Questions tagged [minimal-surfaces]

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

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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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92 views

Why are products of spheres integrable?

Let $n+1 \geq 3$ be an integer and $p + q = n$. Inside the unit sphere $\mathbf{S}^{n+1}$ the product \begin{equation} \mathbf{S}^p(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n}) = \{ (X,Y) \in \mathbf{R}...
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84 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}...
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79 views

Finding metric to make a submanifold minimal

I would like to ask a question, which I am not sure if there exist some known results (that I could not locate). Given a smooth embedded submanifold $M$ in $\mathbb R^n,$ is it always possible to find ...
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71 views

Which stationary varifolds have non-integer density?

A central object in geometric measure theory are the generalised, and weakly defined minimal surfaces called stationary varifolds. Let me recall some definitions. Given an open subset $U \subset \...
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73 views

Examples of bundles with minimal fibers

There is a result of Chen https://link.springer.com/article/10.1007/s00605-007-0451-y A Riemannian submersion $\pi : F \hookrightarrow (E,g) \to B$ with minimal fibers $F$ and such that $g$ has ...
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99 views

Extensions of minimal hypersurfaces

Let $B \subset \mathbf{R}^{n+1}$ be the unit ball, and $M \subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We ...
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119 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 ...
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90 views

Umbilic points of minimal hypersurfaces and distributional Simons inequality

Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel ...
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70 views

Is this family of minimal tori compact?

Let $\Sigma$ be a smooth $2$-sphere and let $M = \Sigma \times \mathbb{S}^1$. Fix an integer $n \geq 0$. Is there generic set $\mathcal{S}$ of Riemannian metrics on $\Sigma$ such that the following ...
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1answer
112 views

Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold

Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and ...
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66 views

Infinitely many distinct minimal tori

Let $M = \Sigma_g \times \mathbb{S}^1$ be endowed with the product metric, where $\Sigma_g$ is a compact orientable surface of genus $g$ with an arbitrary fixed metric. Is it true that there are ...
4
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1answer
179 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 ...
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70 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 ...
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1answer
174 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 ...
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1answer
263 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 \...
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127 views

Does the minimal surface system in the plane have the weak unique continuation property?

Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
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1answer
158 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 ...
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2answers
157 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, ...
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1answer
148 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 ...
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25 views

Outer Minimizing Horizon for a Perturbed Metric

I am reading this paper which proves the Riemannian Penrose inequality in general relativity. On pages 10 - 11, it is stated that the Riemannian $3$-manifold $(M^3, g_t)$ has a strictly outer ...
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173 views

Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions

In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := \Delta_g + |A|^2$ (usually called Jacobi ...
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55 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 ...
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1answer
589 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 ...
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67 views

Embeddedness and homology of a limit of minimal surfaces

Consider the following theorem, proved in this paper: Theorem (Theorem 6.1). Suppose we have a sequence $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal $...
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90 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 ...
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1answer
140 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 ...
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1answer
219 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 ...
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217 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 \...
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159 views

Special Riemannian metric on the product

Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
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1answer
121 views

maximal surface, parametric approach

I am interesting in maximal surfaces: space-like surface in Minkowski $\mathbb{R}^{2,1}$ (or De Sitter $dS^3$). Of course space-like implies locally graph and almost all the literature is interested ...
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1answer
282 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. ...
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59 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 $M$...
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187 views

Can this integral be made nonpositive?

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
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166 views

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{...
3
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1answer
101 views

Stability of minimal hypersurface with flat directions

Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a minimal complete hypersurface. Let's consider $\bar{\Sigma} := \Sigma \times \mathbb{R}^k \subseteq \mathbb{R}^{n+k +1}$. Clearly $\bar{\Sigma}$ is a ...
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1answer
261 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?
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69 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 $C^1$ ...
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1answer
552 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 ...
8
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1answer
254 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, ...
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75 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 ...
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78 views

about the compactness of minimal surfaces

If a Caccioppoli set $A$ is of minimal perimeter in every compact set $K$ contained in some open set, can we say that $A$ is of minimal perimeter in the open set? If not, please construct a ...
2
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1answer
243 views

On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite ...
8
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111 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 ...
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2answers
102 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
6
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1answer
333 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 ...
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1answer
251 views

Is the intersection of two minimal surfaces minimal?

Consider two $n$-dimensional minimal surfaces without boundary. Suppose they are embedded in some $\mathbb{R}^m$ in such a way that they intersect transversally. Is their intersection a minimal ...
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2answers
195 views

Mean curvature upper bounds and area, or geodesic curvature upper bounds and length

Let $M$ be a closed manifold with non-torsion $\pi_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean ...
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0answers
102 views

Contour of soap film intersection on a rigid doubly curved surface

When soap/shampoo bubble/film enclosing a small pressure in a small volume is formed on a (relatively hard) flat surface or inside parts of a circular cone/funnel, it forms a hemispherical bubble or ...
8
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1answer
626 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 ...