Questions tagged [minimal-surfaces]

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

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Visualise a non-orientable minimal surface, given a representation for the orientable double cover surface (Meeks' Mobius strip)

Suppose that $S$ is a non-orientable surface in $\mathbb{R}^n$, represented by $X:M\rightarrow\mathbb{R}^n$ and $\tilde{S}$ is its orientable double cover surface, represented by $\tilde{X}:\tilde{M}\...
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Lifting Minimal Surfaces

Let $f(z)=z+\frac{\overline{z}}{4}$ where $\left|z\right|<1$. Does the function lift to minimal surface? $h(z)=z$ , $g(z)=\frac{z}{4}$ since $f(z)=h(z)+\overline{g(z)}$ so $h\text{'}(z)=1$ and $g\...
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Seeking a parametrization for the octoid

My undergraduate student and I are working through chapter 2 of the book Complex Analysis Topics for Undergraduates and Beginning Researchers: an Exploration with Unsolved Problems, (http://www....
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Isoperimetric inequality

I'm trying to understand a lemma from this paper. In my understanding, it says that Since $M$ is closed, there exists $\alpha>0$ and a map $\phi:M\to \mathbb{R}^n$ (with some property that I am ...
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Compute Quintic/Cubic polynomial surface patches given a set 3D points

Dear StackExchange Community, My goal is to fit surface patches for known XYZ pointcloud in the 3D Cartesian space, where each point has a known surface normal vector UVW. The intention is to cover ...
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Analyzing a saddle point derived from the difference of two surfaces

I have a generated 3D surface, $z_1 = f^1(w(x),x,y)$ to look at the approximate magnitude of change for different combinations of $x_1$ and $y_1$. Here, $w(x)$ is the series of single positive roots ...
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47 views

Gauss map of a minimal surface converse

It can be shown that the Gauss map of a minimal surface is conformal quite easily, but does the converse of this statement hold? Specifically, what surfaces have a conformal Gauss map?
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Do minimal surfaces imply minimal volume?

I have a set of non-intersecting (approximate) minimal surfaces derived from a parametrization of a gyroid that I've cropped & capped the ends, see image below for 3d print of some of them. Am I ...
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Book on the geometry of rotationally symmetric riemannian manifolds

I would like to find some references where there are specific computations and properties of rotationally symmetric riemannian manifolds, e.g. spectrum of the laplacian, schrödinger operators, ...
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Proof of a theorem on Plateau's problem in “Elementary Differential Geometry” by Andrew Presley

I'm studying the proof of this theorem on Plateau's problem in the textbook "Elementary Differential Geometry" by Andrew Presley, but I don't understand, being more specific, why $\beta^0=\...
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Notion of stability for minimal surfaces

A quote from Danny Calegari's lecture notes about minimal surfaces: "A critical point for a smooth function on a finite dimensional manifold is usually called stable when the Hessian (i.e. the ...
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This (rather long) implicit equation has a short explicit solution, but how can it be found?

I am curious if a method exists for solving for $k$ or $h$ in this implicit equation: $$\frac{k^2}{h} \mathrm{sech}^2(k) \sqrt{1 + \left(\frac{k}{h} \tanh(k)\right)^2} = \ln\left( \frac{k}{h} \tanh(k) ...
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Is there a unique minimal surface with boundary $S_2$?

Take a loop of wire ($S_2$), twisted into a complicated shape. One can dip the wire in soap to form a minimal surface soap film. Is there a unique minimal surface or can there be more than one ...
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Exercise 10 of Colding-Minicozzi's minimal surfaces book

I am doing exercise 10 of Colding-Minicozzi's minimal surfaces book which goes as Find a sequence of functions $u_j:D_1\to\mathbb{R}$ on the unit disk $D_1$ satisfying the minimal surface equation ...
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Weak Sobolev convergence in Mobius group of disk to a constant.

Let $\Gamma \subset \mathbb{R}^3$ be a Jordan curve, $D^2 \subset \mathbb{R}^2$ be the unit disk and define: $$\mathcal{F} = \{u \in W^{1,2}(D^2, \mathbb{R}^3) \ \vert \ u \vert_{\partial D} \ \text{...
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How to understand intrinsic property of surfaces? [duplicate]

This semester began to study classical differential geometry, some conceptual things keep confusing me. When I saw Gauss's Egregium theorem, I didn't feel any special place in this theorem. I thought ...
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Why constant mean curvature surfaces are isothermal surfaces?

