arXiv subjects

Liuwei Gong

Publications and source records attributed to Liuwei Gong.

11 recordsLinked to original sources

arXiv2026

Noncompactness for the constant $Q_{2N}$-curvature problem

For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb S^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ when $N=4,5$, $n\ge2N+19$ when $6\le N\le8$, $n\ge2N+18$ when $9\le N\le17$, and $n\ge2N+17$ when $N\ge18$. For each $L \in \mathbb{N} \cup \{0\}$, the metric can be chosen arbitrarily close to the round metric in the $C^L$ norm. It can also be chosen to have a coercive GJMS operator, a strictly positive Green function, and positive Green function mass wherever the Weyl tensor and its covariant derivatives through order $\lfloor(n-2N-4)/2\rfloor$ vanish. Together with the known cases $N=1,2,3$, this establishes noncompactness at every order and proves the sharpness of the companion compactness ranges under the same operator and mass hypotheses. The construction relies on our formula for the fixed-volume Hessian of total $Q_{2N}$-curvature in transverse-traceless (TT) directions at closed Einstein metrics with $\mathrm{Ric}=(n-1)g$. For every $N \in \mathbb{N}$ and $n>2N$, this Hessian is represented by a degree-$N$ polynomial in the Lichnerowicz Laplacian, extending Matsumoto's formula for $n=2N$. For the metric perturbations generated by a fixed algebraic Weyl tensor, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation in the bubble scale yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real $n$, this matrix changes from negative definite to indefinite at $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$ as $N\to\infty$, with $a_*\approx16.2018$ and $c_*\approx13.5128$, explaining the eventual bound $n\ge2N+17$.

math.DG↗

arXiv2026

An ambient metric framework for sharp compactness of the constant $Q_{2N}$-curvature problem

We develop a geometric method for studying $C^{2N}$-compactness of positive solutions to the constant $Q_{2N}$-curvature equation on closed Riemannian $n$-manifolds, for every integer $N\ge1$ and every dimension $n>2N$. Assuming the GJMS operator is invertible, its Green function is positive, and the positive mass condition holds, we establish sharp dimension ranges for the compactness of the set of all positive solutions with prescribed positive constant $Q_{2N}$-curvature. The largest dimension for compactness is $24$ for $N=1$ ([21]), $2N+20$ for $N=2,3$ ([11]), $2N+19$ for $N=4,5$, $2N+18$ for $N=6,7,8$, $2N+17$ for $9\le N\le17$, and $2N+16$ for all $N\ge18$. The constructions in [4,5,33,11] for $N=1,2,3$ and in the companion paper [12] for $N\ge4$ give noncompact examples in every dimension exceeding the corresponding compactness bound. We show that these $L^{\infty}$-unbounded examples can be chosen to satisfy the same operator and mass hypotheses, proving sharpness within this class. Our method relates the Pohozaev quadratic forms to the Hessian $D^2\mathscr{F}(g_c)$, where $\mathscr{F}$ is the volume-normalized total $Q_{2N}$-curvature functional and $g_c$ is the standard round metric on $\mathbb{S}^n$. The ambient metric argument of [12] supplies an explicit formula for $D^2\mathscr{F}(g_c)$ on transverse-traceless tensors. The York decomposition projects metric perturbations onto this space. The scalar--vector--tensor decomposition represents the Pohozaev quadratic forms by explicit finite-dimensional matrices. We prove that these matrices are positive definite throughout the stated dimension range, a crucial step in establishing compactness.

math.DG↗

arXiv2026

Sharp local scalar-curvature rigidity of spherical caps

Let $B(δ)\subset\mathbb S^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\geq3$ and $0<δ<π/2$. We prove local scalar-curvature rigidity under preservation of the induced boundary metric and a one-sided lower bound for the boundary mean curvature. More precisely, every sufficiently small $C^{2,α}$ perturbation $g$ of $\bar{g}$ satisfying $R(g)\geq n(n-1)$ and $H(g)\geq(n-1)\cotδ$ is round up to a diffeomorphism fixing the boundary pointwise. Inspired by the scalar--vector--tensor decomposition used in the study of Yamabe compactness, we establish positivity of the Brendle--Marques quadratic form in the divergence-free gauge for variations preserving the induced boundary metric and satisfying the joint linearized constraints $D R_{\bar{g}}(h)=D H_{\bar{g}}(h)=0$. The hemisphere counterexamples of Brendle--Marques--Neves show that the radius range is sharp.

math.DG↗

arXiv2026

On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II

We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.

math.DG↗

arXiv2026

On critical dimensions for compactness in the boundary Yamabe problem, I

We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.

math.DG↗

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 Hörmander-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 Łojasiewicz-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↗

arXiv2025

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↗