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Kwok-Kun Kwong

Publications and source records attributed to Kwok-Kun Kwong.

At least 19 recordsLinked to original sources
arXiv2026

Higher order isoperimetric inequalities and rigidity for surfaces in $\R^n$

We establish a family of sharp higher order isoperimetric type inequalities for smooth compact surfaces in Euclidean spaces of arbitrary codimension. These inequalities involve arbitrarily high derivatives of the boundary curve, together with the mean curvature vector of the surface. For example, the first and second order inequalities yield the sharp two-sided estimate for the isoperimetric deficit: \begin{align*} & \fracπ{R}\int_{\partial M}|x-Rν|^2\, ds +2π\int_M x\cdot H\, dA \le L^2-4πA\\ \le& \fracπ{R}\int_{\partial M}|x-Rν|^2\, ds +2π\int_M x\cdot H\, dA +\fracπ{3R}\int_{\partial M}|x+R^2κ|^2\, ds, \end{align*} where $R=\frac{L}{2π}$ and the origin is chosen to be the centroid of the boundary. For minimal surfaces, equality in the lower bound holds precisely for flat disks, whereas the second inequality admits non-round equality cases. Our results extend that of Carleman, Sachs, Chakerian, and Li-Schoen-Yau. A distinctive feature of our work is the unexpected rigidity structure arising at higher orders. At the first and second orders, equality curves admit decompositions into circular Fourier modes supported in mutually orthogonal planes. Remarkably, this phenomenon fails at higher orders. We establish a multiplicative factorization of the unit tangent map into elementary planar rotations acting on a fixed unit vector, providing a characterization of equality cases at arbitrary order. As a result, we construct a third order equality curve in $\mathbb R^3$ whose Fourier modes cannot be decomposed into mutually orthogonal circular modes.

math.DG↗
arXiv2026

Jacobian estimates and geometric inequalities under intermediate Ricci curvature

We prove a Jacobian estimate underlying both the Heintze--Karcher comparison and the Alexandrov--Bakelman--Pucci method for submanifolds of arbitrary codimension. The estimate retains the contributions of ambient curvature along geodesics emanating from the submanifold and requires no curvature sign assumption. As applications, we obtain a quantitative Fenchel--Willmore inequality with explicit curvature remainders under nonnegative $n$-intermediate Ricci curvature, Michael--Simon Sobolev and isoperimetric inequalities under nonnegative $(n-1)$-intermediate Ricci curvature, and corresponding extensions under quadratic curvature decay. In each result, the required intermediate Ricci curvature condition depends only on the dimension of the submanifold and is independent of its codimension.

math.DG↗
arXiv2026

An improved volume bound under Ricci and scalar curvature lower bounds

We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert. $$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{-\frac{1}{2}} d \mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.

math.DG↗
arXiv2026

Bayesian Sphere-on-Sphere Regression with Optimal Transport Maps

Spherical regression, in which both covariates and responses lie on the sphere, arises in many scientific applications and has attracted considerable methodological attention in recent years. Despite this progress, constructing flexible and expressive regression models between spherical domains remains challenging, particularly because a single global mapping is often insufficient to capture complex relationships across the entire sphere. A natural strategy is therefore to partition the spherical domain and allow distinct mappings within each region, though this introduces the additional challenge of modeling the partition structure itself. To address these issues, we propose an approach based on optimal transport to model spherical partitions, combined with parametric mappings defined locally within each region. We adopt a Bayesian framework to jointly model both the partitioning and the associated regression maps. This framework enables the identification of heterogeneous regions on the sphere while providing principled uncertainty quantification. Through real-data applications, we demonstrate that the proposed method achieves strong predictive performance, yields meaningful uncertainty estimates, and reveals interpretable clustering structure in spherical data.

stat.ME↗
arXiv2025

Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on this connection, we establish extensions of the Fenchel-Willmore inequality to submanifolds with boundary and to complete non-compact submanifolds. In addition, we derive a sharp Sobolev-type inequality for submanifolds in the same setting. These Sobolev-type inequalities admit a number of applications, including topological consequences in the surface case.

math.DG↗
arXiv2025

Fenchel-Willmore inequality for submanifolds in manifolds with non-negative $k$-Ricci curvature

We establish a sharp Fenchel-Willmore inequality for closed submanifolds of arbitrary dimension and codimension immersed in a complete Riemannian manifold with non-negative intermediate Ricci curvature and Euclidean volume growth. In the hypersurface case, this reduces to non-negative Ricci curvature. We also characterize the equality case. This generalizes the recent work of Agostiniani, Fogagnolo, and Mazzieri \cite{Agostiniani-Fogagnolo-Mazzieri}, as well as classical results by Chen, Fenchel, Willmore, and others.

