arXiv subjects

Teng Huang

Publications and source records attributed to Teng Huang.

At least 19 recordsLinked to original sources

arXiv2026

Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions

In this article, we investigate conformal Killing vector fields on closed Riemannian manifolds under a curvature pinching condition. By establishing a new Bochner-type identity for the $1$-form dual to a conformal Killing vector field, we derive a sharp gradient estimate via a Moser iteration procedure. Based on this estimate, we prove that, under a suitable upper bound on the Ricci curvature, every nontrivial conformal Killing vector field must be nowhere vanishing. Consequently, on even-dimensional manifolds with non-zero Euler characteristic, every conformal Killing vector field vanishes identically, which in turn implies that the conformal transformation group of such a manifold is finite. Our results extend the classical rigidity theorems of Yano and Bochner from the setting of non-positive Ricci curvature to that of small positive Ricci curvature, and generalize the recent results of Chen and Han from Killing vector fields to conformal Killing vector fields.

math.DG↗

arXiv2026

$L^{1}$-Integrability of $L^{2}$-Harmonic Forms and the Hopf Conjecture

In this note, we study $L^{2}$-harmonic forms on complete simply-connected Riemannian manifolds with non-positive sectional curvature. We first establish an a priori $L^{\infty}$-estimate for such forms via Moser iteration, under the curvature bounds $-K\leq\mathrm{sec}_{g}\leq0$. We then prove that any $L^{2}$-harmonic form which is also $L^{1}$-integrable must vanish identically. Consequently, on the universal cover of a closed non-positively curved manifold, the $k$-th $L^{2}$-Betti number vanishes if and only if every $L^{2}$-harmonic $k$-form is $L^{1}$-integrable. This criterion reformulates a topological vanishing statement as an analytic integrability condition.

math.DG↗

arXiv2026

Vafa-Witten Equations and Conformal Geometry

In this article, we establish geometric and analytic constraints imposed by the existence of nontrivial solutions to the Vafa-Witten equations on closed 4-manifolds. Using conformal invariance and refined Bochner-type estimates, we first prove an inequality relating the Yamabe constant $Y(g)$ to the $L^{2}$-norm of the self-dual Weyl tensor: $Y(g)\leq 2\sqrt{6}\|W_{g}^{+}\|_{L^2}$; when $Y(g)>0$, this yields a topological lower bound $\int_{M} |W_{g}^{+}|^{2} \geq \frac{4}{3}\pi^{2}(2\chi(M)+3\sigma(M))$. In the equality case, we show that the manifold must be K\"{a}hler with nonnegative scalar curvature and that the connection is reducible. As an application, for positive Einstein manifolds with $\operatorname{Ric}=3g$ admitting an irreducible Vafa-Witten solution, we obtain a sharp volume bound and prove the manifold cannot be K\"{a}hler. Through dimensional reduction $S^{1}\times N$, we establish a one-to-one correspondence between stable flat connections on a closed 3-manifold $N$ and $S^{1}$-invariant Vafa-Witten solutions, which yields a new estimate for the Yamabe constant $Y(g_{S^{1}\times N})\leq 2\sqrt{6\pi}\big(\int_{N}|\operatorname{Ric}(g_{N})-\frac{1}{3} R_{g_{N}}g_{N}|^2\big)^{1/2}$. Finally, under a regularity assumption that every anti-self-dual connection in the compactified moduli space is regular, we prove an energy gap: there exists $\varepsilon(g,P)>0$ such that any Vafa-Witten solution satisfies either $F_{A}^{+}\equiv0$ or $\|F_{A}^{+}\|_{L^{2}}\geq\varepsilon$.

math.DG↗

arXiv2026

Hirzebruch $\chi_{y}$-genus of compact almost K\"{a}hler manifold with negative sectional curvature

Let $(X,J,\omega)$ be a closed $2n$-dimensional almost K\"{a}hler manifold with negative sectional curvature. We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch $\chi_{y}$-genus satisfy the inequality $(-1)^{n-p}\chi_{p}(X)\geq 1$ for all $p=0,1,\cdots,n$. In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies $(-1)^{n}\chi(X)\geq n+1$. The proof is based on new $L^{2}$-estimates for harmonic forms on the universal covering, combined with a refined vanishing theorem for the operator $\bar{\partial}+\bar{\partial}^{*}$ and Atiyah's $L^{2}$-index theorem. This work extends the classical result of Gromov [J. Differential Geom., 1991] from the K\"{a}hler to the almost K\"{a}hler setting under the stated smallness condition.

