# Questions tagged [minimal-surfaces]

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

104
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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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### 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 ...

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### 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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### '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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### 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\...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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 ...

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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 ...

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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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### 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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### 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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### 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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### 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 $B_2$...

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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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### 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 ...

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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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### 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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### 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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### 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 $n$-...

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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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### 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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### 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 $n$-...

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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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### 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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### 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 ...

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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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### 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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### 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 ...

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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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### 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 &...