Questions tagged [minimal-surfaces]
For questions about minimal surfaces in the sense of Riemannian geometry (as opposed to complex geometry).
121
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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 ...
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answer
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What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?
Note: I asked this on Mathematics SE and even though @TheSimpliFire offered a bounty on it, no-one had a good answer
Find the optimal shape of a coffee cup for heat retention. Assuming
A constant ...
33
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3
answers
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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 ...
23
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1
answer
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Is every minimal hypersurface in $S^n$ algebraic?
Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question
Is every minimal ...
16
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3
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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 ...
11
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1
answer
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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 ...
10
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1
answer
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A counterexample to a conjecture of Lawson
Yau quotes Lawson as having formulated the following conjecture [1]:
Let $M$ be an embedded minimal surface in $\mathbf{S}^3$. Prove that the two domains in $\mathbf{S}^3$ divided by $M$ have equal ...
9
votes
2
answers
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Almgren's regularity Theorem ; a simple example?
Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...
9
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Non-calibrated area-minimising surface
Let $(M^{n+k},g)$ be a Riemannian manifold. Call a surface $\Sigma^n \subset M$ calibrated if there is a closed $n$-form $\omega$ defined on a neighbourhood $U \subset M$ of $\Sigma$ so that $\omega \...
9
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2
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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, ...
9
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Interesting geometric flow of space curves with non-vanishing torsion
Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by
\begin{equation}
\partial_t \gamma = \tau^{-\frac{1}{2}} n,
\end{...
9
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0
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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 ...
8
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2
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Plateau's Problem from an annulus
Let $(M^n,g)$ be a complete Riemannian manifold with bounded geometry, that is, it has bounded curvature and positive injectivity radius. Given two disjoint smoothly embedded homotopically trivial ...
8
votes
2
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Work on triply periodic minimal surfaces
I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, ...
8
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3
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How to interpret this quote of Lin?
I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].
It is a well-known fact that a weakly converging sequence of stationary integral currents may have a ...
8
votes
1
answer
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Minimal surface enclosing two congruent balls
Let $B_1$ and $B_2$ be two unit-radius balls in $\mathbb{R}^3$ whose centers are separated
by a distance $d \ge 2$.
Q. For sufficiently small $d$,
is the minimal area surface enclosing $B_1$ and $B_2$...
6
votes
2
answers
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How many minimal surfaces do we have if the metric in the target space is not flat
It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point,
and any two othogonal vectors in this plane, and any ...
6
votes
1
answer
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What is the current status on bad tangent cones at isolated singularities?
Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
Question.
...
6
votes
1
answer
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Compactness theorem for minimal surfaces
I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here.
Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold. ...
6
votes
1
answer
369
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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
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0
answers
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Convergence of free boundary minimal surfaces
I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...
6
votes
0
answers
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Minimizing area in relative homology class
A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...
6
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0
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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 ...
5
votes
1
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Which punctured Riemann surface are the complex structures of complete minimal surfaces in $\mathbb{R}^3$?
Question: Let $\Sigma$ be a punctured Riemann surface(i.e. a closed Riemann surface with several points removed). Is there always a complete conformal minimal immersion $X: \Sigma \to \mathbb{R}^3$?
...
5
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1
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Manifolds with boundary admitting no closed embedded minimal hypersurface
The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
5
votes
2
answers
356
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Flapping wings: on a question of Kapouleas
The Lawson minimal surfaces $\xi_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [Law70]
these were constructed from geodesic triangulations. An ...
5
votes
1
answer
393
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A question on the monotonicity formula for minimal submanifolds
I'm reading the proof of monotonicity formula from A Course in Minimal Surfaces by Colding-Minicozzi. The theorem says
Suppose $\Sigma^k \subset \mathbb{R}^n$ is a minimal submanifold and $x_0\in\...
5
votes
1
answer
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Complete stable minimal hypersurface in positively curved manifolds
Let $(M^n,g)$ be a complete noncompact orientable Riemannian manifold with positive sectional curvature. Can we find an orientable stable minimal hypersurface $N$ in $M$?
It follows from R. Schoen's ...
5
votes
1
answer
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Stable region of minimal hypersurfaces with finite Morse index
In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):
Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
5
votes
1
answer
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What is the Morse index of the Scherk surfaces?
The Scherk surfaces are properly embedded, complete minimal surfaces
\begin{equation}
S_\alpha \subset \mathbf{R}^3
\end{equation}
that are asymptotic at infinity to the union of two planes $\Pi_1, \...
