arXiv subjects

Jiaxu Ma

Publications and source records attributed to Jiaxu Ma.

3 recordsLinked to original sources

arXiv2026

Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces

We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de~Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de~Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect \(n|h|^2-H^2\) with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.

math.DG↗

arXiv2024

Anisotropic capillary hypersurfaces in a wedge

We investigate anisotropic capillary hypersurfaces within a wedge in Euclidean space. In this study, we generalize the Minkowski norm \(F\), traditionally employed to define the anisotropic surface energy, to a gauge on the unit sphere \(S^n\). This generalization helps to illuminate a significant relationship between capillary hypersurfaces and hypersurfaces with free boundary. Our main results include new Minkowski formulae and a Heintze-Karcher type inequality. As an application, we prove an Alexandrov-type theorem, thereby extending the known results to the anisotropic setting.

math.DG↗

arXiv2018

Annotation-Free and One-Shot Learning for Instance Segmentation of Homogeneous Object Clusters

We propose a novel approach for instance segmen- tation given an image of homogeneous object clus- ter (HOC). Our learning approach is one-shot be- cause a single video of an object instance is cap- tured and it requires no human annotation. Our in- tuition is that images of homogeneous objects can be effectively synthesized based on structure and illumination priors derived from real images. A novel solver is proposed that iteratively maximizes our structured likelihood to generate realistic im- ages of HOC. Illumination transformation scheme is applied to make the real and synthetic images share the same illumination condition. Extensive experiments and comparisons are performed to ver- ify our method. We build a dataset consisting of pixel-level annotated images of HOC. The dataset and code will be published with the paper.

cs.CV↗