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Questions tagged [minimal-surfaces]

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

308 questions
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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 it appears in the picture. Now before you jump up…
Mario Carneiro
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11
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1 answer

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) + \sqrt{1 + \left(\frac{k}{h}…
David Brock
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11
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2 answers

Is there a minimal graph in $\mathbb{R}^3$ which is not area-minimizing?

Let $\Omega\subset\mathbb{R}^2$ be an open subset such that $\partial\Omega$ is a closed, simple curve. I'm trying to find an example of an $u:\overline{\Omega}\to\mathbb{R}$ such that $\Sigma:=\text{graph}(u)$ is a minimal surface and, yet, there…
rmdmc89
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10
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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 tent roof. What shape/equation does it have…
Narasimham
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10
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1 answer

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 has non-positive sectional curvature. If $f$ is…
Glen Wheeler
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6 answers

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!
Lazywei
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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 $H^2 \leq C$? Context: A polarization of degree $d$ on a…
Tom
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8
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1 answer

Lemma $1.30$ - A course in minimal surfaces by Colding and Minicozzi

This is a lemma which appears in A course in minimal surfaces by Colding and Minicozzi, section $8.1$ The second variation formula on page $39$. Before the lemma, I put some notations of this section. Suppose now that $\Sigma^k \subset M^n$ is a…
George
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7
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2 answers

Derivative of area gives mean curvature?

My lecturer made a comment today about how 'the derivative of the area gives you the mean curvature' but I'm not really sure what he meant? I guess what I mean is that I don't understand this definition: "A surface $M \in \mathbb R^3$ is minimal if…
Sarah Jayne
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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 and its associates (Schwarz P and gyroid). I…
Parcly Taxel
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The Gauß map of a minimal surface: is it holomorphic or antiholomorphic?

I'm reading A survey on classical minimal surface theory, by William H. Meeks and Joaquín Pérez. In the early beginning, they start giving eight definitions of minimal surfaces. The last of them is Definition 2.1.8 A surface $M\subset \Bbb R^3$ is…
Derso
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7
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1 answer

Computing the first variation of volume: all around confusion

$\DeclareMathOperator{\vol}{vol}$I've been working through the computation of the first variation of volume presented in Jost's Riemannian Geometry and Geometric Analysis (page 196 in the sixth edition, section titled: Minimal Submanifolds), and…
pomegranate
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Does this infinite isohedron with the surface topology of the gyroid have a name?

I was experimenting with discretizations of the gyroid minimal surface in Rhino (3d modelling software), and modelled this infinite polyhedron, and wondered what it is called. I didn't find any documentation of it yet, but perhaps I am not…
DPKR
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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 minimisers for higher genera have not been proven, there…
Parcly Taxel
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6
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1 answer

Minimal surfaces

Among the definitions of minimal suraface I found these two: (1) A surface $M\subset\mathbb{R}^3$ is minimal if for any point $p\in M$ there is a neighborhood $U$ of $p$ in $M$ that minimizes the area relatively to its boundary. (2) A surface…
Puzzled
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