Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness
Abstract
For 1≤ r≤ n−1 we study shrinking self-similar ends of the r-mean curvature flow, Sr=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x−(r+1), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.
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BibTeX
@misc{alencar2026asymptotically,
title = {{Asymptotically conical shrinking ends of the $r$-mean curvature flow: rotational construction and uniqueness}},
author = {Hilário Alencar and Márcio Batista and Gregório Silva Neto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14227},
eprint = {2610.14227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14227},
}
RIS
TY - UNPB
TI - Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness
AU - Hilário Alencar
AU - Márcio Batista
AU - Gregório Silva Neto
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14227
UR - https://arxiv.org/abs/2610.14227
AB - For 1≤ r≤ n−1 we study shrinking self-similar ends of the r-mean curvature flow, S_r=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x^−(r+1), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.
ER -