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Junbang Liu

Publications and source records attributed to Junbang Liu.

10 recordsLinked to original sources
arXiv2026

The relationship between spacetime singularities and regions at infinity

Ideal attached points are a core concept in general relativity for pseudo-Riemannian manifolds, and whether the spacetime can be extended with certain properties is a central consideration in their choice. This paper establishes a sufficient condition for the separability between singularities and points at infinity for any maximally extended pseudo-Riemannian manifold. We focus on the incomplete geodesics of $(\mathcal{M},g)$, and produce an envelopment $(\mathcal{M},g,\hat{\mathcal{M}})$ of the spacetime such that an incomplete geodesic $\gamma:[0,1) \rightarrow \mathcal{M}$ has an endpoint $q$ in $\hat{\mathcal{M}}$. If there is no pair of geodesics approaching $q$ which is intertwined, then $q$ is a singularity. Additionally, $q$ will not be approached by any geodesic with infinite affine parameter, and therefore cannot cover a point at infinity, thereby rendering it a {\it pure singularity} in the abstract boundary framework. We apply the Endpoint Theorem to the maximal g-boundary introduced by Graf and Beld-Serrano in arXiv:2307.11034, and also provide a result on the separability between directional singularities and pure singularities. This analysis is then applied to the Schwarzschild spacetime.

gr-qc↗
arXiv2026

Semistable $J$-equation on K\"ahler threefold

For a $3$-dimensional compact K\"ahler manifold $X$ with a pair of K\"ahler class $(\alpha,\beta)$ which is $J$-semistable, we show that the $J$-null locus of $(\alpha,\beta)$ forms a subvariety of $X$. We also give an analyic characterization of the $J$-null locus using K\"ahler currents with analytic singularities analogous to the theorem of Collins-Tosatti \cite{CT15}. As an application, we show the smooth convergence of the $J$-flow off the $J$-null locus. It gives higer order partial regularity for weak solutions of semistable $J$-equations obtained in \cite{M26b}.

math.DG↗
arXiv2026

The $J$-equation on K\"ahler manifolds under a smooth boundary cone condition

For a $n$-dimensional compact K\"ahler manifold $X$ with a pair of K\"ahler classes $(\alpha,\beta)$, we show that the algebraically defined $J$-null locus is equal to the analytically defined $J$-non-ample locus under the assumption of $J$-bigness(which is automatic for semistable pair up to dimension $3$ \cite{3d}), and boundary cone condition $c_{\alpha,\beta}\omega^{n-1}-(n-1)\omega^{n-2}\wedge\chi\geq 0$ for some K\"ahler form $\omega\in \alpha,\chi\in \beta$. The key tool is the generalized Khovanskii-Teissier inequality associated to $J$ equation formulated by Collins \cite{CT21} and its extension to singular K\"ahler space.

math.DG↗
arXiv2025

On the uniqueness of even $L^p$ Minkowski problem

We prove that there is a unique $p_0\in [0,1)$, which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even $L^p$ Minkowski problem has a unique solution for $p\geq p_0$, and the uniqueness fails for infinitely many convex bodies if $p p_0$.

math.MG↗
arXiv2025

ContrastiveGaussian: High-Fidelity 3D Generation with Contrastive Learning and Gaussian Splatting

Creating 3D content from single-view images is a challenging problem that has attracted considerable attention in recent years. Current approaches typically utilize score distillation sampling (SDS) from pre-trained 2D diffusion models to generate multi-view 3D representations. Although some methods have made notable progress by balancing generation speed and model quality, their performance is often limited by the visual inconsistencies of the diffusion model outputs. In this work, we propose ContrastiveGaussian, which integrates contrastive learning into the generative process. By using a perceptual loss, we effectively differentiate between positive and negative samples, leveraging the visual inconsistencies to improve 3D generation quality. To further enhance sample differentiation and improve contrastive learning, we incorporate a super-resolution model and introduce another Quantity-Aware Triplet Loss to address varying sample distributions during training. Our experiments demonstrate that our approach achieves superior texture fidelity and improved geometric consistency.

cs.CV↗
arXiv2024

On relative $L^\infty$ estimate for complex Monge-Amp\`ere equations

We prove a relative $L^\infty$ estimate for a class of complex Monge-Amp\`ere type equations on K\"ahler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about modulus of continuity, stability estimates, and $W^{1,1}$-estimates of Green's functions. The argument is based on the PDE method developed by Guo-Phong-Tong and constructing appropriate comparison metrics from entropy bound.

math.DG↗
arXiv2024

Complex Alexandrov-Bakelman-Pucci estimate and its applications

We prove an Alexandrov-Bakelman-Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $2^{2n}|\det(u_{i\bar{j}})|^2\geq |\det(\nabla^2u)|$) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge-Amp\`ere equations. The approach is based on the De Giorgi iteration method developed by Guo-Phong-Tong for equations of complex Monge-Amp\`ere type.

math.DG↗
arXiv2024

Depth Estimation Algorithm Based on Transformer-Encoder and Feature Fusion

This research presents a novel depth estimation algorithm based on a Transformer-encoder architecture, tailored for the NYU and KITTI Depth Dataset. This research adopts a transformer model, initially renowned for its success in natural language processing, to capture intricate spatial relationships in visual data for depth estimation tasks. A significant innovation of the research is the integration of a composite loss function that combines Structural Similarity Index Measure (SSIM) with Mean Squared Error (MSE). This combined loss function is designed to ensure the structural integrity of the predicted depth maps relative to the original images (via SSIM) while minimizing pixel-wise estimation errors (via MSE). This research approach addresses the challenges of over-smoothing often seen in MSE-based losses and enhances the model's ability to predict depth maps that are not only accurate but also maintain structural coherence with the input images. Through rigorous training and evaluation using the NYU Depth Dataset, the model demonstrates superior performance, marking a significant advancement in single-image depth estimation, particularly in complex indoor and traffic environments.

cs.CV↗
arXiv2022

Twist and Turn Squeezing in a Multi-Mode Bose-Einstein Condensate

Here we examine the generation of Twist and Turn (TNT) Squeezing in a large atom-number Bose-Einstein Condensate for the purposes of generating quantum-enhanced states for atom interferometry. Unlike previous analysis, we examine situations where the multi-mode dynamics is significant, and cannot be captured by a simple single-mode model. We find that in some regimes, with careful choice of the rotation parameter, we can still obtain squeezing much more rapidly than via one-axis twisting (OAT).

quant-ph↗