Shrinking Doughnuts

Abstract

Let M ⁿ be a smooth compact oriented manifold, and let X ∶ M × [0, T) → Rⁿ⁺¹ be a smooth family of immersions of M in n+1 dimensional Euclidean space. The orientation of M allows one to define a unique smooth unit normal vector field v _x ∶ M × [0, T) → Rⁿ⁺¹. Given this choice of v _x , we can define the principal curvatures, k ₁ ,⋯, K _n , of the immersion X(·, t) and the mean curvature H _x = (k ₁ + ⋯ + K _n )/n in the usual way. By definition, the family of immersions X(·, t) “moves by its mean curvaturex201D; if the normal velocity satisfies

$$\left\langle {\frac{{\partial X}}{{\partial t}},{v_X}} \right\rangle = n{H_X}$$

(1)

at all (p, t) ∈ M × [0,T). Here (x,y) = x_oy_o + ⋯ + x _n y _n denotes the Euclidean inner product on R ⁿ⁺¹.