From the Encyclopedia of Mathematics website, I read that constant mean curvature (CMC) surfaces, in particular minimal surfaces, are isothermal surfaces, i.e. a surface whose curvature lines form an ...
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Is there a connection between minimal surfaces and tree graphs?

For any connected graph $G=(V,E)$, we must have $|E|\geq|V|-1$. Any edges less, and the graph wouldn't be connected. It's sorta like saying a tree graph is a "local minimum" in the space of ...
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minimal graphs converge to a harmonic function

Suppose $\{u_i\}$ is a sequence of positive $C^2$-functions on the unit ball $B,$ satisfying the minimal surface equation, i.e., $${\rm div}\left(\frac{Du_i}{\sqrt{1+|Du_i|^2}}\right)=0$$ for all $i.$ ...
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How to show the second fundamental form of a flat minimal surface is zero?

The surface is flat if its Guassian curvature vanishes everywhere. The surface is minimal if ist mean curvature vanishes everywhere. Therefore, for a function f(x,y), its corresponding surface's ...
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Catenary solution of minimum surface area of revolution

To get a curve having minimum surface area of revolution passing through end points ( $x_1$, $y_1$) and ($x_1$,$-y_1$), we got the equation: $$ \DeclareMathOperator{\arccosh}{\operatorname{arcosh}} y =...
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Can a graph of a function $\xi$ with non-positive curvature have 5 connected components in which $\xi \neq 0$?

The following problem arises from some trouble I've had regarding a paper by E. Hopf. We have a function $\xi(x,y)$ such that: $\xi$ is of class $C^2$ in $\mathbb{R}^2$. $\xi_{xx}\xi_{yy} - \xi_{xy}^...
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Simplifying $F(\sin^{-1}\sqrt{2/(2-p)},1-p^2/4)$ (for a minimal surface)

The tetragonal distortion of the Schwarz P minimal surface has a Weierstrass–Enneper parametrisation of $g(w)=w,f(w)=(w^8+\lambda w^4+1)^{-1/2}$ and a Bonnet angle of $\pi/2$ where $\lambda<-2$. ...
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Inverting a function in the parametrisation of Riemann's minimal surface

While writing code to render Riemann's minimal surface – which is now available here as part of my Malibu STL surface generator – the following problem came up. Consider the following function: $$y(\...
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Which $\rho$ minimizes the mean curvature of $\Psi$?

For $x,y\in(0,1)$ $$\Psi_\rho(x,y)=\exp\bigg( \frac{-\rho}{\log(x)\log(y)}\bigg).$$ Which $\rho$ minimizes the mean curvature of $\Psi?$ If my calculations are somewhat accurate, then $\rho \approx ...
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Creating a smooth height map for an irregular n-gon.

Lets say we initially have a collection of points in a 2D space which form an irregular n-gon. We then give each of the points a z value to make the polygon 3D. In my case, there are only 2 possible z ...
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Rendering minimal surfaces defined on complex manifolds

Costa's minimal surface was originally defined using a Weierstrass–Enneper parametrisation of $$f=\wp(t)\qquad g=\frac{2\sqrt{2\pi}\wp(1/2)}{\wp'(t)}=\frac A{\wp'(t)}$$ where $\wp$ has periods $1,i$ ...
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On the cruelty of really solving hyperelliptic integrals

I want to derive closed-form parametric expressions describing the Schwarz H minimal surface starting from the Weierstrass–Enneper parametrisation, much like Gandy et al. did for the Schwarz D surface ...
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Rendering Lawson's comparison surfaces for the Willmore problem

The sphere minimises Willmore energy for genus $0$; the stereographic projection of the Clifford torus – major radius $1$, minor radius $1/\sqrt2$ – does so for genus $1$. While corresponding ...
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Doubt about the metric tensor of K-Noids

My question is quite short, but unfortunatelly I didn't find the answer in a quick search. So: Is it possible to write the metric tensor $$ds^{2} = g_{ab}dx^{a}dx^{b} \tag{1}$$ for surfaces like K-...
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Compare the area of two different catenoids spanned by the same two circles

A catenoid is obtained by rotating the graph of the function $f(x)=a \cosh (x/a)$ around the $x$-axis. Consider catenoids that satisfy the boundary condition $f(c)=f(-c)=r>0$. We have $a\cosh(c/a)=...
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Minimal Surfaces of Revolution in Higher Dimensions

I know that a surface of revolution can be parameterized in the following way: $x(s,\theta)=(r(s)\cos(\theta),r(s)\sin(\theta),s)$. In addition, the only minimal surfaces of revolution are planes and ...
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Minimal Manifold, $\mathbb{R}^4$

We know that all the minimal surfaces of revolution in $\mathbb{R}^3$ are the Catenoid and the euclidean plane $\mathbb{R}^2$ in $\mathbb{R}^3$. Do we know what are the minimal surfaces of ...
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Inequality involving Second Fundamental Form

I'm stuck in the second part (the inequality part) of the following: let $S$ be a compact regular surface, $f: S \longrightarrow \mathbb{R}$ given by $f(p)=\langle p,p \rangle$, and $p_0 \in S$ a ...
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Find the area of the minimal surface between two loops?

