arXiv subjects

Mingxiang Li

Publications and source records attributed to Mingxiang Li.

At least 19 recordsLinked to original sources

arXiv2026

A Cheng-Yau type estimate for positive biharmonic functions

We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies $Ric_g \ge -(n-1)K g$. If $u$ is a positive biharmonic function in $B_{2R}(p)$, then $$ -\frac{\Delta_g u}{u} +\frac{1}{8n}\frac{|\nabla u|_g^2}{u^2} \le C_n\left(R^{-2}+K\right) \quad\text{on }B_R(p). $$ Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies $\Delta_g u\equiv c$ and the sharp estimate $|\nabla u|_g^2\le2cu$ for some nonnegative constant $c$. We also show that every positive $k$-polyharmonic function has nonnegative constant $(k-1)$-st Laplacian and growth of order at most $2k-2$.

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

On the proof of Bray's conjecture

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

math.DG↗

arXiv2026

$\tau_0$-VLA: a Hierarchical Robot Foundation Model with World-Model-Guided Test-Time Computation

Long-horizon robot manipulation requires a robot to both execute individual skills reliably and sequence them coherently over extended tasks. Most hierarchical vision-language-action (VLA) models make each such decision with a single forward pass, leaving no mechanism to allocate additional computation to difficult or consequential choices. We introduce $\tau_0$-VLA, a hierarchical robot foundation model that formulates high-level subtask generation as a compute-scalable inference problem through world-model-guided test-time computation. At each inference step, the high-level policy uses execution memory to generate a subtask and, when needed, searches over alternatives before committing to its output. A low-level policy then executes the generated subtask across multiple robot embodiments. The policy is trained on 40,115 hours of heterogeneous real-world data with multimodal co-training. Across in-domain and distribution-shifted settings, allocating additional test-time computation substantially improves next-subtask prediction accuracy, and these gains translate into higher closed-loop success on long-horizon robot manipulation tasks.

cs.RO↗

arXiv2026

On the positivity of Yamabe invariant and Paneitz operator

Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. We show that the existence of a conformal metric with positive $Q$-curvature $Q_g$ and positive scalar curvature $R_g$ is equivalent to the positivity of both the Yamabe invariant $Y(M^n,[g])$ and the Paneitz operator $P_g$. For $n=5$, this equivalence confirms a conjecture of Gursky-Hang-Lin (2016, IMRN). Furthermore, assuming $Y(M^n,[g])>0$, $Q_g\ge 0$, and $Q_g\not\equiv 0$, we prove that both $R_g$ and $P_g$ are positive which resolves a problem of Hang-Yang (2016, CPAM). As a corollary, we show that the hypotheses of Gursky-Malchiodi (2015, JEMS) are equivalent to those of Hang-Yang (2016, CPAM).

math.DG↗

arXiv2026

Bonnet-Myers type theorems for $Q$-curvature on four-manifolds

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies $|\pi_1(M^4)|\leq 16\pi^2/(\int_{M^4}Q_g d\mu_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\geq \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi/\sqrt{15\theta}$.

math.DG↗

arXiv2026

A sharp isoperimetric inequality and the top order $Q$-curvature

For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$

math.DG↗

arXiv2026

Resonant tunneling diode-integrated terahertz transceiver module for wireless communications

Terahertz bands enable ultra-broadband wireless communications but require compact, low-cost, and efficient transceiver modules. Conventional implementations based on metallic waveguides or silicon lenses suffer from high loss, bulkiness, and fabrication complexity. Here, we present a compact terahertz transceiver module enabled by a resonant tunneling diode (RTD) integrated with a photonic-electronic antenna chain. The RTD on InP is coupled to a modified Vivaldi antenna and an all-silicon effective-medium-clad waveguide, terminating in a rod antenna interfaced with a 3D-printed cyclic olefin copolymer lens. This architecture enables broadband directive radiation without matching networks or anti-reflection coatings. Packaged in a low-cost 3D-printed PLA enclosure, the module achieves realized gains of 28-33 dBi (E11x) and 30-33 dBi (E11y) across 220-330 GHz. As a receiver, it exhibits a noise voltage density of 5.6 x 10^-9 V/sqrt(Hz), a minimum noise equivalent power of 1.8 pW/sqrt(Hz), and an average responsivity of 6.8 kV/W. It supports error-free transmission up to 30 Gbit/s (OOK) and 80 Gbit/s (16-QAM) over 10 cm, and enables real-time uncompressed high-definition video streaming over 1 m. As a transmitter, it achieves error-free OOK transmission up to 12 Gbit/s at 332 GHz. These results demonstrate a promising terahertz transceiver architecture for 6G systems.

physics.optics↗

arXiv2025

Obstructions to prescribed Q-curvature of complete conformal metrics on $\mathbb{R}^n$

