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Questions tagged [minimal-surfaces]

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43
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2answers
1k views

What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent ...
32
votes
3answers
3k views

Do bubbles between plates approximate Voronoi diagrams?

For example, soap bubbles:                   Image from UPenn: "A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours ...
10
votes
1answer
377 views

What is the minimal surface connecting two circles that don't lie in parallel planes?

I'm curious about a general answer for oblique planes, but specifically, I'm interested in the case where one circle's axis is perpendicular to the other's, and its center lies on the other's axis. To ...
9
votes
1answer
373 views

Do minimal submanifolds minimize area locally?

A few days ago I asked this question on math.stackexchange: Consider $(\tilde{M},g)$ a riemannian manifold and $M \subset \tilde{M}$ riemannian submanifold. Is it true that if $M$ is a minimal ...
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
1answer
290 views

A possible generalization of the exponential map

Let $M$ be a $n$-dimensional Riemannian Manifold, fix $p\in M$, and $1<k<n$. Do we know if the following is true? For any $k$-dimensional subspace $V$ of $T_p M$, there exists a minimal ...
6
votes
0answers
96 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
1answer
159 views

Are complete minimal submanifolds closed?

Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?
4
votes
0answers
48 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
1answer
179 views

On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite ...
2
votes
2answers
145 views

Mean curvature upper bounds and area, or geodesic curvature upper bounds and length

Let $M$ be a closed manifold with non-torsion $\pi_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean ...
2
votes
0answers
64 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
50 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
1answer
127 views

Is the intersection of two minimal surfaces minimal?

Consider two $n$-dimensional minimal surfaces without boundary. Suppose they are embedded in some $\mathbb{R}^m$ in such a way that they intersect transversally. Is their intersection a minimal ...
0
votes
2answers
74 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
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 ...