Questions tagged [minimal-surfaces]

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

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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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Scherk minimal surface

I want to show that the Scherk surface is minimal. I found some results in witch the coefficients of the first and second fundamental form are calculated directly.I tried to calculate and I don't get ...
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2 answers
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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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Convergence of minimal surfaces

I'm studying the proof of the Positive Mass Theorem given by Schoen-Yau ("On the Proof of the Positive Mass Conjecture in General Relativity"), but there's a detail I cannot understand. When ...
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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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Deformation of minimal surface actually decreases area

I'm working on an exercise from Kristopher Tapp's Differential Geometry of Curves and Surfaces that's asking me, for a compact regular surface $\sigma$ with orientation $N$ and Gaussian and mean ...
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Using the Divergence theorem to show that a graph has least area among all surfaces with the same boundary

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(u):=\{(x,y,u(x,y))\,:\,(x,y\in\...
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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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Minimal immersion of projective plane in sphere

Problem is to build minimal immersion of $RP^2$ with canonical metric of curvature $K=1$ in $S^n$ of some radius, using first eingefunctions. From Takahashi theorem it is clear that radius of sphere ...
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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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Is there a theory for minimal weighted surfaces?

There is a rich theory around minimal surfaces, i.e. the critical points of the area functional $$A(X) = \int \sqrt{g(u)}d^nu$$ with $g(u):= det(g_{ij}(u))$ and $g_{ij} = \left(\frac{\partial X}{\...
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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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4 votes
1 answer
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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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2 votes
2 answers
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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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Is the Goldschmidt solution a minimum, or saddle point?

Consider minimizing the surface area of the surface spanning two parallel rings. It is well known that the global minimizer is either a catenoid, or the so-called Goldschmidt solution consisting of ...
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Projective plane does not minimally immerse into $S^3$

Can someone please give me a reference (or a proof) of the fact that the projective plane cannot be minimally immersed into the 3-sphere? My reference is: Lawson, “Complete Minimal Surfaces in $\...
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How to show that a surface is minimal

I am reading Differential Geometry of Curves and Surfaces from M.P. Do Carmo. In the chapter about minimal surfaces he says that, (1) A regular parametrized surface is called minimal if its mean ...
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The Gauss map is angle preserving for minimal surface

Let x:U $\rightarrow \mathbb{R}^3$ a be angle-preserving parametrization of a minimal surface with gauss curvature K(p) $\neq$ 0. Then, the unit normal vector field $N_{\textbf{x}}$:U$\rightarrow$ S$^...
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Is there a "simple" factoring of Costa's minimal surface piercing a compact surface?

I've only just heard of Costa's minimal surface. I'm interested in the fact that it may be formed by puncturing a compact surface. Another, more simple, minimal surface is the Catenoid. This also may ...
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Uniqueness of a Complete Minimal Surfaces

Let $R$ be a complete Riemannian Manifold of dimension $n$, let $S \subseteq R$ be homeomorphic to the sphere of dimension $k < n$, and let $M$ be a minimal surface of $R$ which is homeomorphic to ...
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Lifting Minimal Surfaces

Let $f(z)=z+\frac{\overline{z}}{4}$ where $\left|z\right|<1$. Does the function lift to minimal surface? $h(z)=z$ , $g(z)=\frac{z}{4}$ since $f(z)=h(z)+\overline{g(z)}$ so $h\text{'}(z)=1$ and $g\...
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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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3 votes
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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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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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1 answer
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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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2 votes
1 answer
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Proof of a theorem on Plateau's problem in "Elementary Differential Geometry" by Andrew Presley

I'm studying the proof of this theorem on Plateau's problem in the textbook "Elementary Differential Geometry" by Andrew Presley, but I don't understand, being more specific, why $\beta^0=\...
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1 vote
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Notion of stability for minimal surfaces

A quote from Danny Calegari's lecture notes about minimal surfaces: "A critical point for a smooth function on a finite dimensional manifold is usually called stable when the Hessian (i.e. the ...
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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) ...
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2 votes
1 answer
39 views

Is there a unique minimal surface with boundary $S_2$?

Take a loop of wire ($S_2$), twisted into a complicated shape. One can dip the wire in soap to form a minimal surface soap film. Is there a unique minimal surface or can there be more than one ...
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2 votes
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Exercise 10 of Colding-Minicozzi's minimal surfaces book

I am doing exercise 10 of Colding-Minicozzi's minimal surfaces book which goes as Find a sequence of functions $u_j:D_1\to\mathbb{R}$ on the unit disk $D_1$ satisfying the minimal surface equation ...
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Weak Sobolev convergence in Mobius group of disk to a constant.

Let $\Gamma \subset \mathbb{R}^3$ be a Jordan curve, $D^2 \subset \mathbb{R}^2$ be the unit disk and define: $$\mathcal{F} = \{u \in W^{1,2}(D^2, \mathbb{R}^3) \ \vert \ u \vert_{\partial D} \ \text{...
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Why constant mean curvature surfaces are isothermal surfaces?

From the Encyclopedia of Mathematics website, I read that constant mean curvature (CMC) surfaces, in particular minimal surfaces, are isothermal surfaces, i.e. a surface whose curvature lines form an ...
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1 vote
1 answer
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Is there a connection between minimal surfaces and tree graphs?

For any connected graph $G=(V,E)$, we must have $|E|\geq|V|-1$. Any edges less, and the graph wouldn't be connected. It's sorta like saying a tree graph is a "local minimum" in the space of ...
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1 vote
1 answer
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minimal graphs converge to a harmonic function

Suppose $\{u_i\}$ is a sequence of positive $C^2$-functions on the unit ball $B,$ satisfying the minimal surface equation, i.e., $${\rm div}\left(\frac{Du_i}{\sqrt{1+|Du_i|^2}}\right)=0$$ for all $i.$ ...
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How to show the second fundamental form of a flat minimal surface is zero?

The surface is flat if its Guassian curvature vanishes everywhere. The surface is minimal if ist mean curvature vanishes everywhere. Therefore, for a function f(x,y), its corresponding surface's ...
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Can a graph of a function $\xi$ with non-positive curvature have 5 connected components in which $\xi \neq 0$?

The following problem arises from some trouble I've had regarding a paper by E. Hopf. We have a function $\xi(x,y)$ such that: $\xi$ is of class $C^2$ in $\mathbb{R}^2$. $\xi_{xx}\xi_{yy} - \xi_{xy}^...
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Simplifying $F(\sin^{-1}\sqrt{2/(2-p)},1-p^2/4)$ (for a minimal surface)

The tetragonal distortion of the Schwarz P minimal surface has a Weierstrass–Enneper parametrisation of $g(w)=w,f(w)=(w^8+\lambda w^4+1)^{-1/2}$ and a Bonnet angle of $\pi/2$ where $\lambda<-2$. ...
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2 votes
1 answer
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Inverting a function in the parametrisation of Riemann's minimal surface

While writing code to render Riemann's minimal surface – which is now available here as part of my Malibu STL surface generator – the following problem came up. Consider the following function: $$y(\...
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5 votes
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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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4 votes
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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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