Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

Abstract

Given a smooth, symmetric, homogeneous of degree one function f (λ 1 , · · · , λn) satisfying ∂ i f > 0 for all i = 1, · · · , n, and a rotationally symmetric cone C in R n+1 , we show that there is a f self-shrinker (i.e. a hypersurface Σ in R n+1 which satisfies f (κ 1 , · · · , κn) + 1 2 X · N = 0, where X is the position vector, N is the unit normal vector, and κ 1 , · · · , κn are principal curvatures of Σ) that is asymptotic to C at infinity.