Questions tagged [calibrated-geometry]

A calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n)

Filter by
Sorted by
Tagged with
2
votes
0answers
33 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 \...
0
votes
1answer
71 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 ...
0
votes
1answer
51 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 ...
9
votes
1answer
131 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
votes
1answer
104 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-...
3
votes
0answers
86 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
votes
1answer
321 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 ...
1
vote
1answer
57 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-...
11
votes
2answers
657 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
vote
0answers
151 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 ...