All Questions
Tagged with geometric-analysis minimal-surfaces
31 questions
2
votes
0
answers
72
views
Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
- 337
4
votes
1
answer
181
views
Is every area-minimizing cone a level set of a least-gradient function?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
- 5,038
3
votes
0
answers
100
views
Asymmetric minimal surfaces in $H^3$
Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 ...
- 337
1
vote
0
answers
67
views
Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
- 5,038
2
votes
1
answer
148
views
Minimal graph with confusing (?) property
Let $n \geq 2$ and $C = \{ (x,y) \in \mathbf{R}^{2n} \mid \lvert x \rvert = \lvert y \rvert \} \subset \mathbf{R}^{2n}$ be the Simons cone. (Whether this is area-minimizing or not does not seem to ...
- 5,038
2
votes
0
answers
207
views
Tangent cones at infinity and the regularity of minimal submanifolds
In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
- 21
1
vote
1
answer
123
views
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 $\Sigma$ ...
- 5,038
1
vote
1
answer
120
views
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 ...
- 5,038
5
votes
1
answer
145
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, \...
- 5,038
2
votes
0
answers
134
views
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 ...
- 5,038
2
votes
0
answers
65
views
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 ...
- 5,038
1
vote
0
answers
40
views
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 ...
- 5,038
2
votes
0
answers
78
views
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 \...
- 5,038
1
vote
0
answers
59
views
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{...
- 5,038
3
votes
0
answers
100
views
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 ...
- 5,038
4
votes
0
answers
192
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 ...
- 5,038
2
votes
0
answers
175
views
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 ...
- 5,038
5
votes
1
answer
194
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 $\...
- 5,038
1
vote
0
answers
236
views
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 ...
- 5,038
10
votes
1
answer
490
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 ...
- 5,038
1
vote
1
answer
196
views
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} \, \...
- 5,038
3
votes
0
answers
102
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 $n$-...
- 5,038
1
vote
0
answers
179
views
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 ...
- 5,038
5
votes
0
answers
120
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 ...
- 5,038
5
votes
0
answers
130
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}...
- 5,038
5
votes
0
answers
165
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 ...
- 5,038
5
votes
0
answers
307
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 \...
- 823
5
votes
1
answer
147
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 $M$...
- 823
3
votes
0
answers
92
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 $C^1$ ...
- 308
4
votes
0
answers
89
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 ...
- 1,630
52
votes
3
answers
2k
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 ...
- 573