Questions tagged [minimal-surfaces]
Question on minimal surfaces, or surfaces that have zero mean curvature.
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Help understanding smoothness classes. What exactly are the conditions to be a twice continuously differentiable, closed surface?
I know that a $C^2$ surface means the 2nd partial derivatives exist and are smooth, but I'm a bit unfamiliar with math notation/colloquia. I want to make sure I understand correctly.
Say we have an ...
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Inequality regarding minimal perimeter sets
My professor was teaching out of Connor Mooney's notes on Minimal Surfaces and I missed a couple classes and am trying to catch up and I am having trouble understanding the proof of theorem 5.12 (...
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Holomorphic one form and its association to Minimal surface
I am new to studying holomorphic one form. While studying minimal surface, it is written that in The Weierstrass-Enneper Representation $\mathbf{X}(z)=\Re\int \left( \frac{1}{2} (1 - g^2) \, dh, \frac{...
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What is the Equation for the Batista-Costa Minimal Surface?
The Batista-Costa surface is a triply periodic minimal surface. Three photos of part of the same surface are below:
where the first two were taken form the research paper: The New Boundaries of 3D-...
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Minimal surface stretched over four points
Let $x_0, \ldots, x_3 \in \mathbb{R}^3$. Let $C$ be a pathwise affine curve connecting these points in order $x_0 \to x_1 \to x_2 \to x_3 \to x_0$. (That is, $C$ consists of four segments that connect ...
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Proof that the solution to Plateau's problem for two concentric circles is a surface of revolution
Plateau's problem for concentric circles
Visualisation figure set (will be refered to with fig.1-5)
Given two concentric circles $C_1$ and $C_2$ in $\mathbb{R}^3$ with the center points $(x_1,0,0)$ ...
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Why the local mass-ratio in a complex submanifold of the Euclidean space is nonzero? [closed]
Okay, I'm not very comfortable with either complex geometry or minimal surfaces, so bear with me. I've encountered the following theorem, but I can't find a proof:
Suppose $V\subset \mathbb{C}^n$ is a ...
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Regarding first variation of area functional for a regular surface
I have been reading Dierkes-Hildebrandt's, "Minimal Surfaces" recently and I came across this proof for showing that a minimal surface has zero mean curvature at all points. I had a doubt ...
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Second fundamental form of minimal surface under Weierstrass representation.
Consider the Weierstrass representation $$f(z)=\text{Re}\int ((1-g^2),i(1+g^2),2g)\omega$$, where $g(z)$ is a meromorphic function and $\omega(z)$ is a holomorphic 1 form.
I'm trying to derive the ...
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Intuition of minimal surfaces in the class of sets of locally finite perimeter.
I'm reading the book Minimal Surfaces and Functions of Bounded Variation by Enrico Giusti. I'm wondering if anyone could help me understand the intuition of the theorem below.
We define
$$\...
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asymptotic directions are perpendicular then the mean curvature is zero?
I just did exercise 7 from Manfredo's section 3.2, which says that if the mean curvature is zero at a non-planar point then the asymptotic directions are perpendicular.
My question is: Is the converse ...
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Computing minimal surface numerically
I am interested in computing a minimal surface on a domain $\Omega = [0,1]^{2}$. Specifically, I would like to fit some Ansatz function $z(x, y) = ax^{2}y^{2} + b x y + c x + d y$ with parameters $a, ...
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What does it mean that the helicoid can 'glide' over itself?
In the Wolfram definition it says
The helicoid is the only non-rotary surface which can glide along itself.1
1Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231-232, 1999.
I ...
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On Schoen and Yau's proof of the positive mass theorem: extracting a minimal surface in the limit
I'm reading Schoen and Yau's 1979 paper on the Positive Mass theorem. I'm having trouble understanding the proof of how they extracted a minimal surface as the limit of solutions to the Plateau ...
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The distribution of normal vectors of minimal surfaces
Is there a minimal surface such that its normal vectors are distributed everywhere on the unite sphere?
I've got the results above:
Let $S$ be a complete regular minimal surface in $\mathbb{E}^3$. ...
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