Questions tagged [minimal-surfaces]
Question on minimal surfaces, or surfaces that have zero mean curvature.
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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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Marginally Outer Trapped Surface as a Minimal Surface
In general relativity, a marginally outer trapped surface is defined to be a spacelike, two-dimensional surface in a space-time such that outgoing null rays perpendicular to the surface are do not ...
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Verifying a Scaled Variable Function Satisfies the Minimal Surface Equation
Let $u = u(x, y)$ be a known solution of the minimal surface equation
$$(1 + u_y^2) u_{xx} + (1+u_x^2) u_{yy} - 2 u_x u_y u_{xy} = 0$$
Now, say $$v(x, y) = \frac{1}{\lambda} u(\lambda x, \lambda y)$$
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Minimal surfaces, but 1 dimension lower
There is a set of points. For simplicity, let's say that those are 2D points (although the problem works in more dimensions as well). The goal is to find the minimum possible length of a connected 1-...
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Why is the minimal surface of revolution not a cylinder?
I'm trying to find the curve $r=f(z)$ that links two parallel circles of same radii, aligned on the same axis $ (Oz) $, and that minimizes the surface of revolution between both circles.
The solution ...
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Lemma 1.34 in Colding-Minicozzi
Recall the stability operator $L=\Delta_\Sigma+Ric(N,N)+|A|^2$ for a minimal hypersurface $\Sigma^n\subset M^{n+1}$. Stability of $\Sigma$ is equivalent to non-negativity of the Rayleigh quotient $$\...
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Is any minimal hypersurface also a minimal submanifold?
Recently I learned that a minimal submanifold is one whose mean curvature vector $\vec{H}$ vanishes identically, which makes me wonder whether a minimal hypersurface qualifies as a minimal submanifold ...
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Lemma 1.19 in Colding-Minicozzi
The statement is as follows. See page 30 of CM or below for an image of the proof.
If $u:\Omega\subset \mathbb{R}^2\to \mathbb{R}$ is a solution to the minimal surface equation, then for all ...
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Is the surface $z=\ln \cos x-\ln \cos y$ minimal?
Is the surface $z=\ln \cos x-\ln \cos y$ minimal?
I found the partial derivatives:
$$\frac{\partial z}{\partial x}=-\mathrm{tg}x,\frac{\partial z}{\partial y}=\mathrm{tg}y,\frac{\partial ^2z}{\partial ...
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Does any compact boundary become a minimal hypersurface iff each of the boundary component is a minimal hypersurface?
There are researchers who made the following assumptions in their paper on the Penrose inequality:
Let $M$ be a complete, connected Riemannian $3$-manifold and suppose that the boundary $\partial M$ ...
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Why $\langle x_i,x_{jk}\rangle=0$ holds true?
I am reading the paper "Minimal Immersions of 2-Manifolds into Spheres". In (1.1), page 379, the author wrote that $\langle x_i,x_{jk}\rangle=0$. I don't know why this equality holds true ...
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How to calculate the mean curvature of clifford torus immersed in the standard sphere $\mathbb{S}^3$?
We know that clifford torus is a minimal surface immersed in $\mathbb{S}^3$, but how to express this immersion in parametrization and how to calculate the mean curvature of this immersion? If anyone ...
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Are minimal area and mean curvature definitions of minimal surfaces really equivalent?
I am reading Lecture Notes on Minimal Surfaces, which can be accessed here and I am trying to find a proof of the following:
Mean curvature of a surface $S$ is zero everywhere if and only if
surface $...
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How can the boundary of a Riemannian $3$-manifold consist of minimal surfaces?
I'm studying a research paper, and there is a theorem whose statement looks very strange to me. In 2001, Gerhard Huisken and Tom Ilmanen proposed a theorem in their collaborative work THE INVERSE MEAN ...
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Energy Function Expansion in Deform by Laplacian Coordinates
I am currently working on the strain energy function for a particular graph.
The paper I am currently referencing is "Spatial Relations Preserving Character Motion Adaptation".
I am asking ...
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reference on minimal surfaces
I'm looking for an existence theorem of minimal surface.
