# Questions tagged [minimal-surfaces]

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

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### Log resolution and a divisor of pullback of function

Let $(X,x)$ be a three fold singularity
$m_{X,x}$ a ideal sheaf correspoinding to $x$.
$\sigma:Y_1\rightarrow X$ blow up at by $m_{X,x}$
$\phi:Y\rightarrow Y_1$ resolution of $Y_1$
Set $f:=\phi*\sigma$...

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### About the definition of cDV singularity

M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...

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### Finiteness of rational double point

Let $(R,\mathfrak{m
})$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are ...

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### Min-max theory on non-trivial homology class

The min-max theory for minimal surface is developed for the area functional on the space of cycle $Z_n(M)$, producing an unstable minimal surface with area equal to the width. Of course, this is ...

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### Contraction of $(-1)$ curve and extremal ray

I want to prove Castelnuovo's contraction theorem by Mori's contraction theorem.
Question. How can one show that a $(-1)$ curve on a smooth surface is an extremal ray?

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### About the contractability

Let $X\subset \mathbb{P}^3$ be the surface defined by the equation $xy-zw=0$, and consider the curve $E \subset X$ defined by the equation $x=z=0$.
Question. Can $E$ be contracted to a point?

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### Canonical model and the existence of general hyperplane

A variety is a separated reduced scheme of finite type over an algebraically closed field $k$ not necessarily affine.
Let $X$ be a normal variety and $X$ has only one isolated singularity at $x\in X$ ...

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### A property of canonical singularity

Let $X$ be a normal variety with only one singularity at $x$ and $(X,x)$ is a canonical singularity i.e. $(X.x)$ satisfies $(i)$ and $(ii)$.
$(i)$ $(X,x)$ is a $\mathbb{Q}$ Goreinstein singularity.
$(...

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### Reference request The support of $f$-nef divisor

I'm seaching for a proof of the theorem below.
Do you know any reference?
Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...

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### Boundary behavior for submanifolds with bounded second fundamental form

I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with ...

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### Construction of Scherk's surface using soap films

I am currently interested in the differential geometry of minimal surfaces, and I have a rather trivial question regarding Scherk's surface (the one which can be parametrised by the real function $(x,...

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### Well posedness of the Plateau problem under lack of uniqueness

The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not.
Premises
I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...

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### Approximation of triply periodic minimal surfaces with trigonometric level sets

Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately.
To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...

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### Total curvature of a conjugate minimal surface

Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends).
What is exactly meant by a proper annular end? It is an ...

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### Elliptic surfaces with monodromy in Borel subgroup

Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...

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### Does the strong maximum principle for minimal surfaces hold in Riemannian manifolds?

In Euclidean spaces, the following maximum principle for minimal surfaces are well known.
Theorem: If $\Sigma_1$, $\Sigma_2 \subset \mathbb{R}^n$ are complete connected minimal hypersurfaces, $\...

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### 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 ...

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### 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 ...

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### Flat norm of currents and minimal surfaces

Let $A$ be a $k \leq n$ integral current with compact support over $\mathbb{R}^n$ (for conciseness).
Its flat norm $F(A)$ can be defined via
$ F(A) = \inf \{ M(T) + M(S) \, | A = T + \partial S \}$
...

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### 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 ...

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### Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?

Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...

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### 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 ...

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### Harnack inequality for the minimal surface equation

We consider the minimal surface equation $$
(1+|\nabla u|^2) \, \Delta u=\sum_{i,j=1}^n\partial_iu \, \partial_ju \, \partial_{ij}u\quad\hbox{in $B_1\subset\mathbb R^n.$}
$$
If $u\in C^2(B_1)$ is a ...

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### 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 ...

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### Excess function and mean curvature

I'm reading Savin's lecture notes on nonlocal minimal surfaces (available here) and he defines what he calls the excess function of a smooth set $E$ with $0\in\partial E$ by$$e(r)=\frac{\int_{\partial ...

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### Almgren's regularity Theorem ; a simple example?

Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...

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### Minimal surface on $R^3$ with with non Euclidean metric

I am wondering if there is an analog of the following theorem by Morgan and White:
Suppose $g_E$ is the Euclidean metric on $\mathbf R^3$. If $\gamma$ is a closed $C^{k,\alpha}$ curve in $(\mathbf R^3,...

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### Applications of maximal surfaces in Lorentz spaces

I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces.
I can clearly see the mathematical motivations. But I wonder if zero-...

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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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### 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' $\...