Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.14227

Asymptotically conical shrinking ends of the $r$-mean curvature flow: rotational construction and uniqueness

Hilário Alencar·Márcio Batista·Gregório Silva Neto
View original source ↗Read source document ↗Export metadata ↓

Abstract

For $1\leq r\leq n-1$ we study shrinking self-similar ends of the $r$-mean curvature flow, $S_r=c\langle X,N\rangle$ 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.

Explore related subjects

math.DGmath.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hilário Alencar, Márcio Batista, Gregório Silva Neto. 2026-10-09. Asymptotically conical shrinking ends of the $r$-mean curvature flow: rotational construction and uniqueness. https://arxiv.org/abs/2610.14227

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Complete harmonic metrics and subharmonic functions on the unit disc

Let $X$ be a Riemann surface with canonical bundle $K_X$ and holomorphic tangent bundle $T_X=K_X^{-1}$. For each integer $r\geq 2$, each $q\in H^0(K_X^r)$, and each choice of a square root of $K_X$, we obtain a cyclic Higgs bundle. A diagonal harmonic metric $h=(h_1,\ldots,h_r)$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\ldots,r-1$, together with a possibly degenerate Hermitian metric $H_r$ determined by $h_1,h_r$, and $q$. We call $h$ complete if the Kähler metrics induced by $H_1,\ldots,H_{r-1}$ are all complete. Li--Mochizuki proved that if $q$ is nonzero unless $X$ is hyperbolic, there exists a unique complete harmonic metric on the corresponding cyclic Higgs bundle satisfying $\det(h)=1$. The Hitchin equation depends on $q$ only through the subharmonic weight $ϕ_q=\frac{1}{r}\log|q|^2$ on $K_X$. We establish uniqueness of complete solutions with determinant $1$ for general subharmonic weight functions $φ$ whose exponentials are $C^2$ outside a compact subset of $X$. On the unit disc $\mathbb D=\{z\in\mathbb C:|z|<1\}$, we also prove that a real complete solution exists for every subharmonic weight function. We further establish a monotone approximation theorem. If $φ_ε$ are smooth subharmonic weight functions on the discs $\mathbb D_ε=\{z\in\mathbb C:|z|<1-ε\}$ decreasing to $φ$ as $ε\searrow0$, the corresponding complete solutions $h_ε$, normalized by $\det(h_ε)=1$, converge monotonically to a complete solution $h$ associated with $φ$.

math.DG↗
arXiv2026

The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics

This paper reviews the generalised Fried conjecture for Morse--Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray--Singer torsion. On the dynamical side, we characterise Morse--Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and {reviewing the Thom--Smale complex for purely gradient flows. In general, the local contributions of the critical elements of the flow are globally assembled via a Smale filtration to construct the Milnor metric} on the determinant line of the twisted cohomology. Finally, under the assumption of a unitary representation, we present the theorem proving the conjecture: the Ray--Singer metric coincides with the Milnor metric. Furthermore, we discuss the structure of the Milnor metric, which factors into the Ruelle Zeta function evaluated at zero and a specific contribution from the fixed points. We clarify that this object reduces to either the classical algebraic Thom--Smale torsion or the Ruelle Zeta function solely in the limiting cases of purely gradient or fixed-point-free flows.

math.DG↗
arXiv2026

Type I estimates for the Kähler-Ricci flow II

In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial $C^0$ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

math.DG↗