Questions tagged [minimal-surfaces]

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

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Asymptotic of positive solution to elliptic equation

I am reading the paper "Area minimizing hypersurfaces with isolated singularities" by Hardt and Simon (https://eudml.org/doc/152770) and I get stuck on equation 1.9 on page 106. The ...
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1 answer
124 views

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
176 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 $B_2$...
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5 votes
1 answer
151 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 $\...
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1 vote
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106 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 ...
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4 votes
0 answers
79 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 ...
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9 votes
1 answer
310 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 ...
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1 vote
1 answer
95 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} \, \...
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3 votes
0 answers
86 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 $n$-...
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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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4 votes
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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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3 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 $n$-...
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2 votes
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91 views

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
2 answers
478 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 ...
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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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5 votes
1 answer
201 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\...
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3 votes
0 answers
68 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
111 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 ...
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4 votes
2 answers
227 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) \...
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2 votes
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81 views

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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2 votes
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109 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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5 votes
0 answers
107 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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1 vote
0 answers
83 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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115 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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2 votes
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79 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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2 votes
0 answers
108 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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5 votes
0 answers
142 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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4 votes
0 answers
101 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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2 votes
0 answers
71 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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2 votes
1 answer
145 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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4 votes
0 answers
69 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 ...
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4 votes
2 answers
286 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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5 votes
1 answer
109 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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6 votes
1 answer
200 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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8 votes
1 answer
355 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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3 votes
0 answers
150 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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  • 1,002
4 votes
1 answer
236 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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8 votes
2 answers
227 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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5 votes
1 answer
191 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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2 votes
0 answers
195 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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6 votes
0 answers
63 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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23 votes
1 answer
640 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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2 votes
0 answers
69 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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6 votes
0 answers
129 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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4 votes
1 answer
178 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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4 votes
1 answer
294 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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5 votes
0 answers
261 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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2 votes
0 answers
163 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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3 votes
1 answer
135 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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  • 812
6 votes
1 answer
355 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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