arXiv subjects

Liuwei Gong

Publications and source records attributed to Liuwei Gong.

6 recordsLinked to original sources

arXiv2026

A counterexample to a strong maximum principle for the sixth-order GJMS operator

We exhibit an explicit closed seven-dimensional Riemannian manifold \[ (M,g)=\mathbb S^2(1)\times \mathbb S^5\left(\frac1{100}\right), \] where the displayed parameters denote sectional curvatures, for which $\Ric_g>0$, and hence $Q_g^{(2)}>0$. Moreover, \[ Q^{(4)}_g>0,\qquad Q^{(6)}_g>0, \] and the sixth-order GJMS operator $P_{6,g}$ is strictly positive as a self-adjoint operator, but nevertheless $P_{6,g}$ fails the strong maximum principle. The failure is caused by a nonconstant positive eigenvalue of $P_{6,g}$ lying strictly below the eigenvalue of the constant mode. The example also has $Y_2(M,[g])>0$ and $Y_4(M,[g])>0$, while $P_{6,g}$ does not have a positive Green function. It disproves Conjecture~1 of Andrade, Piccione, and Wei and its general-order formulation by Case and Gover.

math.DG↗

arXiv2026

The (local) geometry of oscillatory integrals on manifolds: Dimension three

Sogge studied Kakeya problems on two extreme types of three dimensional Riemannian manifolds: Manifolds with the most symmetries (manifolds of constant sectional curvature) and manifolds with the least symmetries, which he called manifolds with chaotic curvature and variably curved manifolds. In the same paper, Sogge proposed studying manifolds with intermediate symmetry, such as (locally) symmetric spaces. In the current paper, we propose a classification of curvature conditions in the spirit of Sogge's program. In particular, these curvature conditions give a complete geometric characterization of the contact order conditions (for Riemannian distance functions), introduced when people were studying H\"ormander-type oscillatory integral operators. One of these conditions generalizes Sogge's chaotic curvature condition to all finite orders: The chaotic curvature condition of order $\le k$ for every $k\ge 1,$ with the case $k=1$ corresponding to Sogge's original condition for variably curved manifolds. As byproducts of our main results, we show that there are no manifolds satisfying the chaotic curvature condition of order $\le 1$. We also show that both the chaotic curvature condition of order $\le 2$ and its failure can occur robustly under small smooth perturbations, and for every $k\ge 3$, a ``generic" manifold satisfies the chaotic curvature condition of order $\le k$. It turns out that the chaotic curvature condition of order $\le k$ is precisely the same as the notion of non-$(k+2)$-exceptional, where $k$-exceptional is introduced by Lytchak and Petrunin \cite{LP22} when studying convex sets and the non-existence of totally geodesic sub-manifolds. Thus our results imply, in particular, that every manifold is $3$-exceptional.

math.CA↗

arXiv2026

Global Convergence of the Gursky-Malchiodi $Q$-curvature Flow

In their seminal work, Gursky and Malchiodi introduced a non-local conformal flow in dimensions $n \geq 5$ to resolve the constant $Q$-curvature problem. They proved sequential convergence of the flow for initial metrics with positive scalar curvature and $Q$-curvature, provided the energy was sufficiently small. In this paper, we prove the global convergence of the flow for arbitrary initial energy under the same positivity assumptions by establishing a non-local version of the {\L}ojasiewicz-Simon inequality for the Paneitz-Sobolev quotient along the flow. We construct test bubbles and estimate their Paneitz-Sobolev quotients, a strategy that was carried out in the celebrated work of Brendle in the context of the Yamabe flow. We develop a more geometric and systematic proof that addresses the algebraic and computational complexity inherent in the $Q$-curvature and the Paneitz operator. Along the way, we derive a stability inequality for the Paneitz-Sobolev quotient using a higher-order Koiso-Bochner formula established in recent work of Bahuaud, Guenther, Isenberg, and Mazzeo.

math.DG↗

arXiv2025

Compactness and non-compactness theorems of the fourth- and sixth-order constant $Q$-curvature problems

We provide a complete resolution to the question of compactness for the full solution sets of the fourth-order and sixth-order constant $Q$-curvature problems on smooth closed Riemannian manifolds not conformally diffeomorphic to the standard unit $n$-sphere, provided the associated conformally covariant differential operator has a positive Green's function. Firstly, we prove that the solution set of the fourth-order constant $Q$-curvature problem is $C^4$-compact in dimensions $5 \le n \le 24$. For $n \ge 25$, an example of an $L^{\infty}$-unbounded sequence of solutions has been known for over a decade (Wei and Zhao). Additionally, the compactness result for $5 \le n \le 9$ was established by Li and Xiong. Secondly, we demonstrate that the solution set of the sixth-order constant $Q$-curvature problem is $C^6$-compact in dimensions $7 \le n \le 26$, whereas a blow-up example exists for $n \ge 27$. Our main observation is that the linearized equations associated with both $Q$-curvature problems can be transformed into overdetermined linear systems, which admit nontrivial solutions due to unexpected algebraic structures of the Paneitz operator and the sixth-order GJMS operator. This key insight not only plays a crucial role in deducing the compactness result for high-dimensional manifolds, but also reveals an elegant hierarchical pattern with respect to the order of the conformally covariant operators, suggesting the possibility of a unified theory of the compactness of the constant $Q$-curvature problems of all admissible even integer orders.

math.AP↗

arXiv2024

Conformal metrics of constant scalar curvature with unbounded volumes

For $n\geq 25$, we construct a smooth metric $\tilde{g}$ on the standard $n$-dimensional sphere $\mathbb{S}^n$ such that there exists a sequence of smooth metrics $\{\tilde{g}_k\}_{k\in\mathbb{N}}$ conformal to $\tilde g$ where each $\tilde g_k$ has scalar curvature $R_{\tilde{g}_k}\equiv 1$ and their volumes $\text{Vol}(\mathbb{S}^n,\tilde{g}_k)$ tend to infinity as $k$ approaches infinity.

math.AP↗

arXiv2024

Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems

We consider Carleson-Sjölin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of Hörmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sjölin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to Hörmander-type oscillatory integral operators, prove that if a Hörmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures.

math.DG↗