arXiv subjects

Tsz-Kiu Aaron Chow

Publications and source records attributed to Tsz-Kiu Aaron Chow.

16 recordsLinked to original sources

arXiv2026

Rigidity and flexibility under positive isotropic curvature

For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $λ_1^{(2)}\ge(n-1)σ/2$ under $σ$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

math.DG↗

arXiv2026

Nonexistence of complete metrics with uniformly positive scalar curvature

Let $X$ be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that $X$ admits no complete metric with uniformly positive scalar curvature when either infinite relative $K$-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset with compact complement. The proof compares twisted Dirac operators on spin covers of compact manifolds with boundary and combines a heat-kernel argument on fundamental domains with a long-neck estimate.

math.DG↗

arXiv2026

Minimal Two-Spheres and Manifolds with Positive Isotropic Curvature

We improve the Micallef--Moore index estimate for harmonic two-spheres in $n$-manifolds with positive isotropic curvature to the sharp bound $n-3$. Combining with recent work, this completes the diffeomorphism classification of closed manifolds with positive isotropic curvature in the remaining dimensions five and six.

math.DG↗

arXiv2026

Small Normal Curvature and Three-Manifold Topology

For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $κ(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $κ(F)\le\sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}\mathbb{P}^3$, or $S^2\times S^1$. All three possibilities occur, while $κ(F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality \[ (\min_Y R_g)\text{sys}(g)^2<6π^2 \] for every spherical three-space form $Y$ with $|π_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for $\mathbb{R}\mathbb{P}^3$ of Bray--Brendle--Eichmair--Neves.

math.DG↗

arXiv2026

Maximal Normal Curvature and Veronese Rigidity

We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(Σ,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ κ(F):= \sup_{x\inΣ} \sup_{\substack{v\in T_xΣ\\ |v|_g=1}} |A_x(v,v)|. \] If $Σ^{2n}$ is almost Hermitian with harmonic fundamental two-form, or $Σ^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge2$, then \[ κ(F)\ge \sqrt{\frac{2n}{n+1}} . \] In the equality case the harmonic form is parallel and the immersion is, up to a totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of projective spaces. The key input is a Bochner--Gauss mechanism that turns the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators.

math.DG↗

arXiv2026

Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:Σ^m\to M^{m+1}$ with $b_p(Σ;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\fracπ{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt2)\times S^{m-p}(1/\sqrt2)\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec\ge1$ for $p=1$, from normalized $\mathrm{PIC1}\ge1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in $\mathbb{CP}^n$ and $\mathbb{HP}^n$, with projective Clifford rigidity, without any Betti-number assumption.

math.DG↗

arXiv2026

A Spinorial Perelman's Functional: Critical Points and Gradient Flow

In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.

math.DG↗

arXiv2025

Rigidity results for initial data sets satisfying the dominant energy condition

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

math.DG↗

arXiv2025

Scalar curvature rigidity for products of spheres and tori

We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\text{dist}(\partial_-M,\partial_+M)\le 2π\sqrt{n/((n+1)σ)}$ when $R_M\ge (n-m)(n-m-1)+σ$. The method combines stable weighted slicing with a spectral Dirac operator argument.

math.DG↗

arXiv2024

Bandwidth and focal radius with positive isotropic curvature

This paper investigates quantitative metric inequalities for manifolds with positive isotropic curvature (PIC). Our results include upper bounds on the bandwidth and focal radius of hypersurfaces in PIC manifolds, contingent on boundary convexities and Betti numbers. The proof is based on exploiting the spectral properties of a twisted de Rham-Hodge operator on manifolds with boundary.

math.DG↗

arXiv2023

Preserving positive intermediate curvature

Consider a compact manifold $N$ (with or without boundary) of dimension $n$. Positive $m$-intermediate curvature interpolates between positive Ricci curvature ($m = 1$) and positive scalar curvature ($m = n-1$), and it is obstructed on partial tori $N^n = M^{n-m} \times \mathbb{T}^m$. Given Riemannian metrics $g, \bar{g}$ on $(N, \partial N)$ with positive $m$-intermediate curvature and $m$-positive difference $h_g - h_{\bar{g}}$ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive $m$-intermediate curvature interpolating between $g$ and $\bar{g}$. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive $m$-intermediate curvature and strictly $m$-convex boundaries.

math.DG↗

arXiv2022

Equivariant $3$-manifolds with positive scalar curvature

In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the generalized Smale conjecture for spherical three-orbifolds. Moreover, for connected $G$, we make a classification of all PSC $G$-invariant three-manifolds.

math.DG↗

arXiv2021

Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

math.DG↗

arXiv2021

Area and boundary length of surfaces diffeomorphic to annuli

In this paper, we give a proof to a statement in Perelman's paper for finite extinction time of Ricci flow. Our proof draws on different techniques from the one given in Morgan-Tian's exposition and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions.

math.DG↗

arXiv2021

Positivity of Curvature on Manifolds with Boundary

Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.

math.DG↗

arXiv2018

Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows.

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