Questions tagged [calibrated-geometry]

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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 ...
7
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1answer
89 views

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 ...
3
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1answer
76 views

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-...
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0answers
74 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}+...
6
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1answer
230 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 ...
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1answer
49 views

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-...
8
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2answers
441 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:=\...
1
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0answers
123 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 ...