dg.differential geometry - Example for an integral, rectifiable varifold with unbounded first variation

Sunday, 1 January 2012

dg.differential geometry - Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.

Recapitulation

for every $m$ -rectifiable varifold $mu$ exists a $m$ -rectifiable set $E$ in $mathbb R^n$ , meaning $E=E_0 cup bigcup_{kinmathbb N} E_k$ with $mathcal H^m(E_0)=0$ and $E_ksubseteq F_k$ for some $mathcal C^1$ -manifolds $F_k$ of dimension $m$ , and a non-negativ function $thetain L^1_{text{loc}}(mathcal H^m|_E)$ such that $mu=theta mathcal H^m|_E$ . This is a characterisation of $m$ -recitifiable varifolds.
The first variation $deltamu$ of a varifold $mu$ is for $etainmathcal C^1_c(mathbb R^n;mathbb R^n)$ given by

$deltamu(eta)=int div_mueta,dmu,$
where

$div_mu(eta)(x) = sum_{i=1}^n tau_i^T(x)cdot Deta(x)cdot tau_i(x)$ where

$tau_i(x)$ is a orthogonal basis of the tangentspace of

$mu$ in

$x$ , which coinsidence

$mu$ -almost everywhere with

$T_xF_i$ for

$xin E_isubseteq F_i$ as above. So

$div_mueta(x)$ is just the divergence in the manifold

$F_i$ , with

$xin E_isubseteq F_i$ .

We say $mu$ has an locally bounded first variation, if for all $Omega'subseteq Omega$ there exists $c(Omega')<infty$ such that

$deltamu(eta) le C(Omega',Omega) Vert etaVert_{L^infty(Omega)} qquadforall;etainmathcal C^1_c(Omega').$
See for more explanation for example http://eom.springer.de/G/g130040.htm.

For a $mathcal C^2$ -manifold $M$ in $mathbb R^n$ with mean curvature $H_M$ the first variation is

$delta M(eta)=-int_M H_M cdot eta ,dvol_M -int_{partial M} tau_0 cdot eta ,dvol_{partial M} qquadforall;etainmathcal C_c^1(mathbb R^n)$
with the inner normal

$tau_0in T_xMcap(T_xpartial M)^bot$ and where the mean curvature is the trace of the second fundamental form

$A$ by the meaning of

$H_M(x)=sum_{i=1}^m A_x(tau_i,tau_i)$ in the normal space of

$M$ . As obviouse in this case the first variation is locally bounded.

AnswerDesk

Sunday, 1 January 2012