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

Given a smooth, symmetric, homogeneous of degree one function f(λ1,⋯,λn) satisfying ∂if>0 for all i=1,⋯,n, and a rotationally symmetric cone C in Rn+1, we show that there is a f self-shrinker (i.e. a hypersurface Σ in Rn+1 which satisfies f(κ1,⋯,κn)+12X⋅N=0, where X is the position vector, N is the unit normal vector, and κ1,⋯,κn are principal curvatures of Σ) that is asymptotic to C at infinity.