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Questions tagged [minimal-surfaces]

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 it appears in the picture. Now before you jump up…
Mario Carneiro
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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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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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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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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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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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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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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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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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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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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…
japin
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inequality for second fundamental form in distribution sense

I'm trying to read this paper (Schoen-Simon-Yau '74) and I'm struggling to understand the comment to (1.34). (See also here for another question on this paper.) SSY are showing an estimate for the second fundamental form of an immersed…
cesare borgia
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Does minimal submanifolds minimize area locally?

Consider $(\tilde{M},g)$ a riemannian manifold and $M \subset \tilde{M}$ riemannian submanifold. Is it true that if $M$ is a minimal submanifold of $\tilde{M}$ then for every $p \in M$ there exists a neighborhood $W$ of $p$ in $\tilde{M}$ such that…
elsati
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Generalized Jacobi Equation for Minimal Surface Deviation

For background, recall that the Jacobi equation (also known as the equation of geodesic deviation) determines the evolution of the Jacobi field, interpreted as a deviation vector between two "infinitesimally nearby" geodesics. Specifically, let…
Sebastian
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