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

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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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14 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
54 views

Gaussian curvature 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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9 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
20 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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36 views

How find minimum of the $|PA|+2|PB|+3|PC|+4|PD|$ [closed]

Let $P(x,y)(0\le x,y\le 1)$ and such $A(0,0),B(1,0),C(1,1),D(0,1)$, Find the minimun of the value $$|PA|+2|PB|+3|PC|+4|PD|$$ In general,find the minimum of the value $$a|PA|+b|PB|+c|PC|+d|PD|$$ ...
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1answer
38 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
50 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
55 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
62 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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1answer
73 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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2answers
43 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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2answers
953 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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1answer
53 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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35 views

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
62 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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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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33 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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45 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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16 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 ...
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1answer
102 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
106 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
33 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
81 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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1answer
40 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
33 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
45 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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54 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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36 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 ...
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1answer
109 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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2answers
272 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
60 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
103 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
152 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
61 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
54 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
66 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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97 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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1answer
163 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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237 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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59 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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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 ...
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1answer
51 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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148 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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56 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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3answers
888 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
75 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
41 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 ...