We provide some obstructions to the prescribed Q-curvature problem for the complete conformal metrics on $\mathbb{R}^n$ with finite total Q-curvature. One of them is a Bonnet-Mayer type theorem with respect to Q-curvature. Others are related to the decay rate of the prescribed functions.

math.DG↗

arXiv2025

On geometry of $Q^{(2k)}_g$-curvature

The main purpose of current article is to study the geometry of $Q$-curvature. For simplicity, we start with a simple model: a complete and conformal metric $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$. Assuming that the metric $g$ has non-negative $nth$-order $Q$-curvature and non-negative scalar curvature, we show that the Ricci curvature is non-negative. If we further assume that the isoperimetric ratio near the end is positive, we show that the growth rate of $kth$ elementary symmetric function $σ_k(g)$ of Ricci curvature over geodesic ball of radius $r$ is at most polynomial in $r$ with order $n-2k$ for all $1 \leq k \leq \frac{n-2}{2}$. Similarly, we are able to show that the same growth control holds for $2kth$-order $Q$-curvature. Finally, we show that for $k=1$ or $2$, the gap theorems for $Q^{(2k)}_g$ hold true.

math.DG↗

arXiv2025

On a new region for the Lane-Emden conjecture in higher dimensions

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -Δu = v^p, \quad -Δv = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By employing an Obata-type integral inequality, Picone's identity, and exploiting the scaling invariance of the system, we prove that the conjecture holds for any dimension $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}. \]

math.AP↗

arXiv2025

On positivity of the Q-curvatures of conformal metrics

We mainly show that for a conformal metric $g=u^{\frac{4}{n-2m}}|dx|^2$ on $\mathbb{R}^n$ with $n\geq 2m+1$, if the higher order Q-curvature $Q^{(2m)}_g$ is positive and has slow decay barrier near infinity, the lower order Q-curvature $Q^{(2)}_g$ and $Q^{(4)}_g$ are both positive if $m$ is at least two.

math.DG↗

arXiv2025

Conformal metrics with finite total Q-curvature revisited

Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we provides a necessary and sufficient condition for the metric to be normal without assuming metric completeness. As applications, we derive volume comparison theorems and prove a positive mass type theorem related to Q-curvature.

math.DG↗

arXiv2025

Existence of complete conformal metrics on $\mathbb{R}^n$ with prescribed Q-curvature

Given a smooth function $f(x)$ on $\mathbb{R}^n$ which is positive somewhere and satisfies $f(x)=O(|x|^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx|^2$ with finite total Q-curvature such that its Q-curvature equals to $f(x)$.

math.DG↗

arXiv2025

Beam Manipulation for Terahertz Communications: A New Quality Productive Force

The terahertz frequency band, ranging from 0.1 to 10 THz, offers extensive spectral resources for next-generation wireless communication systems. To compensate for the limited transmission power of terahertz transceivers and the significant propagation losses in terahertz channels, high-gain directional antennas are essential. Dynamic beam manipulation is therefore crucial for enabling practical communication applications. Moreover, the stringent gain requirements in terahertz systems result in an expanded Fresnel region, highlighting the critical need for efficient beam manipulation techniques in both near-field and far-field conditions. This article provides a comprehensive overview of terahertz beam manipulation techniques. It begins with an introduction of diffraction theory as the foundational propagation model for beam manipulation. Detailed examples tailored to specific communication scenarios are then presented. Experimental verifications using 3-D printed lenses are included for three distinct beam manipulation cases. Alternative approaches for achieving beam manipulation, such as metasurfaces and reconfigurable intelligent surfaces, are briefly discussed.

physics.app-ph↗

arXiv2024

A Remark On Case-Gursky-Vétois identity and its applications

Based on the works of Gursky (CMP, 1997), Vétois (Potential Anal., 2023) and Case (Crelle's journal, 2024), we make use of an Obata type formula established in these works to obtain some Liouville type theorems on conformally Einstein manifolds. In particular, we solve Hang-Yang conjecture (IMRN, 2020) via an Obata-type argument and obtain optimal perturbation.

math.DG↗

arXiv2024

The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II)

This is a continuation of our previous work (Advances in Mathematics 450 (2024), Paper No. 109768). In this paper, we characterize complete metrics with finite total Q-curvature as normal metrics for all dimensional cases. Secondly, we introduce another volume entropy to provide geometric information regarding complete non-normal metrics with finite total Q-curvature. In particular, we show that if the scalar curvature is bounded from below, the volume growth of such complete metrics is controlled.

math.DG↗

arXiv2024

Higher order Bol's inequality and its applications

In the conformal class of Euclidean space, we give some volume comparison theorems with help of Q-curvature. Meanwhile, for compact four dimensional manifolds with non-negative scalar curvature, we give a volume rigidity theorem with respect to Q-curvature. Finally, we make use of these results to give some sufficient and necessary conditions for the existence of solutions to some conformally invariant equations which answers an open problem raised by Hyder-Martinazzi (2021, JDE).

math.DG↗