math.DG↗

arXiv2026

Schr\"odinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds

We establish new connections between integral curvature bounds and the Euler characteristic of closed Riemannian manifolds through the perspective of Schr\"odinger-type operators. Central to our approach is the twisted Dirac operator $\mathcal{D}_{\theta}$, whose index equals $\chi(M)$. Under integral smallness conditions on the negative part of a potential $V$ and a Sobolev--Poincar\'e inequality, we show that a suitable scaling of $\theta$ forces the kernel of $\mathcal{D}_{t\theta}$ to vanish, thereby implying $\chi(M)=0$. Applying this framework to geometrically natural potentials yields several topological consequences. In even dimensions, sufficiently small integral bounds on partial sums of curvature operator eigenvalues force $\chi(M)$ either to vanish or to have a sign determined by the middle dimension. For four-manifolds, a small $L^{p}$-norm of the negative Ricci curvature relative to the diameter guarantees $\chi(M)\ge 0$. Moreover, when $\chi(M)\neq 0$ we obtain a Li--Yau type lower bound for the first eigenvalue of the rough Laplacian on $1$-forms in terms of the diameter and an integral curvature quantity. Subsequently, we provide an explicit lower bound for the first eigenvalue of the Laplacian on $1$-forms under almost nonnegative curvature conditions, thereby giving an affirmative answer to Yau's Problem 79.

math.DG↗

arXiv2025

Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic. In contrast to the case of functions, such a Li-Yau-type estimate does not hold in general, as evidenced by existing counterexamples. Under assumptions including a lower bound on Ricci curvature, an upper bound on diameter, and an $L^{2p}$-norm bound on the Riemann curvature tensor, we prove that $\lambda^{(1)}_{1}$ is bounded below by a positive constant depending on these parameters. As applications, we derive vanishing results for the Euler characteristic under certain Ricci curvature bounds and the presence of a nonzero Killing vector field, extending classical Bochner-type theorems.

math.DG↗

arXiv2025

We Can Hear You with mmWave Radar! An End-to-End Eavesdropping System

With the rise of voice-enabled technologies, loudspeaker playback has become widespread, posing increasing risks to speech privacy. Traditional eavesdropping methods often require invasive access or line-of-sight, limiting their practicality. In this paper, we present mmSpeech, an end-to-end mmWave-based eavesdropping system that reconstructs intelligible speech solely from vibration signals induced by loudspeaker playback, even through walls and without prior knowledge of the speaker. To achieve this, we reveal an optimal combination of vibrating material and radar sampling rate for capturing high-quality vibrations using narrowband mmWave signals. We then design a deep neural network that reconstructs intelligible speech from the estimated noisy spectrograms. To further support downstream speech understanding, we introduce a synthetic training pipeline and selectively fine-tune the encoder of a pre-trained ASR model. We implement mmSpeech with a commercial mmWave radar and validate its performance through extensive experiments. Results show that mmSpeech achieves state-of-the-art speech quality and generalizes well across unseen speakers and various conditions.

cs.SD↗

arXiv2025

Active Domain Adaptation for mmWave-based HAR via Renyi Entropy-based Uncertainty Estimation

Human Activity Recognition (HAR) using mmWave radar provides a non-invasive alternative to traditional sensor-based methods but suffers from domain shift, where model performance declines in new users, positions, or environments. To address this, we propose mmADA, an Active Domain Adaptation (ADA) framework that efficiently adapts mmWave-based HAR models with minimal labeled data. mmADA enhances adaptation by introducing Renyi Entropy-based uncertainty estimation to identify and label the most informative target samples. Additionally, it leverages contrastive learning and pseudo-labeling to refine feature alignment using unlabeled data. Evaluations with a TI IWR1443BOOST radar across multiple users, positions, and environments show that mmADA achieves over 90% accuracy in various cross-domain settings. Comparisons with five baselines confirm its superior adaptation performance, while further tests on unseen users, environments, and two additional open-source datasets validate its robustness and generalization.

cs.HC↗

arXiv2025

Decomposition of the Curvature Operator and Applications to the Hopf Conjecture

In this article, we investigate the interplay between the curvature operator, Weyl curvature, and the Hopf conjecture on compact Riemannian manifolds of even dimension. By decomposing the curvature operator into Hermitian components, we develop eigenvalue criteria for sectional curvature and prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor. Our results confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature, including locally conformally flat cases, and provide new rigidity theorems under harmonic Weyl curvature conditions.

math.DG↗

arXiv2025

Garding cones and positivity of curvature operators

This article explores the relationship between Garding cones, demonstrating that the shift cone $\overline{\Gamma}^{+}_{2}(\alpha)$ is contained in $\overline{\mathcal{P}}_{m}$. By combining these results with the study of positivity properties of curvature operators, we establish several new connections between algebraic positivity conditions and the geometry of underlying Riemannian manifolds. Our main theorems reveal how shifted cone conditions on curvature operators-both standard and of the second kind-constrain topology, including vanishing theorems for Betti numbers and characterizations of spherical space forms.

