Searcharxiv⌕ Search

arXiv subjects

Chao Xia

Publications and source records attributed to Chao Xia.

At least 19 recordsLinked to original sources
arXiv2026

Alexandrov-Type Rigidity for Minimal Capillary Hypersurfaces on Rotational Supports

In this paper, we prove that, under suitable conditions on the generating profile, every connected, compact embedded minimal capillary hypersurface supported on a one-ended rotational hypersurface in the Euclidean space is a horizontal slice. We also construct a smooth one-ended rotational support carrying a non-horizontal planar capillary $n$-ball, showing that the main slope condition cannot in general be omitted. For obtuse contact angles, we further obtain a catenoid-band classification under a reverse profile inequality.

math.DG↗
arXiv2026

Stable constant weighted mean curvature hypersurfaces in anti-Gaussian space

We prove that in the anti-Gaussian space $\bigl(\R^{m+1},\bar g_{\Euc},e^{|x|^2/4}\,dx\bigr)$, $m\ge2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.

math.DG↗
arXiv2026

An Alexandrov-type theorem in warped product manifolds with radial density

In this paper, we establish a Heintze--Karcher inequality for closed embedded hypersurfaces in a class of warped product manifolds endowed with radial density. As a consequence, we prove Alexandrov-type theorem for constant weighted mean curvature hypersurfaces in such spaces. In particular, we prove that a closed embedded $\lambda$-self-expander in the Euclidean space must be a round sphere centered at the origin.

math.DG↗
arXiv2026

Multi-Granularity Conformal Prediction for Reliable Neural-Operator Automotive Aerodynamic Surrogates

High-fidelity computational fluid dynamics (CFD) provides detailed aerodynamic data for vehicle design, but its cost limits design iteration. Neural-operator surrogates reduce this cost, yet their deterministic predictions do not indicate when a geometry or surface region is reliable. This study develops a conformal-prediction framework for reliability-aware automotive aerodynamic surrogate modeling on the DrivAerML dataset. GeoTransolver is the main backbone, while Transolver assesses transfer across neural-operator architectures. For drag coefficient prediction, conformalized quantile regression constructs calibrated case-level intervals. For surface pressure and wall shear stress (WSS), point prediction is combined with residual-scale estimation and residual-normalized conformal calibration to obtain spatially adaptive intervals. Global absolute, point-adaptive normalized, and case-wise normalized calibration are compared under split and cross-validation-assisted out-of-fold protocols. All experiments target 90% nominal coverage. Conformal calibration corrects the under-coverage of raw drag-coefficient quantile intervals, while out-of-fold score aggregation reduces the Monte Carlo coverage standard deviation from 10.41 to 3.10 percentage points. For surface fields, point-adaptive normalized calibration yields the narrowest near-nominal intervals, reducing mean width by 22.68% for pressure and 25.35%--27.09% for WSS under the out-of-fold protocol. Case-wise normalized calibration is more conservative but improves vehicle-level reliability. Smoothness regularization reduces the residual-scale local-variation score by 74.29% and lowers interval widths without material coverage loss. The framework converts deterministic neural-operator outputs into calibrated reliability indicators for prioritizing uncertain vehicle geometries and surface regions in follow-up CFD verification.

physics.flu-dyn↗
arXiv2026

U3DWind: A Low Altitude Wind Field Dataset and Benchmark for Urban Air Mobility

Urban Air Mobility (UAM) requires reliable assessment of low-altitude wind hazards, because winds, gusts, and building-induced turbulence have been recognized as critical factors affecting vehicle stability, route feasibility, vertiport siting, and airspace management. While wind-tunnel experiments, computational fluid dynamics (CFD), multiscale downscaling, reduced-order models, and UAV planning datasets have advanced wind-aware analysis, public resources for data-driven, city-scale UAM planning remain limited in geographic coverage, scenario diversity, vertical extent, building realism, and task-oriented benchmarking. To address this gap, we introduce U3DWind, a building-resolved low-altitude wind-field dataset generated using our GPU-accelerated Lattice Boltzmann Method--Large-Eddy Simulation (LBM-LES) framework for rapid urban flow simulation. U3DWind covers five megacities in China: Beijing, Shanghai, Guangzhou, Shenzhen, and Hong Kong. It contains 720 simulations, with 16 inflow directions, three reference wind speeds, and three seasonal atmospheric scenarios (annual, summer, and winter) for each city. At a 10 m grid resolution, the dataset provides three-dimensional three-component (3D3C) velocity, turbulent kinetic energy (TKE), flow density, and fluid--solid masks. To support operationally relevant evaluation, we further define five baseline tasks: wind-field prediction, sparse-sensor wind-field reconstruction, site wind-exposure ranking, airworthiness wind-compliance risk scoring, and noise propagation modeling. As a multi-city, building-resolved 3D urban wind-field dataset, U3DWind enables systematic evaluation of wind-induced impacts in low-altitude traffic scenarios and provides an open benchmark for urban airspace management and data-driven high-fidelity urban flow simulation.

physics.flu-dyn↗
arXiv2026

Near-real-time, meter-scale 3D urban wind modeling for low-altitude micrometeorology: numerical verification of a GPU-accelerated lattice Boltzmann framework

This study presents a near-real-time, meter-scale three-dimensional urban wind simulation framework for low-altitude flight events in complex urban meteorological environments. It reconstructs high-resolution wind fields by combining sparse observations with efficient microscale flow modeling. The framework integrates lattice Boltzmann method large-eddy simulation (LBM-LES), high-fidelity urban morphology reconstruction that explicitly resolves real building details, and observation-driven boundary assimilation into a rapid end-to-end pipeline for realistic urban domains. Multi-site Doppler lidar measurements from dense urban Guangzhou, China, are used for evaluation. The system reconstructs three-dimensional wind fields at 5 m resolution over kilometer-scale domains within minutes. Robustness and accuracy are tested through controlled observation reduction, independent validation against withheld lidar stations, and sensitivity analyses of grid resolution and precursor domain extent. Results show stable reproduction of vertical wind structures and key local flow features under complex morphology and limited observations, providing a scalable pathway for near-real-time urban wind reconstruction.

physics.flu-dyn↗
arXiv2026

Mass-$p$-Capacity Inequalities in Asymptotically Flat Half-Spaces

In this paper, we establish general monotone quantities and sharp mass-capacity inequalities related to $p$-capacitary functions in $3$-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere.

math.DG↗
arXiv2026

A half-space Liouville theorem for anisotropic minimal graph with free boundary

In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.

math.DG↗
arXiv2024

The Logarithmic Sobolev inequality on non-compact self-shrinkers

In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds.

math.AP↗
arXiv2024

Monotonicity Formulas for Capillary Surfaces

In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. \href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $\mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. \href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. \href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).

math.DG↗
arXiv2024

Willmore-type inequality in unbounded convex sets

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $\Sigma\subset K$ with boundary $\partial\Sigma\subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac1{n+1}\int_{\Sigma}\vert{H}\vert^n{\rm d}A\ge{\rm AVR}(K)\vert\mathbb{B}^{n+1}\vert.$$ Moreover, equality holds if and only if $\Sigma$ is a part of a sphere and $K\setminus\Omega$ is a part of the solid cone determined by $\Sigma$. Here $\Omega$ is the bounded domain enclosed by $\Sigma$ and $\partial K$, $H$ is the normalized mean curvature of $\Sigma$, and ${\rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.

math.DG↗
arXiv2024

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II

In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture [16, Conjecture 1.5] follows as its consequence.

math.MG↗
arXiv2024

Stable anisotropic capillary hypersurfaces in a half-space

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape. On the other hand, we prove a Bernstein-type theorem for stable anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area growth assumption.

math.DG↗
arXiv2024

CMFDFormer: Transformer-based Copy-Move Forgery Detection with Continual Learning

Copy-move forgery detection aims at detecting duplicated regions in a suspected forged image, and deep learning based copy-move forgery detection methods are in the ascendant. These deep learning based methods heavily rely on synthetic training data, and the performance will degrade when facing new tasks. In this paper, we propose a Transformer-style copy-move forgery detection network named as CMFDFormer, and provide a novel PCSD (Pooled Cube and Strip Distillation) continual learning framework to help CMFDFormer handle new tasks. CMFDFormer consists of a MiT (Mix Transformer) backbone network and a PHD (Pluggable Hybrid Decoder) mask prediction network. The MiT backbone network is a Transformer-style network which is adopted on the basis of comprehensive analyses with CNN-style and MLP-style backbones. The PHD network is constructed based on self-correlation computation, hierarchical feature integration, a multi-scale cycle fully-connected block and a mask reconstruction block. The PHD network is applicable to feature extractors of different styles for hierarchical multi-scale information extraction, achieving comparable performance. Last but not least, we propose a PCSD continual learning framework to improve the forgery detectability and avoid catastrophic forgetting when handling new tasks. Our continual learning framework restricts intermediate features from the PHD network, and takes advantage of both cube pooling and strip pooling. Extensive experiments on publicly available datasets demonstrate the good performance of CMFDFormer and the effectiveness of the PCSD continual learning framework.

cs.CV↗
arXiv2024

Capillary hypersurfaces, Heintze-Karcher's inequality and Zermelo's navigation

In this paper, we establish a Heintze-Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect to this Finsler metric. Our results indicate that the relationship between capillary hypersufaces and hypersurfaces with free boundary is similar to the one between Finsler geometry and Riemannian geometry.

math.DG↗
arXiv2023

Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces

In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the second part, we prove quantitative stability results for the Serrin-type partially overdetermined problem, as well as capillary almost constant mean curvature hypersurfaces in the half-space.

math.AP↗