5
votes
1
answer
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3-manifolds with all minimal surfaces closed
Question. Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\...
5
votes
0
answers
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Approximation of triply periodic minimal surfaces with trigonometric level sets
Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately.
To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...
5
votes
0
answers
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What normal variational vector fields are allowed for the area to be preserved?
Let $(M^3,g)$ be a Riemannian manifold with boundary and let $\varphi : \Sigma \to M$ be a two-sided proper embedding of a compact surface with boundary, with unit normal $N$.
If $\varphi$ is not ...
5
votes
1
answer
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Tangent cones at zero and infinity to minimal surfaces
Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth:
$\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \...
5
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0
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How does the topology of minimal surfaces depend on the radius?
Let $M^n \subset \mathbf{R}^{n+k}$ be a smooth, properly embedded minimal surface, with boundary $\partial M$. The convex hull property states that $M$ is contained inside the convex hull of its ...
5
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0
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Minimal cones and homology spheres
Let $\Sigma \subset \mathbf{S}^{n}$ be a codimension one, embedded minimal surface in the round $n$-dimensional sphere. Let moreover $\mathbf{C} = \mathbf{C}(\Sigma)$ be the minimal cone in $\mathbf{R}...
5
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0
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Singularities of phase interfaces in closed surfaces
Let $(\Sigma,g)$ be a compact surface without boundary. Given $\epsilon > 0$, the $\epsilon$-Allen-Cahn equation is the semilinear elliptic PDE $\epsilon \Delta_g u - \epsilon^{-1} W'(u) = 0$, with ...
5
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0
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On Colding-Minicozzi limit lamination theorem
Colding and Minicozzi proved the following limit lamination theorem (see Theorem 8.26 in "A course in Minimal Surfaces" by Colding and Minicozzi for the complete statement)
THEOREM: Let $\Sigma_i \...
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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 ...
4
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2
answers
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Area-minimising hypersurface with unbounded area growth
Let $T$ be an $n$-dimensional area-minimising hypersurface in $\mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $\mathcal{H}^n(T \cap B_R) \...
4
votes
1
answer
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Curvature estimate for minimal surfaces
I am a bit confused about Theorem 2.16 in the book "A Course in Minimal Surfaces" by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in this paper by Schoen and Simon in the more ...
4
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1
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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
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1
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Is the intersection of such a triple of minimal surfaces in the 3-ball a single point?
Let $S_1,S_2,S_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption)
For each $i$, let $M_i$ denote the minimal surface (i.e. disc) bounded ...
4
votes
1
answer
354
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The minimal surface equation in a Riemannian metric
Let $\Omega \subset \mathbf{R}^2$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is not assumed invariant in the vertical ...
4
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1
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Nearest point is always regular for isoperimetric hypersurfaces
In his paper "Paul Levy's Isoperimetric Inequality" (published as appendix C in Metric Structures for Riemannian and Non-riemannian Spaces), Gromov claims that if $H$ is a minimal $n$-...
4
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1
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Exhaustion of a noncompact manifold by domains with mean convex boundary
Let $(M,g)$ be a complete noncompact Riemannian manifold. Can we find an exhaustion $M=\bigcup_{i \ge 1} U_i$ such that each $U_i$ is a bounded domain with smooth boundary $\partial U_i$ which is mean ...
4
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1
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Minimal graph over convex domain is area-minimizing
I am looking for a reference stating that
If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.
5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in ...
4
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0
answers
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is there a co-Stokes theorem? (with the codifferential)
I am trying to read through J. Simmons, Minimal Varieties in Riemannian Manifolds, and in the proof of Proposition 1.2.2 he calls "Stokes theorem" to the following result:
$$
\int_M\delta\...
4
votes
0
answers
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What are the next-simplest area-minimizing cones?
The simplest area-minimizing, codimension one cones $\mathbf{C} \subset \mathbf{R}^{n+1}$ are the Simons cones. I am trying to understand the behavior of area-minimizing cones a bit better, but these ...
4
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0
answers
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$1$-parameter family of minimal embeddings and the maximum principle
Let $M^3$ be a closed orientable smooth manifold and let $g_t$ be a (smooth) $1$-parameter family of Riemannian metrics on $M$, $t \in \mathbb{R}$. Let $P \subset M$ a fixed closed orientable embedded ...