Questions tagged [minimal-surfaces]

For questions about minimal surfaces in the sense of Riemannian geometry (as opposed to complex geometry).

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52 votes
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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 ...
Steve McCormick's user avatar
42 votes
1 answer
3k views

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 ...
Michael McLaughlin's user avatar
33 votes
3 answers
5k 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 ...
Joseph O'Rourke's user avatar
23 votes
1 answer
744 views

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 ...
JSCB's user avatar
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16 votes
3 answers
1k 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 ...
elsati's user avatar
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11 votes
1 answer
684 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 ...
monguin's user avatar
  • 225
10 votes
1 answer
457 views

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 ...
Leo Moos's user avatar
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9 votes
2 answers
316 views

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 ...
Denis Serre's user avatar
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9 votes
2 answers
602 views

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 \...
Leo Moos's user avatar
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9 votes
2 answers
448 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, ...
JSCB's user avatar
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9 votes
0 answers
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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{...
Jiří Minarčík's user avatar
9 votes
0 answers
141 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 ...
Onil90's user avatar
  • 823
8 votes
2 answers
368 views

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 ...
Totoro's user avatar
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8 votes
2 answers
288 views

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, ...
Hollis Williams's user avatar
8 votes
3 answers
772 views

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 ...
Leo Moos's user avatar
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8 votes
1 answer
206 views

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 $B_2$...
Joseph O'Rourke's user avatar
6 votes
2 answers
433 views

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 ...
Vladimir S  Matveev's user avatar
6 votes
1 answer
487 views

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. ...
Leo Moos's user avatar
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6 votes
1 answer
497 views

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. ...
Onil90's user avatar
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6 votes
1 answer
369 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 ...
JSCB's user avatar
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6 votes
0 answers
85 views

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 ...
Eduardo Longa's user avatar
6 votes
0 answers
193 views

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 ...
Eduardo Longa's user avatar
6 votes
0 answers
180 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 ...
Stefano's user avatar
  • 105
5 votes
1 answer
209 views

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 $X: \Sigma \to \mathbb{R}^3$? ...
gaoqiang's user avatar
  • 379
5 votes
1 answer
254 views

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 ...
Eduardo Longa's user avatar
5 votes
2 answers
356 views

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 ...
Leo Moos's user avatar
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5 votes
1 answer
393 views

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\...
cokernel's user avatar
5 votes
1 answer
158 views

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 ...
Totoro's user avatar
  • 2,515
5 votes
1 answer
128 views

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 $M$...
Onil90's user avatar
  • 823
5 votes
1 answer
112 views

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, \...
Leo Moos's user avatar
  • 4,968
5 votes
1 answer
181 views

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 $\...
Leo Moos's user avatar
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5 votes
0 answers
50 views

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 ...
Greg Hurst's user avatar
5 votes
0 answers
101 views

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 ...
Eduardo Longa's user avatar
5 votes
1 answer
380 views

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 \...
Leo Moos's user avatar
  • 4,968
5 votes
0 answers
115 views

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 ...
Leo Moos's user avatar
  • 4,968
5 votes
0 answers
127 views

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}...
Leo Moos's user avatar
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5 votes
0 answers
160 views

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 ...
Leo Moos's user avatar
  • 4,968
5 votes
0 answers
294 views

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 \...
Onil90's user avatar
  • 823
5 votes
0 answers
90 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 ...
Hao Chen's user avatar
  • 2,541
4 votes
2 answers
271 views

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) \...
Leo Moos's user avatar
  • 4,968
4 votes
1 answer
442 views

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 ...
Math_tourist's user avatar
4 votes
1 answer
355 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?
Valentino's user avatar
  • 369
4 votes
1 answer
162 views

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 ...
Agelos's user avatar
  • 1,844
4 votes
1 answer
354 views

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 ...
Leo Moos's user avatar
  • 4,968
4 votes
1 answer
115 views

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 $n$-...
user7868's user avatar
  • 620
4 votes
1 answer
352 views

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 ...
Totoro's user avatar
  • 2,515
4 votes
1 answer
260 views

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 ...
Anton Petrunin's user avatar
4 votes
0 answers
198 views

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\...
Elías Guisado Villalgordo's user avatar
4 votes
0 answers
137 views

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 ...
Leo Moos's user avatar
  • 4,968
4 votes
0 answers
88 views

$1$-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 ...
Eduardo Longa's user avatar