In a paper by L. Andersson and J Metzger, 'The Area of Horizons and the Trapped Region', they used Jang's equaiton to prove existence of stable ...
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About the variation of volume for submanifolds in Jost's book
I'm reading Jost's Riemannian Geometry and Geometric Analysis and having some questions about the variation formula for submanifolds (page 242 in 7-Ed).
For a local variation $\Phi: M \times (-\...
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what is the size of the smallest box that includes two ellipses/elipsoids of the same center?
How to find the size of the smallest box that includes two ellipses of the same center but arbitrarily tilted ? in $n$ dimensions.
The maximum is $(2b_1\times2b_2 \times ... \times 2b_n)$, but Im ...
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Existence of a minimal cross sectional sphere in $S^2\times [-1,1]$.
Suppose that $(M,g)$ is a Riemannian manifold with $M\approx S^2\times [-1,1]$ and where $g$ is a metric where the area of the `ends' $S^2\times \{-1\}$ and $S^2\times \{1\}$ have "large" ...
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Trying to verify saddle surface is a minimal surface
I am reading a book on calculus of variation, and it talks about minimal surface, it gives the equation of minimal surface of $z(x,y)$, it should satisfy:
$$(1 + z_y^2)z_{xx} - 2 z_x z_y z_{xy} + (1+...
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Normals at the end
I am reading a research paper "Embedded minimal surface with an infinite number of ends" by Callahan etc. In this paper they said "Since we are interested in embedded surfaces, the ...
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Minimal spheres in rank one symmetric spaces
Let $M^n$ be a compact rank one symmetric space endowed with its canonical metric. For a given point $p \in M$, does there exist a positive number $r$ such that the sphere
$$S_r(p) = \{ x \in M: d(x,p)...
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Why are $S_1,S_2$ oriented boundary of least area?
I am trying to understand the paper by Bombieri and Giusti on Harnack inequality on minimal surfaces: https://link.springer.com/article/10.1007/BF01418640.
In particular, I am trying to understand the ...
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Should minimal hypersurface have nonpositive scalar curvature?
Let $\Sigma^n$ be an embedded minimal hypersurface of an orientable Riemannian manifold $M^{n+1}$, then the mean curvature $H$ of $\Sigma$ vanishes everywhere. Let $\lambda_1,…\lambda_n$ be those ...
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Book for Schwarz P and D surfaces
Is there any english translation of the book "Gesammelte Mathematische Abhandlungen" by H. A. Schwarz? I am looking for book which has explained Schwarz P, Schwarz D triply periodic surface. ...
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Intrinsic and extrinsic distance on minimal submanifolds
Let $\Sigma\subset \mathbb R^n$ be a $k$-dimensional complete minimal submanifold. Let $d(x,y)$ be the intrinsic distance between $x,y\in \Sigma$ and $|x-y|$ be the Euclidean distance. Let $y\in\Sigma$...
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Questions about J-holomorphic curves as minimal surfaces
I am a beginner in symplectic geometry. The questions I ask may be so trivial.
In McDuff's "What is Symplectic Geometry?", she writes:
We saw earlier that the symplectic area of a surface ...
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Unit normal vector field to a surface of revolution and its second fundamental form
Suppose $\Sigma^2\subset \mathbb R^3$ is a surface of revolution, i.e. we consider \begin{align}I &\xrightarrow{(r,h)} \mathbb R^+\times\mathbb R\\ t&\mapsto (r(t),h(t))=:(r,h)\end{align}
And ...
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How to Visualize a compressible surface in 3-manifold $M$
I am now reading 'A Course in Minimal Surfaces' written by Tobias Colding & William Minicozzi with helping the following lecture. The lecturer explains an essential sphere during the lecture but I ...
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Using integration by parts to show $\int_{\Sigma} |\nabla^N_{\Sigma} X|^2 = - \int_{\Sigma} \langle X, \Delta^N_{\Sigma} X \rangle$
I'm trying to work through a derivation of the stability operator from minimal surface theory. Suppose $\Sigma^k$ is a minimal submanifold of $\mathbb{R}^n$, and suppose $X$ is a normal vector field ...
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An affine invariant notion of minimal surface?
The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $...