math.DG↗

arXiv2025

Hodge decomposition and Hard Lefschetz Condition on almost K\"{a}hler manifolds

In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost K\"{a}hler manifold $(X, J,\omega)$. We show that if the almost complex structure $J$ on the almost K\"{a}hler manifold $X$ is not too non-integrable in some sense, then the spaces $\mathcal{H}^{\bullet}_{d}$ have the Hodge decomposition $\mathcal{H}^{k}_{d}=\oplus_{p+q=k}\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\mathcal{H}^{\bullet}_{d}$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost K\"{a}hler manifold with $b^{+}_{2}(X)\geq2$.

math.DG↗

arXiv2025

Dolbeault-Morse-Novikov Cohomology on Complex manifolds and its applications

In this article, we investigate the topological properties of complex manifolds by studying Dolbeault-Morse-Novikov cohomology. By establishing an integral inequality, we obtain two main results: (1) When a closed complex manifold admits a nonzero parallel $(0,1)$-form, the Dolbeault-Morse-Novikov cohomology must be trivial, which implies that the Hirzebruch $\chi_{y}$-genus vanishes. (2) When a closed complex manifold admits a nowhere vanishing $(0,1)$-form, we establish a vanishing theorem for a certain class of twisted Dirac operators, which also forces the Hirzebruch $\chi_{y}$-genus to be zero. In particular, we prove that the Hirzebruch $\chi_{y}$-genus of a closed complex manifold vanishes if and only if the manifold admits a nowhere vanishing real vector field. These results generalize some classical theorems from Riemannian manifolds to the complex setting. As a culminating application, we prove that the Hirzebruch $\chi_{y}$-genus must vanish on closed Gauduchon manifolds admitting positive holomorphic scalar curvature.

math.DG↗

arXiv2025

One Snapshot is All You Need: A Generalized Method for mmWave Signal Generation

Wireless sensing systems, particularly those using mmWave technology, offer distinct advantages over traditional vision-based approaches, such as enhanced privacy and effectiveness in poor lighting conditions. These systems, leveraging FMCW signals, have shown success in human-centric applications like localization, gesture recognition, and so on. However, comprehensive mmWave datasets for diverse applications are scarce, often constrained by pre-processed signatures (e.g., point clouds or RA heatmaps) and inconsistent annotation formats. To overcome these limitations, we propose mmGen, a novel and generalized framework tailored for full-scene mmWave signal generation. By constructing physical signal transmission models, mmGen synthesizes human-reflected and environment-reflected mmWave signals from the constructed 3D meshes. Additionally, we incorporate methods to account for material properties, antenna gains, and multipath reflections, enhancing the realism of the synthesized signals. We conduct extensive experiments using a prototype system with commercial mmWave devices and Kinect sensors. The results show that the average similarity of Range-Angle and micro-Doppler signatures between the synthesized and real-captured signals across three different environments exceeds 0.91 and 0.89, respectively, demonstrating the effectiveness and practical applicability of mmGen.

cs.CV↗

arXiv2024

Learning Decisions Offline from Censored Observations with {\epsilon}-insensitive Operational Costs

Many important managerial decisions are made based on censored observations. Making decisions without adequately handling the censoring leads to inferior outcomes. We investigate the data-driven decision-making problem with an offline dataset containing the feature data and the censored historical data of the variable of interest without the censoring indicators. Without assuming the underlying distribution, we design and leverage {\epsilon}-insensitive operational costs to deal with the unobserved censoring in an offline data-driven fashion. We demonstrate the customization of the {\epsilon}-insensitive operational costs for a newsvendor problem and use such costs to train two representative ML models, including linear regression (LR) models and neural networks (NNs). We derive tight generalization bounds for the custom LR model without regularization (LR-{\epsilon}NVC) and with regularization (LR-{\epsilon}NVC-R), and a high-probability generalization bound for the custom NN (NN-{\epsilon}NVC) trained by stochastic gradient descent. The theoretical results reveal the stability and learnability of LR-{\epsilon}NVC, LR-{\epsilon}NVC-R and NN-{\epsilon}NVC. We conduct extensive numerical experiments to compare LR-{\epsilon}NVC-R and NN-{\epsilon}NVC with two existing approaches, estimate-as-solution (EAS) and integrated estimation and optimization (IEO). The results show that LR-{\epsilon}NVC-R and NN-{\epsilon}NVC outperform both EAS and IEO, with maximum cost savings up to 14.40% and 12.21% compared to the lowest cost generated by the two existing approaches. In addition, LR-{\epsilon}NVC-R's and NN-{\epsilon}NVC's order quantities are statistically significantly closer to the optimal solutions should the underlying distribution be known.

