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

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Apply “thickness” to a minimal surface

By definition a minimal surface has no volume. But my goal is to give the minimal surface some volume. Is there a mathematical way to do this? I found a thread in a forum of a math visualization tool ...
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44 views

Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface

Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface I'm really lost on how to do this question. I know we have to use the Euler equation to show this, but other than that ...
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1answer
27 views

About totally umbilical hypersurfaces

Suppose $\tilde{M} \subset M$ is a hypersurface sitting inside a Riemannian manifold $(M,g)$. The second fundamental form of $M$ evaluated on $u,v \in T_pM$ is denoted $II(u,v)$ and defined as the ...
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nonlinear Poisson equation - finite elements

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
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1answer
47 views

A surface doubly ruled by orthogonal lines is a plane

The question was originally asked here Doubly Ruled Surfaces and I am following the hint provided by the OP. That is, first show that $K\equiv0$ and then deduce that the surface is a plane. Let the ...
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1answer
33 views

Associated family of isometric surfaces of a helicoid (or a catenoid)

This question is to understand the local isometry between a helicoid and a catenoid using complex analysis. Given the usual parametrizations, $$x(u,v)=(-\cosh{u}\sin{v},\cosh{u}\cos{v},u)$$ ...
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73 views

Minimum sum of the squares

Find the smallest value of the expression $$(x_1-x_2)^2+(x_2-x_3)^2+...+(x_{n-1}-x_n)^2+(x_n-x_1)^2,$$ if $x_1,x_2,...,x_n -$ pairwise different integers My work so far: I have a hypothesis, that the ...
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38 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
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25 views

Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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1answer
74 views

Gaussian curvature of minimal surfaces expressed by torsion and curvature of its geodesics

Let $S$ be a minimal surface, and let $\gamma$ be a geodesic parametrized by arc length, with curvature $k$ and torsion $\tau$,show that in the points of $\gamma$ $$ -K=k^2+\tau^2, $$ where, $K$ is ...
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12 views

Formulation of boundary constrained minimal surface

Using standard notation of classical surface theory how is the standard Plateau problem formulated as an iso-perimetric one minimizing area for given boundary length $$ \int \sqrt{ E \, du^2 + 2 F ...
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1answer
32 views

Is every totally geodesic surface minimal?

Let $M$ be a Riemannian manifold, and let $S$ be a hypersurface (codimension $1$). If $S$ is completely geodesic, does that imply that it is minimal? If not, what are the conditions? If yes, is there ...
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1answer
40 views

Lipschitz Continuity for a function on stable minimal hypersurface immersed in $\mathbb{R}^n$

I'm going through a proof of Schoen-Simon-Yau's $L^p$ bound on the norm squared of the 2nd f.f., $|A|^2$, for stable (orientable) minimal hypersurfaces in $\Sigma \subset \mathbb{R}^n$, from Minicozzi ...
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1answer
53 views

what is the minimum surface area shape required in order to contain a 1 meter line at all angles

been stuck on solving/proving the following puzzle: You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that ...
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1answer
68 views

Understanding an example for “minimal surface doesn't imply least area”

I can't understand two things regarding the following example: 1- Why the minimal surface $S$ will not minimize the area among all surfaces with boundary the two circles if $S_0 < S$? I don't ...
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1answer
69 views

Conformal immersions from surfaces into 3-manifolds

Let $f:(S,g) \to (M,h) $ be a smooth immersion of a compact surface into a 3 - manifold. Is it true that there exists a diffeomorphism $\phi: S \to S$, such that the metric $(f \circ \phi)^*(h)$ is ...
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2answers
49 views

components of normal field are Jacobi fields?

Given a minimal surface $\Sigma$ in $R^3$ with associated normal field $N$, I am told that each of the components of $N$ is a Jacobi field, meaning that $Lu=0$ where L is the stability (Jacobi) ...
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1answer
65 views

Does any minimal surface which is regular has a conformal parameterization?

Since given surface is regular and minimal, I have $$H=\frac{Eg-2Ff+Ge}{2(EG-F^2)}=0$$ and $$X_u\times X_v\ne0$$ Can I derive $E=G,\ F=0$ from these conditions?
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Parametrization of helicoid like surface for Faraday's law of induction of a solenoid?

I want to visualize with mayavi a possible surface for Faraday's law of induction in the electrodynamics of a solenoid. I.e. something like a helicoid with a smooth transition to a rectangular area, ...
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1answer
146 views

How to prove the ruled minimal surface is helicoid or plane?

This is an exercise in elementlary differential geometry (named as Catalan's theorem). Though there are many proofs of this problem, I meet some trouble to prove it. My idea is as follows: (1) ...
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1answer
88 views

Isothermic Surface

What is an Isothermic Surface intuitively? There are a couple of definitions, but I really don't understand what it means if a surface is isothermic. What are ist properties, what is it used for?
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34 views

Required minimum number of points on boundary of minimal surface

What is the minimum number of points required to uniquely determine a minimal surface in 3-Space? Four? Five? If six or more points on boundary are given it gives rise to over-determination.. right?
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35 views

Scaling of minimal surfaces

After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact. Is the statement ...
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68 views

Parametrization for intersection curve of catenoid and cylinder

Required to obtain equation of intersection line of two surfaces.. the catenoid of revolution and displaced or eccentric cylinder..in a parameterized form. $$ (x^2 + y^2) = c^2 \cosh ^{2} (z/c) ; \, ...
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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 ...
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1answer
144 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 ...
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27 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 ...
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1answer
113 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 ...
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1answer
36 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 ...
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1answer
85 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, ...
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44 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 $x(u,v)$, ...
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1answer
40 views

Scherk’s fifth minimal surface

Scherk’s fifth minimal surface is defined implicitly by $$ \sin(z)=\sinh(x) \sinh(y). $$ How can I show that this surface is minimal?
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1answer
52 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 ...
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1answer
102 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 ...
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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 ...
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117 views

Relative minimal surface and minimal surface in the algebraic geometry

Liu's book define regular fibered surface X$\to$S is relative minimal surface if it does not contain any exceptional divisor, regular fibered surface X $\to$ S is minimal surface if every birational ...
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334 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 ...
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1answer
66 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 $div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=0$, ...
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1answer
111 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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188 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 ...
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1answer
63 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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1answer
58 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
70 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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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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301 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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64 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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166 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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58 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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46 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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43 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 ...