answer
110 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 ...
2
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2answers
59 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
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0answers
121 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 ...
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1answer
41 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
105 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
52 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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1answer
45 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 ...
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3answers
422 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!
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1answer
276 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 ...
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2answers
159 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
2
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0answers
58 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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22 views

Comparation of values in minimal submanifold

Let $\phi: M^m\to H^n(k)$ be minimal immersion. Show that ...
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2answers
78 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. ...
2
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0answers
87 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 ...
7
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1answer
115 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 ...
3
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1answer
199 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 $ ...
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1answer
85 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
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1answer
315 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: ...
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2answers
515 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 ...
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2answers
1k views

Minimizing (and maximizing) the area of triangles

How would one solve questions like this one here in general? I have gotten an answer for that question, but I don't understand what's the intuition behind it. Can smoeone clearly explain how to ...
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1answer
299 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 ...
4
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3answers
112 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 ...
7
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1answer
272 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 ...