arXiv subjects

Shibing Chen

Publications and source records attributed to Shibing Chen.

At least 19 recordsLinked to original sources

arXiv2026

The Cartan-Hadamard conjecture in dimension five

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

math.DG↗

arXiv2026

Nonradial stable solutions near the Joseph--Lundgren threshold

We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.

math.AP↗

arXiv2026

Global $W^{2,p}$ Regularity in Optimal Transport

In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good-bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.

math.AP↗

arXiv2026

The many-body Blaschke-Santal\'o type inequality via optimal transport

Let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric measurable sets of finite volume such that \[ \sum_{1\le i<j\le k}\langle x_i,x_j\rangle\le \binom{k}{2}, \qquad \forall\,x_i\in K_i, x_j\in K_j. \] We prove the sharp many-body Blaschke--Santal\'o type inequality \[ \prod_{i=1}^k |K_i|\le |B^n|^k \] proposed by Kalantzopoulos and Saroglou, and characterize all equality cases. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate. Using the geometric--functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional version inequality proposed by Kolesnikov and Werner.

math.AP↗

arXiv2026

Symmetry breaking for the complex sine-Gordon equation

We consider the existence of vortex solutions to the complex sine-Gordon II (CSG2) equation, which can be viewed as an analogy of the Ginzburg-Landau (GL) equation. Using the nontrivial kernels $\eta_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $\Psi_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $\Psi_{2,\alpha}$. Explicit formulas of these solutions are also available, from which we propose a new bilinear system for this equation. Our method can be generalized to higher degree case. Nondegenracy and stability of degree-1 solution are also proved. Finally, we formally discuss the Lyapunov-Schmidt reduction procedure for the multivortex solutions of the CSG2 and GL equation.

math.AP↗

arXiv2026

The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body $K=-K\subset \mathbb{R}^3$, \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Rold\'an-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies. In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}.

math.MG↗

arXiv2026

The Mahler Conjecture in Three Dimensions

The Mahler conjecture dates back to 1938. This paper solves the conjecture for general convex bodies in three dimensions by developing a method called the shadow flow. The equality case is characterized as well. This method is also applied to give a new proof of the three-dimensional symmetric case, which was first proved by Iriyeh--Shibata.

math.MG↗

arXiv2026

On the monotonicity of affine quermassintegrals

Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn--Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov--Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts that \[ I_{m,-n}(K)^{1/m}\ge I_{k,-n}(K)^{1/k}, \qquad 1\le m (m+2)(k+2)-2$, there exists an origin-symmetric $C^2_+$ convex body $K\subset\mathbb R^n$ such that \[ I_{m,-n}(K)^{1/m} < I_{k,-n}(K)^{1/k}. \] The example is obtained from the Euclidean ball by an arbitrarily small degree-four spherical harmonic perturbation. On the positive side, we prove that the endpoint chain is true in dimension three: for every convex body $K\subset\mathbb R^3$, \[ I_{1,-3}(K)\ge I_{2,-3}(K)^{1/2}\ge I_{3,-3}(K)^{1/3}=1. \] The equality cases in both non-trivial inequalities are exactly ellipsoids, up to translation and nonsingular affine transformations.

math.AP↗

arXiv2026

Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem

We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of $\partial\{|\nabla u|>0\}$, the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss--Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of $\nabla u$, rather than by a sign condition on $u$. The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up.

math.AP↗

arXiv2026

Global regularity and free boundary geometry in the planar Chon\'e-Rochet model

In this paper, we study minimizers of the Chon\'e--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly convex domains or, more generally, whenever the zero set of $u$ has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global $C^1$ regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a $C^1$ embedded curve, significantly strengthening previously known regularity results.

math.AP↗

arXiv2025

The $L_p$ chord Minkowski problem for super-critical exponent

The $L_p$ chord Minkowski problem was recently introduced by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the $L_p$ chord measure of a convex body. In this paper, we solve the $L_p$ chord Minkowski problem for the super-critical exponents by combining a nonlocal Gauss curvature flow introduced in \cite{HHLW exi} and a topological argument developed in \cite{GLW2022}. Notably, we provide a simplified argument for the topological part.

math.AP↗

arXiv2025

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.

math.AP↗

arXiv2025

Uniqueness in the near isotropic Lp dual Minkowski problem

For n>1 and -1 0.

math.AP↗

arXiv2023

Optimal (partial) transport to non-convex polygonal domains

In this paper, we investigate optimal (partial) transport problems for which the target is a non-convex polygonal domain in $\mathbb{R}^2$. For the complete optimal transport problem, we prove that the singular set is locally a smooth one-dimensional curve away from finitely many points. For the optimal partial transport problem, we prove that the free boundary is smooth away from finitely many singular points. In higher dimensions, we formulate two conjectures concerning the structure of singularities when the target is a non-convex polytope.

math.AP↗

arXiv2023

Global regularity in the Monge-Amp\`ere obstacle problem

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Amp\`ere obstacle problem: $(Du)_{\sharp}f\chi{_{\{u>\frac{1}{2}|x|^2\}}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$ uniformly convex domains $\overline{\Omega}$ and $\overline{\Omega^*}$, respectively. Furthermore, we assume that $\int_{\Omega}f\geq \int_{\Omega^*}g$. The main result shows that $Du:\overline U\rightarrow\overline{\Omega^*}$, where $ U=\{u>\frac{1}{2}|x|^2\}$, is a $W^{1, p}$ diffeomorphism for any $p\in(1,\infty)$. Previously, it was only known to be a continuous homeomorphism according to Caffarelli and McCann \cite{CM}. It is worth noting that our result is sharp, as we can construct examples showing that even with the additional assumption of smooth densities, the optimal map $Du$ is not Lipschitz. This obstacle problem arises naturally in optimal partial transportation.

math.AP↗

arXiv2023

On the planar Gaussian-Minkowski problem

The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this ``uniqueness'' result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.

math.MG↗

arXiv2022

Regularity of singular set in optimal transportation

In this paper, we establish a regularity theory for the optimal transport problem when the target is composed of two disjoint convex domains. This is an important model in which singularities arise. Even though the singular set does not exhibit any form of convexity a priori, we prove its higher order regularity by developing novel methods, which also have many other applications. Notably, our results are achieved without requiring any convexity of the source domain. This aligns with Caffarelli's celebrated regularity theory.

math.AP↗

arXiv2022

A note on the singular set of the no-sign obstacle problem

In this note, we prove the uniqueness of blowups at singular points of the no-sign obstacle problem $Δu=χ_{_{B_1\backslash \{u=|Du|=0\}}}\ \text{in}\ B_1,$ thus give a positive answer to a problem raised in \cite[Notes of Chaper 7, page 149]{PSU12}.

math.AP↗