Questions tagged [minimal-surfaces]
For questions about minimal surfaces in the sense of Riemannian geometry (as opposed to complex geometry).
149 questions
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Counterexamples in Simon-Smith min-max theory
I am reading about the Simon-Smith min-max method for constructing minimal surfaces in 3-manifolds, following this survey by Colding and De Lellis.
Here is what I understand:
Given a min-max sequence ...
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Ways in which minimal surfaces locally minimize area
Let $(M,g)$ be an $n$-dimensional Riemannian manifold.
For a $k$-dimensional submanifold $\Sigma\subseteq M$, say that it is minimal if it is a critical point of the $k$-dimensional area functional ...
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Intersection of minimal surfaces
A classical application of the maximum principle for minimal surfaces gives the following result:
$\textbf{Theorem:}$ Let $M_1,M_2$ be two embedded (disk) least area compact minimal surfaces in $\...
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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 ...
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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....
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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\...
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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, ...
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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
$$A_p(x, ...
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Picard number of surfaces of general type
Let $X$ be a smooth complex surfaces of general type, with $3c_2(X)=c_1(X)^2$ i.e. $X$ is a quotient of the complex ball $\mathbb{B}^2$.
Example. Fake projective planes, i.e. smooth complex projective ...
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How to understand Douglas' solution to the Plateau problem?
Douglas' solution to the Plateau problem in $\mathbb{R}^3$ roughly relies on minimizing the $A$ functional
$$A(g)=\frac{1}{16\pi}\int_{S^1\times S^1}d\theta d\phi \dfrac{\|g(\theta)-g(\phi)\|^2}{\sin^...
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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 ...
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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)} \...
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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 ...
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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 ...
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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 ...
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