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Gauss Bonnet for locally outermost closed minimal surface in Hawkings Theorem
So I'm trying to understand the proof for the following theorem from Lan-Hsuan Huang: "Trapped Surfaces, Topology of Black Holes, and the Positive
Mass Theorem":
Any orientable locally ...
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Bounding diameter of the arc of a closed curve
I was reading chapter 4 of Colding and Minicozzi's A Course in Minimal surfaces and I came across a statement in the proof of Lemma 4.14:
Suppose $\Gamma\subset\mathbb{R}^3$ is a simple closed curve ...
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Continuity of the curve shortening process
I'm studying the shortening process, introduced in the book Course in Minimal Surfaces by T. Colding and W. Minicozzi, which is inspired by the Birkhoff's curve shortening process.
In the book, is ...
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Minimal surface and Gauss map
I am struggling for an exercice in differential geometry :
S is a minimal surface if and only if the Gauss map $N : S \rightarrow S^2$ satisfies for all $p \in S$ and all $\omega_1, \omega_2 \ T_p(S)$ ...
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The definition of sweepout
A continuous map $\sigma:S^1 \times [0,1] \to M$ is called a sweepout on $M$, if
For each $t$ the map $\sigma(\cdot,t)$ is $W^{1,2}$;
The map $t \to \sigma(\cdot,t)$ is continuous from $[0,1]$ to $W^{...
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Existence of surface with given boundary curve and prescribed conormal
Suppose $\Gamma\subset\mathbb{R}^3$ is a smooth, embedded, closed curve and suppose $n(p)\in N_p\Gamma$ is a smooth vector field on $\Gamma$ and normal to $\Gamma$. Does there always exist a surface $\...
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Using the Divergence theorem to show that a graph has least area among all surfaces with the same boundary
$\DeclareMathOperator{\graph}{graph}$I want to show that for $\Omega\subset\mathbb{R}^2$ an open bounded regular domain with a $C^2$ function $u:\overline{\Omega}\to\mathbb{R}$ that the graph $\graph(...
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Almgren and Taylor's movie about soap bubbles
As I know, in 1970's two well-known geometric analysts, Fred Almgren and Jean Taylor, along with mathematician Michele Emmer, produced a film about minimal surfaces entitled "Soap Bubbles". ...
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Liouville's equation is equivalent to the Gauss–Codazzi equations for minimal immersions into the 3-space
I was going through this wiki page Link on the Liouville's equation. In that page there is this comment without any source 'Liouville's equation is equivalent to the Gauss–Codazzi equations for ...
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Area of minimal submanifold in $S^3$
I am asked to prove that lower bound of area of compact minimal submanifold with no boundary in $S^3$ is $4\pi$. Only idea i have is Gauss equation using orthonormal frame $e_1,e_2$. We have
$$1=K+|...
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Is functional $\int_0^a \left( (u')^2 - u^2 \right) {\rm d} x$ convex?
I posted a question here and I think I solved the point two of the question. Is my approach right?
Give that the functional
$$I[u] = \int_0^a \left( (u')^2 - u^2 \right) {\rm d} x$$
is homogenous, i....
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Asymptotic of positive solution to elliptic equation
I am reading the paper "Area minimizing hypersurfaces with isolated singularities" by Hardt and Simon (https://eudml.org/doc/152770) and I get stuck on equation 1.9 on page 106.
The ...
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Non-trivial Examples of Surfaces of Voss
Let $I$ and $J \subset \mathbb{R}$ be two intervals of the real line. A smooth parametrized immersed surface $\sigma: I\times J \rightarrow \mathbb{R}^3$ is called a surface of Voss if its coordinate ...
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Mean curvature vector in do Carmo's Riemannian Geometry
Pictures below are from do Carmo's Riemannian Geometry. The third picture is from the 133th page.
First, I want to show the mean curvature vector does not depend on the chosen frame $E_i$. But I don't ...
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Uniqueness of Bounded Minimal Surfaces
Say you have a complete Riemann manifold $R$ and a complete minimal hypersurface $M$ of dimension $n$ embedded in $R$. Say we remove all but but a patch $P$ from $M$ where $P$ is topologically $B^n$. ...
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