Kowalski and
Preiss showed that the cone
C = { (x,y,z,t) : x² = y²+z²+t² }
has the property that that the 3 dimensional Hausdorff measure of
any ball centred at C intersected with C is the same as the measure
of the same ball intersected with a 3 dimensional plane passing
through the centre of that ball. Moreover, any Radon measure in ℝⁿ
having constant (n-1)-dimensional density ratios must be supported
either on a plane or on C × ℝᵏ, where k=n-4.
Simons showed that the
varifold V associated to the cone
C = { x ∈ ℝ⁸ :
x₁² + x₂² + x₃² + x₄² = ½,
x₅² + x₆² + x₇² + x₈² = ½ }
is stationary and stable (i.e. second variation non-negative).
Allard showed (see §5.3) that if C is a stationary varifold in a ball B centred at the origin and the density ratios of ‖C‖ at 0 do not exceed the density at x for ‖C‖ almost all x ∈ B and 0 is in the support of ‖C‖, then C is associated to a plane with constant density.
In particular, constant density ratios and minimality imply that the varifold is a plane. None of these conditions alone guarantees that the varifold is a plane.
De Philippis, De Rosa, and Ghiraldin found recently a sufficient and necessary condition on the integrand F for the following implication to hold: if V is F-stationary, then V is rectifiable.
| e-mail: initial.lastname --at-- mimuw.edu.pl | Last update: 18-04-2018 |