math.DG↗
arXiv2025

Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper \cite{kwong2023geometric} via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality.

math.DG↗
arXiv2024

Alexandrov-Fenchel type inequalities with convex weight in space forms

In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family of geometric inequalities, as each convex, non-decreasing function yields a corresponding inequality, providing considerable flexibility.

math.DG↗
arXiv2023

A unified approach to higher order discrete and smooth isoperimetric inequalities

We provide a simple unified approach to obtain (i) Discrete polygonal isoperimetric type inequalities of arbitrary high order. (ii) Arbitrary high order isoperimetric type inequalities for smooth curves, where both upper and lower bounds for the isoperimetric deficit $L^2-4πF$ are obtained. (iii) Higher order Chernoff type inequalities involving a generalized width function and higher order locus of curvature centers. The method we use is to obtain higher order discrete or smooth Wirtinger inequalities via discrete or smooth Fourier analysis, by looking at a family of linear operators. The key is to find the right candidate for the linear operators, and to translate the analytic inequalities into geometric ones.

math.CA↗
arXiv2023

Generalized Alexandrov theorems in spacetimes with integral conditions

We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $Σ$ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in \cite{wang2017Minkowski} by relaxing the curvature conditions on $Σ$ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for $Σ$ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.

math.DG↗
arXiv2023

An estimate for the genus of embedded surfaces in the 3-sphere

By refining the volume estimate of Heintze and Karcher \cite{HK}, we obtain a sharp pinching estimate for the genus of a surface in $\mathbb S^{3}$, which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if $g$ is the genus of a closed orientable surface $Σ$ in a $3$-dimensional orientable Riemannian manifold $M$ whose sectional curvature is bounded below by $1$, then $4 π^{2} g(Σ) \le 2\left(2 π^{2}-|M|\right)+\int_Σ f(|\stackrel \circ A|)$, where $ \stackrel \circ A $ is the traceless second fundamental form and $f$ is an explicit function. As a result, the space of closed orientable embedded minimal surfaces $Σ$ with uniformly bounded $\|A\|_{L^3(Σ)}$ is compact in the $C^k$ topology for any $k\ge2$.

math.DG↗
arXiv2023

Geometric inequalities involving three quantities in warped product manifolds

In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family concerns the $k$-th boundary momentum, area, and weighted volume, and has applications to Weinstock-type inequalities for Steklov or Wentzell eigenvalues on star-shaped mean convex domains. This generalizes the main results of [12]. The second family involves a weighted $k$-th mean curvature integral and two distinct quermassintegrals and extends the authors' recent work [33] with G. Wheeler and V.-M. Wheeler.

math.DG↗
arXiv2022

Effect of the average scalar curvature on Riemannian manifolds

We investigate the effect of the average scalar curvature on the conjugate radius, average area of the geodesic spheres, average volume of the metric balls and the total volume of a closed Riemannian manifold $N$ (or more generally $N$ with finite volume whose negative Ricci curvature integral on $SN$ is finite). For example, we prove that if the average scalar curvature is larger than the lower bound of the normalized Ricci curvature, then we can improve the Bishop-Gromov estimate on the average volume of the metric balls of any size. We also prove the monotone decreasing property of a certain geometric integral when the average scalar curvature has a lower bound. This leads to a comparison theorem of the average total mean curvature of geodesic spheres of radius up to $\mathrm{inj}(N)$.

math.DG↗
arXiv2022

Rigidity on non-negative intermediate curvature

In a recent work of Brendle-Hirsch-Johne, a notion of intermediate curvature was introduced to extend the classical non-existence theorem of positive scalar curvature on torus to product manifolds. In this work, we study the rigidity when the intermediate curvature is only non-negative when the ambient dimension is at most $5$.

math.DG↗
arXiv2022

Length-minimizing level curves via calibrations

We present an elementary criterion to show the length-minimizing property of geodesics for a large class of conformal metrics. In particular, we prove the length-minimizing property of level curves of harmonic functions and the length-minimizing property of a family of the conic sections with the eccentricity $\varepsilon$ in the upper half plane endowed with the conformal metric $ \left( {\varepsilon}^{2} + \frac{1}{\;{y^2} \;} \right) \left(dx^{2} + dy^{2} \right)$.

math.DG↗
arXiv2021

Higher order Poincare inequalities and Minkowski-type inequalities

We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz's proof of the Wirtinger's inequality using Fourier theory. We then give some geometric implication of these inequalities by applying them on the sphere. More specifically, by applying them to the support function of a convex hypersurface in the Euclidean space, we obtain some sharp Minkowski-type inequalities, such as a stability inequality for the classical Minkowski inequality and the Alexandrov-Fenchel inequality.

math.DG↗
arXiv2021

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Girão-Pinheiro.

math.DG↗