Say we have two loops given parametrically by $X^\mu(\sigma)$ and $Y^\mu(\sigma)$. We want to find the minimal possible area of a tube-like surface ending on these loops. It would be a functional $F[X,...
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Question on the relationship between the surface bounded by Jordan curve and critical point

I was reading 'Notes on Minimal surface' by Michael Beeson. I am confused on top of page 11 that saying 'A surface of least area bounded by Γ would be a critical point of A, but not necessarily ...
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Checking uniqueness of solution to a Laplace equation. Related to minimal surface modelling

hope you all are doing well. I am working on minimal surfaces (Chemical engineering background), and I am stuck at a particular problem. I need to solve Laplace equation with the following boundary ...
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Outermost Minimal Surface in a Manifold

An outermost minimal surface is a minimal surface which does not contain another minimal surface within it. Interestingly, outer minimal surfaces are always spheres, because the only other ...
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Geometric Meaning of Conditions on Curve Shortening

The following shows the meaning of the notations. Here is the definition of curve shortening. I can understand the geometric meaning of (1) & (2), but not (3) & (4). Can someone explain more ...
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Maximum Principle for Minimal Surface Equation with Dirichlet Boundary Condition

I'm an undergraduate student and I'm currently reading a classical paper for my final project for the course differential geometry on the Bernstein problem of minimal surfaces, namely, the paper: ...
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Minimal embedding of real projective surface into $S^4$

I'm reading a paper, and one result was quoted there: There is a minimal embedding of $\mathbb {RP}^2$ into $S^4$, and the corresponding volume is 6$\pi$. Any reference is welcomed. And thanks for ...
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Minimal Surface has constant Gaussian Curvature After Conformal Change $\tilde{g}=-Kg$

Question: suppose $M\subset \mathbb{R}^3$ is a minimal surface (mean curvature $H \equiv 0$), show that after conformal change $$\tilde{g}=-Kg$$ the Gaussian curvature $\tilde{K}\equiv 1$. Since $...
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Is there any minimal surface that is topologically a sphere?

I am currently studying about Willmore energy and out of my expectation, I produced a simple result that any surface that is topologically a sphere (genus 0) cannot be a minimal surface. (I am not ...
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Area convergence

Let $(M^3,g)$ be a compact, connected and oriented Riemannian manifold with boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of free boundary minimal surfaces embedded in $M$ that converges ...
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On algebraic minimal surfaces

In the theory of minimal surfaces, we could construct various types of minimal surface based on the way we choose a pair of holomorphic & meromorphic fucntion. To be more specific, let $f(\zeta)$ ...
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Reparametrization of an ordinary differential equation

The following is part of the solution of an Exercise in Curves and Surfaces, second edition, by Montiel and Ros, which asks the reader to prove that the only surfaces of revolution with zero mean ...
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Minimal surface terminology

If $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ is smooth, then its graph $G = \{(x,y, f(x,y) | (x,y) \in \Omega\}$ is a minimiser of area if and only if it satisfies the Euler-Lagrange ...
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If $M \subset \mathbb{R}^{n+1}$ is a compact hypersurface, then $M$ is not a minimal hypersurface

I would like to know if there are any conditions on a compact hypersurface $M^n \subset \mathbb{R}^{n+1}$ that ensure that $M$ is not a minimal hypersurface. The motivation for this question is ...
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How can we perturb saddle surface into minimal Enneper's surface?

$$f(u,v) =u^3 - 3 uv^2$$ Here $f$ is a harmonic function. If $u=r\cos\ t,\ v= r\sin\ t$, then $f(u,v)=r^3\cos\ (3t)$ Then we have a parametrization \begin{align*} z(u,v)&:=(u-\frac{1}{3}f(u,v),...
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Locus of fixed points of an involution in a surface

I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I do not know why the following is true. If S is a minimal smooth projective surface with an involution i....

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