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Keaton Naff

Publications and source records attributed to Keaton Naff.

15 recordsLinked to original sources
arXiv2025

Free boundary and capillary minimal surfaces in spherical caps I: Low genus

This is the first of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset\mathbb{S}^3$. In this article, we extend our previous half-space intersection properties to warped products, and extend (non-)umbilicity of discs and annuli to capillary minimal surfaces in high codimension. We establish a dual operation relating free boundary and capillary minimal surfaces. These results are discussed in a continuous, unified framework, particularly in relation to uniqueness of minimal annuli.

math.DG↗
arXiv2025

Linear stability and instability of Kähler Ricci solitons

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of \cite{chi04} and \cite{hm11}, via recent work the \cite{cm21} on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted $L^2$-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

math.DG↗
arXiv2025

Linear stability of the blowdown Ricci shrinker in 4D

We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups. The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its $U(2)$-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a $9$-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations.

math.DG↗
arXiv2025

Planarity and convexity for pinched ancient solutions of mean curvature flow

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle--Huisken--Sinestrari \cite{BHS}. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.

math.DG↗
arXiv2024

Half-space intersection properties for minimal hypersurfaces

We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect in every closed hemisphere. Two approaches are developed: one using classifications of stable minimal hypersurfaces, and the second using conformal change and comparison geometry for $α$-Bakry-Émery-Ricci curvature. Our methods yield the analogous intersection properties for free boundary minimal hypersurfaces in space form balls, even when the interior or boundary curvature may be negative. Finally, Colding and Minicozzi recently showed that any two embedded shrinkers of dimension $n$ must intersect in a large enough Euclidean ball of radius $R(n)$. We show that $R(n) \leq 2 \sqrt{n}$.

math.DG↗
arXiv2023

Canonical metrics and ambiKähler structures on 4-manifolds with $U(2)$ symmetry

For $U(2)$-invariant 4-metrics, we show that the $B^t$-flat metrics are very different from the other canonical metrics (Bach-flat, Einstein, extremal Kähler, etc). We show every $U(2)$-invariant metric is conformal to two separate Kähler metrics, leading to ambiKähler structures. Using this observation we find new complete extremal Kähler metrics on the total spaces of $\mathcal{O}(-1)$ and $\mathcal{O}(+1)$ that are conformal to the Taub-bolt metric. In addition to its usual hyperKähler structure, the Taub-NUT's conformal class contains two additional complete Kähler metrics that make up an ambi-Kähler pair, making five independent compatible complex structures for the Taub-NUT, each of which has a conformally Kähler (1,1) form.

math.DG↗
arXiv2022

The prescribed point area estimate for minimal submanifolds in constant curvature

We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in hyperbolic space, in any dimension and codimension. In certain cases, we also prove the corresponding estimate in the sphere. Our estimates are analogous to those of Brendle and Hung in the Euclidean setting.

math.DG↗
arXiv2022

Moving monotonicity formulae for minimal submanifolds in constant curvature

We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball. These monotonicity formulae involve an energy-like integral over sets which are, in general, not geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by the second author in [Zhu18].

math.DG↗
arXiv2022

A local noncollapsing estimate for mean curvature flow

We prove a local version of the noncollapsing estimate for mean curvature flow. By combining our result with earlier work of X.-J. Wang, it follows that certain ancient convex solutions that sweep out the entire space are noncollapsed.

math.DG↗
arXiv2022

Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

We extend the second part of \cite{Bre20} on the uniqueness of ancient $κ$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient $κ$-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $κ$-noncollapsed. We show that the only noncompact ancient $κ$-solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.

math.DG↗
arXiv2021

Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

In this paper, we study the classification of $κ$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

math.DG↗
arXiv2020

Singularity models of pinched solutions of mean curvature flow in higher codimension

We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq γ_1 |H|^2, |\nabla^2 A| \leq γ_2 |H|^3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) solutions to the mean curvature flow in $\mathbb{R}^N$ that satisfy the pinching condition $|H| > 0$ and $|A|^2 < c(n) |H|^2$, $c(n) = \min\{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}\}$. We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.

math.DG↗
arXiv2019

Shrinking Ricci solitons with positive isotropic curvature

We show that in dimensions $n \geq 12$, a non-flat complete gradient shrinking solitons with uniformly positive isotropic curvature (PIC) must be a quotient of either the round sphere $S^n$ or the cylinder $S^{n-1} \times \mathbb{R}$. We also observe that in dimensions $n \geq 5$, a complete gradient shrinking soliton that is strictly PIC and weakly PIC2 must be a quotient of either the round sphere $S^n$ or the cylinder $S^{n-1} \times \mathbb{R}$.

math.DG↗