# Tagged Questions

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

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### 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 ...
172 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 ...
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 $H^2 \... 1answer 427 views ### why the K3 surfaces are minimal surfaces I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read beauville's book on complex algebraic surfaces: ... 1answer 319 views ### Total mean curvature in$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 ... 2answers 75 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 ... 1answer 137 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 ... 2answers 375 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 ... 3answers 124 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$. I know it's a bit ... 1answer 310 views ### Maximum principle for minimal hypersurfaces The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces$ M_1 $and$ M_2 $of$ R^n $has a common point$ x_0 \in M_1 \cap M_2 $... 2answers 49 views ### components of normal field are Jacobi fields? Given a minimal surface$\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$where L is the stability (Jacobi) ... 1answer 198 views ### How useful is the Weierstrass representation of minimal surfaces? Weierstrass representation of minimal surfaces says that if I have a holomorphic function$f: U \rightarrow \mathbb{C}$and a meromorphic function$g: U \rightarrow \mathbb{C}$such that$f g^2$is ... 1answer 63 views ### Blow-ups followed by contractions Let$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$contains exactly$r$... 1answer 54 views ### what is the minimum surface area shape required in order to contain a 1 meter line at all angles been stuck on solving/proving the following puzzle: You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that ... 2answers 107 views ### Quadric surface as a$\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 $$\mathbb{F_n}=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}\... 1answer 599 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 ... 1answer 58 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 ... 1answer 99 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? 1answer 158 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 ... 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, ... 1answer 71 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 ... 1answer 79 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 ... 2answers 90 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) $$If \textbf{q} is minimal, then r(t) = a\cosh(t)+b\sinh(t) for a,b ... 1answer 124 views ### History of the Enneper Surface I was just wondering whether anyone could tell me more about the Enneper surface and its history (why it is important historically in the development of mathematics), or where to go in order to learn ... 0answers 27 views ### Parametrization admitting conservation of K & H Which mappings admit conservation of K and H ? ( Gauss and mean curvatures). Apart from helicoid/catenoid isometry, which examples can be given of surface bending and distortion so that mean ... 1answer 54 views ### Shortest path in the plane under derivative constraint A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ... 1answer 101 views ### Embedded and Non-Parametric Surface definition What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be 'non-... 0answers 160 views ### Castelnuovo's rationality criterion following Beauville Following Beauville's book "Complex algebraic Surfaces", in order to prove Castelnuovo's rationality criterion i need to prove one lemma and one proposition. There is one point of proof of lemma V.8 ... 0answers 186 views ### How is Euler-Lagrange equation used to find optimal solutions in minimizing a function? How is the Euler-Lagrange equation:$$ 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 ... 0answers 77 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 respectively. ... 0answers 160 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 ... 3answers 1k 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! 1answer 77 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 ... 1answer 70 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 ... 1answer 45 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 ... 1answer 51 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 ... 1answer 123 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) \... 1answer 48 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 ... 1answer 78 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, $$where, K is ... 1answer 40 views ### Is every totally geodesic surface minimal? Let M be a Riemannian manifold, and let S be a hypersurface (codimension 1). If S is completely geodesic, does that imply that it is minimal? If not, what are the conditions? If yes, is there ... 1answer 41 views ### Lipschitz Continuity for a function on stable minimal hypersurface immersed in \mathbb{R}^n I'm going through a proof of Schoen-Simon-Yau's L^p bound on the norm squared of the 2nd f.f., |A|^2, for stable (orientable) minimal hypersurfaces in \Sigma \subset \mathbb{R}^n, from Minicozzi ... 1answer 160 views ### How to prove the ruled minimal surface is helicoid or plane? This is an exercise in elementlary differential geometry (named as Catalan's theorem). Though there are many proofs of this problem, I meet some trouble to prove it. My idea is as follows: (1) ... 1answer 116 views ### The Weierstrass-Enneper representation, the Gauss map Lemma: Let x:S\to\mathbb{R}^3 be a conformal minimal immersion of a Riemann surface. The 1-forms f_k=(x_{k,u}-ix_{k,v})dz satisfy:$$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad \sum_k|f_k|^2\not=0\... 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 ... 1answer 48 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? 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 ... 1answer 54 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 ... 1answer 325 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 ... 0answers 28 views ### For a minimal surface$M$under Mean Curvature Flow, can it evolve between minimal surfaces continually? I don't know much about this subject at all; I'm only just getting into it. As it turns out, a physicist friend of mine asked me a formulation of the following: Suppose$M$is a surface in$\mathbb{R}...
A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...