Questions tagged [minimal-surfaces]
For questions about minimal surfaces in the sense of Riemannian geometry (as opposed to complex geometry).
146 questions
0 votes
1 answer
100 views
In what sense is the minimal surface system elliptic?
Let $F:\mathbb{R}^{n\times k}\to \mathbb{R}$ be given by $P\mapsto \sqrt{\rm{det}(I+P^tP)}$. Is $F$ rank-one convex? The reason for this question is that I want to know whether, in general codimension ...
- 1,580
0 votes
0 answers
93 views
Confusion on Min-Max theory minimal surfacce
I am reading some introduction on Min-Max theory and was confused on some of the following points: The theory can be done on the space of current in either $\mathbb{Z}$ or $\mathbb{Z}_2$ coefficients....
- 147
4 votes
0 answers
197 views
Is $\|V\|(\partial B_\rho)$ always 0 for a stationary varifold $V$?
Let $V=\underline v(M,\theta)$ be a stationary integral $n$-varifold in $\Bbb{R}^{n+k}$ where $M$ is the $n$-rectifiable set of $V$ and $\theta$ is the multiplicity function. We write $\|V\|=H^n\...
- 191
1 vote
1 answer
234 views
Is there a representation for the solutions of the complex valued Liouville equation?
The Liouville equation $\Delta w=ke^{2w}$, where $\Delta$ is the usual Laplacian in $\mathbb{R}^2$, when $w$ is real valued function, has a well known representation in terms of meromorphic functions, ...
2 votes
1 answer
133 views
Can the asymptotic distribution on a minimal surface be generated by the gradient of a function?
Let $S$ be a minimal surface in $\mathbb{R}^3$ and choose a unit normal $N : S \to \mathbb{S}^2$ for $S$. The second fundamental form of $S$ at a point $p \in S$ is the bilinear form given by $$A_p(x, ...
- 2,127
5 votes
1 answer
408 views
Picard number of surfaces of general type
Let $X$ be a smooth complex surfaces of general type, with $3c_2(X)=c_1(X)^2$ i.e. $X$ is a quotient of the complex ball $\mathbb{B}^2$. Example. Fake projective planes, i.e. smooth complex projective ...
- 918
4 votes
0 answers
187 views
How to understand Douglas' solution to the Plateau problem?
Douglas' solution to the Plateau problem in $\mathbb{R}^3$ roughly relies on minimizing the $A$ functional $$A(g)=\frac{1}{16\pi}\int_{S^1\times S^1}d\theta d\phi \dfrac{\|g(\theta)-g(\phi)\|^2}{\sin^...
- 151
6 votes
1 answer
244 views
Stable minimal surface whose Gauss curvature is not everywhere positive
Let $(M, g)$ be a 3-dimensional Riemannian manifold with positive scalar curvature, and let $\Sigma\subset M$ be a compact minimal surface without boundary. It is known that (e.g. by the work of ...
- 339
2 votes
0 answers
76 views
Compactness of Mean Convex Surfaces
Suppose we have a sequence of connected hypersurfaces, $\{Y_i^{n-1}\}$, inside a riemannian manifold $(M^{n}, g)$ such that $C> H_{Y_i} > c > 0$ for all $i$, and $||H_i||_{C^{1,\alpha}(Y_i)} \...
- 453
3 votes
0 answers
110 views
Which function does this cross section optimize?
I often notice tankers of the type illustrated in the figure below. The cross section is neither circular nor elliptical. Is it a "notable" geometric shape? Which function or property does ...
- 285
4 votes
0 answers
148 views
References on smoothness of minimal surfaces in Riemannian manifolds
It's well known that $C^1$ minimal surfaces (surfaces that are locally area minimzing) in $\mathbb{R}^n$ are automatically smooth, and one can prove this result by solving the Dirichlet problem of the ...
- 460
7 votes
1 answer
329 views
Mass minimizing current in real homology class
It is a well-known results by Federer and Fleming that there exists at least one mass-minimizing normal current in every real homology class of a closed $n$-dimensional Riemannian manifold $M$. Their ...
- 71
2 votes
1 answer
395 views
Jacobian fibration of elliptic fibration: basic relations between Enriques invariants
Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic ...
- 3,884
3 votes
1 answer
201 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}(\...
- 453
1 vote
1 answer
121 views
When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?
Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be ...
- 111
2 votes
1 answer
192 views
Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if $(i)$ it it is $\mathbb{Q}$-Gorenstein. and $(ii)$For any resolution of singularity $F:Y\rightarrow X$, $K_Y-f^*K_X>...
- 368
1 vote
0 answers
146 views
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 ...
- 368
2 votes
0 answers
146 views
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 ...
- 368
4 votes
1 answer
399 views
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 ...
- 147
2 votes
1 answer
183 views
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?
- 368
5 votes
1 answer
452 views
Contractibility of a curve on a surface
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?
- 368
2 votes
0 answers
143 views
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$ ...
- 368
2 votes
1 answer
212 views
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. $(...
- 368
2 votes
0 answers
127 views
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 ...
- 368
1 vote
0 answers
111 views
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 ...
- 191
2 votes
1 answer
271 views
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,...
anon
1 vote
1 answer
307 views
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 ...
- 6,830
8 votes
1 answer
411 views
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 ...
- 451
1 vote
0 answers
100 views
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 ...
- 11
1 vote
0 answers
70 views
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 $\...
- 171
3 votes
1 answer
508 views
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, $\...
- 460
4 votes
1 answer
249 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 ...
- 5,163
3 votes
0 answers
132 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 ...
- 453
3 votes
2 answers
401 views
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 \}$ ...
- 93
1 vote
0 answers
89 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 ...
- 5,163
2 votes
0 answers
83 views
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 ...
2 votes
1 answer
202 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 ...
- 5,163
1 vote
1 answer
321 views
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 ...
- 143
3 votes
0 answers
314 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 ...
- 31
3 votes
0 answers
132 views
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 ...
- 121
9 votes
2 answers
528 views
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 ...
- 53.1k
3 votes
1 answer
385 views
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,...
- 147
3 votes
1 answer
205 views
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-...
- 2,621
1 vote
1 answer
186 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 $\Sigma$ ...
- 5,163
42 votes
1 answer
3k views
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 ...
3 votes
0 answers
87 views
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 $M$ ...
- 5,163
2 votes
1 answer
262 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 ...
- 5,163
3 votes
1 answer
191 views
'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 ...
- 5,163
4 votes
1 answer
225 views
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 ...
- 2,086
2 votes
0 answers
91 views
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 ...
- 191