Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [minimal-surfaces]

The tag has no usage guidance.

7
votes
0answers
87 views

Counter-examples to the higher dimensional statement of the half-space theorem

The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$. The higher dimensional ...
6
votes
0answers
93 views

Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?

In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...
6
votes
0answers
118 views

Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

I have a question for you. BACKGROUND Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical ...
4
votes
0answers
47 views

Foliation of cylinders by constant mean curvature spheres

Let the cylinder $S^{n-1}\times \mathbb R$ be equipped with the standard metric $g$. Suppose that there exists a sequence of metrics $g_{\epsilon}$ on $S^{n-1}\times \mathbb R$ such that $g_{\epsilon}$...
4
votes
0answers
68 views

Is every space group the symmetry group of some triply periodic minimal surface?

I know that there are a lot of TPMS with different symmetry groups. It seems like every space group is the symmetry group of some TPMS. But I can not find a reference that confirms this for all the ...
3
votes
0answers
48 views

Morse index of a closed minimal surface with a small disc removed

Consider the following observation: Let $c_1$ be a geodesic on the unit round sphere $S^2$ with length $2\pi-\epsilon$, where $\epsilon$ is sufficiently small. Then $c_1$ has Morse index one ...
2
votes
0answers
63 views

Normal form of volume functional about a minimal surface

Let $S$ be a closed manifold and $(M,g)$ be a Riemannian manifold. Minimal submanifolds are by definition the critical points of the volume functional $$F: \mathcal{Imm}(S,M) \to \mathbb{R} \qquad \...
1
vote
0answers
48 views

Reference question: $C^1$ estimate for a stable minimal surface

I am looking for an answer/reference to the following question: Let $(M,g)$ be a complete Riemannian manifold and $\Sigma\subset M$ a closed, stable minimal surface. Is it possible to prove $C^1$ ...
1
vote
0answers
53 views

about the compactness of minimal surfaces

If a Caccioppoli set $A$ is of minimal perimeter in every compact set $K$ contained in some open set, can we say that $A$ is of minimal perimeter in the open set? If not, please construct a ...
1
vote
0answers
95 views

Contour of soap film intersection on a rigid doubly curved surface

When soap/shampoo bubble/film enclosing a small pressure in a small volume is formed on a (relatively hard) flat surface or inside parts of a circular cone/funnel, it forms a hemispherical bubble or ...
0
votes
0answers
84 views

What is combinatorially possible, for a singular minimal surface in $\mathbf{R}^3$?

A cubic planar graph gives a cell decomposition of a two-sphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex ...