# Questions tagged [minimal-surfaces]

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

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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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### 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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### 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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### 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 ...