Questions tagged [minimal-surfaces]
For questions about minimal surfaces in the sense of Riemannian geometry (as opposed to complex geometry).
104
questions
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Singularities of mean-convex MCF in the sphere?
Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...
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votes
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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 ...
3
votes
0
answers
34
views
Intersection of $n$-dimensional minimal surfaces with two-dimensional planes
Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ ...
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A question related to Ros's two-piece property
In 1995 Ros proved that minimal surfaces in the round three-sphere $S^3$ enjoy a two-piece property: any hyperplane $\Pi \subset \mathbf{R}^4$ divides every minimal surface $\Sigma \subset S^3$ into ...
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3
votes
1
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'Degenerate' tangent point of a minimal graph
Let $u: D_1 \to \mathbf{R}$ be a smooth function defined on the unit disk $D_1 \subset \mathbf{R}^2$ which describes the minimal graph $G$. Suppose that at the origin $G$ is tangent to the horizontal ...
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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 ...
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0
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Are the isoparametric cones the only known examples of minimizing hypercones?
After searching the literature for a long time, it seems to me that the only known examples of area minimizing hypercones are isoparametric cones and their products with $\Bbb R^k$. By isoparametric ...
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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 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, \...
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votes
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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 ...
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3
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179
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Sweeping out the disk: what comes out?
In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...
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Plateau problem in the disk: a question about geodesic nets
Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$:
\begin{equation}
p_1,\dots,p_{2n} \in \partial D.
\end{equation}
We assume that these are all ...
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answers
118
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What prevents spontaneous oscillations in minimal surfaces?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be an unstable minimal cone with an isolated singularity at the origin. Let $\Sigma \subset \partial B$ be its link, and $(\varphi_i)$ be the eigenfunctions ...
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Defining minimality 'through deformations'
Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
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Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
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2
votes
0
answers
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Curvature estimate in terms of the boundary
The curvature of a minimal disc $S^2 \subset \mathbf{R}^3$ can be bounded in terms of the curvature of its boundary via the Gauss–Bonnet formula:
\begin{equation}
\frac{1}{2}\int_S \lvert A \rvert^2 \...
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2
votes
1
answer
203
views
A geometric criterion for uniqueness in the Plateau problem?
Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi_t = \{ z = t\}$ at ...
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vote
0
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Dirichlet-to-Neumann estimate for minimal graphs
Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{...
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3
votes
1
answer
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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 ...
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votes
0
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Is there a simple parametrization of Costa's surface?
The classical parametrization of Costa's surface uses the Weierstrass Zeta function.
Is there a parametrization that uses only elementary functions? The exact surface is not required. A good ...
5
votes
1
answer
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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 $X: \Sigma \to \mathbb{R}^3$?
...
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Are there Lojasiewicz-Simon estimates with boundary?
Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.
Are there Lojasiewicz–Simon estimates ...
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2
votes
0
answers
73
views
How do you construct barriers for minimal surfaces?
There is no comparison principle for minimal surfaces: two minimal surfaces $M_1, M_2 \subset B$ in the unit ball of $\mathbf{R}^3$, with the boundary $\partial M_1 \subset \partial B$ lying 'above' $\...
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2
votes
1
answer
120
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Upper bound on volume growth of area minimizers
Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}...
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votes
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answers
89
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 ...
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vote
1
answer
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A paradox based on Simons cones
Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\...
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3
votes
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answer
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Minimize the area of a maximal surface in R^{2,1} with boundary on the unit sphere
Let $\mathrm{H}$ be the unit sphere in the Minkowski space $\mathbb{R}^{2,1}$ (i.e., a one-sheeted hyperboloid $x_1^2+x_2^2=x_3^2+1$). Assume that $\gamma\subset \mathrm{H}$ is a closed space-like ...
2
votes
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answers
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Why are $S_1,S_2$ oriented boundaries of least area?
I am trying to understand the paper by Bombieri and Giusti on Harnack inequality on minimal surfaces: https://link.springer.com/article/10.1007/BF01418640.
In particular, I am trying to understand the ...
- 131
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votes
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answers
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When are the Schoen-Yau minimal surfaces embedded?
In 1979, Rick Schoen and Shing-Tung Yau published an Annals paper in which they proved the existence of so-called 'incompressible' minimal surfaces in compact Riemannian manifolds.
Question. Under ...
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votes
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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.
...
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3
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 \...
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vote
1
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139
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Minimal surface enclosing balls
(This question is tangentially related to an earlier
question I posed: Minimal surface enclosing two congruent balls.)
Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise
...
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8
votes
1
answer
190
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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 $B_2$...
- 147k
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 $\...
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0
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139
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What does Colding-Minicozzi theory say about convergence with multiplicity?
Let $(M_j \mid j \in \mathbf{N})$ be a sequence of compact minimal surfaces in the unit ball $B \subset \mathbf{R}^3$, with boundaries $\partial M_j \subset \partial B$. Assume that the sequence ...
- 4,536
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votes
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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 ...
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10
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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 ...
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vote
1
answer
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Minimal surfaces with increasing area but bounded Morse index
Question. What restrictions are known on closed manifolds $(M^3,g)$ that contain a sequence of embedded minimal surfaces $(\Sigma_j \mid j \in \mathbf{N})$ with
\begin{equation}
\mathrm{area} \, \...
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3
votes
0
answers
97
views
When is the least-area surface unique?
Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
- 4,536
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vote
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Can min-max be set up around a minimal cone?
Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom.
Question. Given a regular minimal cone $\mathbf{C}$, can one set up a ...
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votes
0
answers
109
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 ...
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4
votes
1
answer
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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 $n$-...
- 610
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Critical points of the area functional restricted to CMC embeddings
For fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,\alpha}$ embeddings $f, f' : M \to N$ are said to be equivalent if there exists $\varphi \in \operatorname{Diff}(M)$ such that $f' = f \...
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8
votes
3
answers
727
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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 ...
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0
votes
0
answers
93
views
Is every minimal graph smooth?
The following result was taken from the book of Gilbarg-Trudinger:
In particular, if the graph is minimal, then $u$ is smooth.
Now comes my question: does the same conclusion hold for graphs over ...
- 2,001
5
votes
1
answer
273
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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\...
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3
votes
0
answers
70
views
Minimal normal graph
Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
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4
votes
0
answers
140
views
Sequence of minimal surfaces with bounded second fundamental form and area
Let $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces ...
- 2,001
4
votes
2
answers
256
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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) \...
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2
votes
0
answers
93
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Intersection of minimal and CMC surfaces
Let $(M,g)$ be a complete and oriented Riemannian 3-manifold without boundary. Given $\Sigma_1$ and $\Sigma_2$ closed surfaces embedded in $M$, where the former is minimal and the latter has CMC $H &...
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