cs.LG↗

arXiv2023

$L^{2}$-Hodge theory on Complete Almost K\"{a}hler Manifolds and the Hopf Conjecture

In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\"{a}hler manifolds $(X,\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a generalized Hard Lefschetz duality, and a Lefschetz decomposition, all restricted to the space $\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ of forms of pure bidegree. In the second part, as applications of these identities, we prove vanishing theorems for $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\"{a}her form $\omega$. We also provide refined $L^{2}$-estimates to sharpen the vanishing theorems in three specific settings. As a final application, the topology of compact almost K\"ahler manifolds with negative sectional curvature is studied. Under a smallness condition on the Nijenhuis tensor depending on the curvature, the authors prove that the Hirzebruch $\chi_{y}$-genus satisfies $(-1)^{n-p}\chi_{p}(X)\geq1$ for all $p=0,1,\cdots,n$, which in particular implies the Hopf conjecture for the Euler number $(-1)^{n}\chi(X)\geq n+1$. This extends a classical result of Gromov [J. Differential Geom., 1991] from the K\"ahler to the almost K\"ahler setting.

math.DG↗

arXiv2022

Multi-Metric AutoRec for High Dimensional and Sparse User Behavior Data Prediction

User behavior data produced during interaction with massive items in the significant data era are generally heterogeneous and sparse, leaving the recommender system (RS) a large diversity of underlying patterns to excavate. Deep neural network-based models have reached the state-of-the-art benchmark of the RS owing to their fitting capabilities. However, prior works mainly focus on designing an intricate architecture with fixed loss function and regulation. These single-metric models provide limited performance when facing heterogeneous and sparse user behavior data. Motivated by this finding, we propose a multi-metric AutoRec (MMA) based on the representative AutoRec. The idea of the proposed MMA is mainly two-fold: 1) apply different $L_p$-norm on loss function and regularization to form different variant models in different metric spaces, and 2) aggregate these variant models. Thus, the proposed MMA enjoys the multi-metric orientation from a set of dispersed metric spaces, achieving a comprehensive representation of user data. Theoretical studies proved that the proposed MMA could attain performance improvement. The extensive experiment on five real-world datasets proves that MMA can outperform seven other state-of-the-art models in predicting unobserved user behavior data.

cs.IR↗

arXiv2022

Auto-Focus Contrastive Learning for Image Manipulation Detection

Generally, current image manipulation detection models are simply built on manipulation traces. However, we argue that those models achieve sub-optimal detection performance as it tends to: 1) distinguish the manipulation traces from a lot of noisy information within the entire image, and 2) ignore the trace relations among the pixels of each manipulated region and its surroundings. To overcome these limitations, we propose an Auto-Focus Contrastive Learning (AF-CL) network for image manipulation detection. It contains two main ideas, i.e., multi-scale view generation (MSVG) and trace relation modeling (TRM). Specifically, MSVG aims to generate a pair of views, each of which contains the manipulated region and its surroundings at a different scale, while TRM plays a role in modeling the trace relations among the pixels of each manipulated region and its surroundings for learning the discriminative representation. After learning the AF-CL network by minimizing the distance between the representations of corresponding views, the learned network is able to automatically focus on the manipulated region and its surroundings and sufficiently explore their trace relations for accurate manipulation detection. Extensive experiments demonstrate that, compared to the state-of-the-arts, AF-CL provides significant performance improvements, i.e., up to 2.5%, 7.5%, and 0.8% F1 score, on CAISA, NIST, and Coverage datasets, respectively.

cs.CV↗

arXiv2022

On harmonic symmetries for locally conformally Kähler manifolds

In this article, we study harmonic symmetries on the compact locally conformally Kähler manifold $M$ of $dim_{\mathbb{C}}=n$. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator $\square$. We observe that the spaces $\ker(\square)\capΩ^{l}=\{0\}$ for any $|l-n|\geq2$ and $\kerΔ_{\bar{\partial}}\cap P^{k,n-1-k}\cap\ker(i_{θ^{\sharp}})\cong\ker(\square^{k,n-1-k})$, $\kerΔ_{\bar{\partial}}\cap P^{k,n-k}\cong\ker(\square^{k,n-k})$. Furthermore, suppose that $M$ is a Vaisman manifold, we prove that (i) $α$ is $(n-1)$-form in $\ker(\square)$ if only if $α$ is a transversally harmonic and transversally effective $\mathcal{V}$-foliate form; (ii) $α$ is a $(p,n-p)$-form in $\ker(\square^{p,n-p})$ if only if there are two forms $β_{1}\in\mathcal{S}^{p-1,n-p}$ and $β_{2}\in\mathcal{S}^{p,n-p-1}$ such that $α=θ^{1,0}\wedgeβ_{1}+θ^{0,1}\wedgeβ_{2}$.

math.DG↗