All Questions
Tagged with calibrations or calibrated-geometry
13 questions
0
votes
0
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27
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Show that $\frac{1}{k!}\omega^k$ is a linear $2k$-calibration
Definitions. Given an oriented vector space $V$ with metric $g$, a linear $k$-calibration on $V$ is a $k$-form $\rho\in\Lambda^k V^*$, such that for every oriented $k$-dimensional subspace $W\subset V$...
- 4,366
9
votes
1
answer
295
views
Stuck on Differential Geometry proof
My concrete questions are (for context see below):
Is is true that $i$ as below embeds $M$ in $T^*\mathbb{R}^n$?
Is it true that $M$ is Lagrangian in $\mathbb{C}^n$ if and only if $i(M)$ is ...
- 155
1
vote
1
answer
177
views
Combining two fundamental matrices
Let $\mathcal{F_{ab}}$ be the fundamental matrix obtained from images $A$ and $B$
$$ \mathcal{F_{ab}} = \begin{bmatrix}
ab_{11} & ab_{12} & ab_{13} \\
ab_{21} & ab_{22} & ab_{23} \\
...
- 11
2
votes
0
answers
77
views
Special Lagrangian inequality from Harvey-Lawson's Calibrated Geometries
I am trying to understand the proof of Theorem 1.7 on page 88 of Harvey-Lawson's Calibrated Geometries. I do not understand how they conclude that $(dz_1 \wedge \dots \wedge dz_n, A(e_1\wedge \dots \...
- 21
0
votes
1
answer
131
views
Wirtinger's theorem fails to hold in the real case
We have a complex manifold $M$ equiped with a hermitian metric, then for a complex submanifold $S \subset W$, the Wirtinger's theorem tells us that the volume form on $S$ is the restriction of a ...
user388493
0
votes
1
answer
120
views
Question on concept of homology in calibrated geometry
The fundamental lemma of calibrated geometry states that calibrated submanifolds are absolutely volume minimising in their homology class. In the proof, homology equivalent is used synonymously with ...
- 456
11
votes
1
answer
235
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Calibrations vs. Riemannian holonomy
I've began to study the relationship between calibrations and holonomy, mainly through D.D. Joyce's Riemannian Holonomy Groups and Calibrated Geometry and partly through internet material.
Pretty ...
- 10.9k
3
votes
1
answer
147
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Inner circle of torus of revolution is calibrated
I'm working on the following problem from Lee's "Introduction to Smooth Manifolds":
Let $D \subseteq \mathbb R^3$ be the surface obtained by revolving the circle $(r-2)^2 + z^2 = 1$ around the z-...
- 4,397
3
votes
0
answers
115
views
Showing that a 7-manifold has $G_{2}$ holonomy
I have to show that the direct product of the multi-center Taub-NUT metric with $\mathbb{R}^{3}$ corresponds to a 7-manifold with G2 holonomy.
The metric of the Taub-NUT is:
$ds_{TN}^{2}=V(r)(dr^{2}+...
- 1,843
6
votes
1
answer
687
views
Does minimal submanifolds minimize area locally?
Consider $(\tilde{M},g)$ a riemannian manifold and $M \subset \tilde{M}$ riemannian submanifold.
Is it true that if $M$ is a minimal submanifold of $\tilde{M}$ then for every $p \in M$ there exists a ...
- 91
1
vote
1
answer
135
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Verify a two-form is calibration
$u: \Omega \subset \mathbb R^2 \rightarrow \mathbb R$ is a $C^2$ function. Graph of $u$ is
$$
G_u=\{(x,y,u(x,y)) : (x,y)\in \Omega\}
$$
And the upward pointing unit normal is $N$. $\omega$ is the two-...
- 6,956
11
votes
2
answers
1k
views
Is there a minimal graph in $\mathbb{R}^3$ which is not area-minimizing?
Let $\Omega\subset\mathbb{R}^2$ be an open subset such that $\partial\Omega$ is a closed, simple curve.
I'm trying to find an example of an $u:\overline{\Omega}\to\mathbb{R}$ such that $\Sigma:=\...
- 10.9k
1
vote
0
answers
276
views
Why is a graph of a function in $\mathbb{R}^n$ satisfying the minimal surface equation actually area minimizing?
I am reading the Colding and Minicozzi book "A course in Minimal Surfaces".
I have a question regarding one of the points mentioned in the book. I want to prove that a minimal hypersurface which is a ...
- 317