arXiv subjects
Publications and source records attributed to Yuguang Shi.
Through a careful analysis of the Yamabe equation on singular spaces, we establish a positive mass theorem for singular asymptotically hyperbolic manifolds with arbitrary ends. We also derive a rigidity result that is novel even in the smooth setting.
In this paper, we prove the Riemannian positive mass theorem up to dimension $19$, building on a combination of torical symmetrization and the singularity blow-up technique developed in [HSY26], together with the generic regularity theory for area-minimizing hypersurfaces established in [CMS23, CMSW25]. Similar ideas are also employed to investigate the Geroch conjecture up to dimension $12$.
We prove the existence of foliations by area-minimizing hypersurfaces in asymptotically flat (AF) manifolds with arbitrary dimension and arbitrary ends. Also we provide behaviors of those hypersurfaces near the infinity of AF ends and demonstrate that the singular set of those area-minimizing hypersurfaces is outside AF ends (cf Theorem \ref{thm: foliation}). Building on the positive mass theorem for AF manifolds with arbitrary ends, we establish a global behavior for free-boundary area-minimizing hypersurfaces inside coordinate cylinders in AF manifolds of dimension less than or equal to $8$ (cf. Theorem \ref{thm: 8dim Schoen conj})
We prove two "Singularity removal rigidity theorems" for minimal hypersurfaces with isolated singularities in manifolds of nonnegative scalar curvature (Theorems \ref{thm: rigidity for minimal surface} and \ref{thm: georch free of singularity}). In particular, we observe a new phenomenon that the extremal scalar curvature condition forces smoothness, which reveals a kind of positive effect of minimal hypersurface singularities in scalar curvature geometry. As an application, we obtain a direct proof of the positive mass theorem (PMT) for asymptotically flat $8$-manifolds with arbitrary ends (Theorem \ref{thm: pmt8dim}), without using N. Smale's generic regularity theorem. A key ingredient is a new spectral version of PMT for AF manifolds with arbitrary ends, whose proof relies on PMT for asymptotically locally flat (ALF) manifolds with $\mathbf{S}^1$-symmetry.
Building upon dimension reduction techniques in the study of positive scalar curvature (PSC) geometry, we prove an effective version of the positive mass theorem (PMT) for asymptotically flat (AF) manifolds of dimension $n\leq 8$ with arbitrary ends (Theorem \ref{thm: 8dim Schoen conj}). Furthermore, we prove two "free of singularity type rigidity theorems" for minimal hypersurfaces with isolated singularities (Theorem \ref{prop: rigidity for minimal surface} and Theorem \ref{thm: georch free of singularity}). Our approach bypasses the need for N. Smale's regularity theorem for minimal hypersurfaces in generic $8$-dimensional compact manifolds, providing a direct derivation of the PMT for such AF manifolds (Theorem \ref{thm: pmt8dim}). Motivated by these developments, we further establish PMT for singular spaces (Theorems \ref{thm:pmt with singularity4}). These results assume only that the scalar curvature is non-negative in a strong spectral sense, a condition naturally aligned with the stability of minimal hypersurfaces in PSC ambient manifolds.
Real-time and high-performance 3D object detection plays a critical role in autonomous driving and robotics. Recent pillar-based 3D object detectors have gained significant attention due to their compact representation and low computational overhead, making them suitable for onboard deployment and quantization. However, existing pillar-based detectors still suffer from information loss along height dimension and large numerical distribution difference during pillar feature encoding (PFE), which severely limits their performance and quantization potential. To address above issue, we first unveil the importance of different input information during PFE and identify the height dimension as a key factor in enhancing 3D detection performance. Motivated by this observation, we propose a height-aware pillar feature encoder, called PillarHist. Specifically, PillarHist statistics the discrete distribution of points at different heights within one pillar with the information entropy guidance. This simple yet effective design greatly preserves the information along the height dimension while significantly reducing the computation overhead of the PFE. Meanwhile, PillarHist also constrains the arithmetic distribution of PFE input to a stable range, making it quantization-friendly. Notably, PillarHist operates exclusively within the PFE stage to enhance performance, enabling seamless integration into existing pillar-based methods without introducing complex operations. Extensive experiments show the effectiveness of PillarHist in terms of both efficiency and performance.
In this paper, we investigate the topological obstruction problem for positive scalar curvature and uniformly positive scalar curvature on open manifolds. We present a definition for open Schoen-Yau-Schick manifolds and prove that there is no complete metric with positive scalar curvature on these manifolds. Similarly, we define weak Schoen-Yau-Shick manifolds by analogy, which are expected to admit no complete metrics with uniformly positive scalar curvature.
In this paper, we demonstrate that any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$ can be foliated by a family of area-minimizing hypersurfaces, each of which is asymptotic to Cartesian coordinate hyperplanes defined at an end of $(M^n, g)$. As an application of this foliation, we show that for any asymptotically flat manifold $(M^n, g)$ with $4\leq n\leq 7$, nonnegative scalar curvature and positive mass, the solution of free boundary problem for area-minimizing hypersurface in coordinate cylinder $C_{R_i}$ in $(M^n, g)$ either does not exist or drifts to infinity of $(M^n, g)$ as $R_i$ tends to infinity. Additionally, we introduce a concept of globally minimizing hypersurface in $(M^n, g)$, and verify a version of the Schoen Conjecture.
Recently, iteration-based stereo matching has shown great potential. However, these models optimize the disparity map using RNN variants. The discrete optimization process poses a challenge of information loss, which restricts the level of detail that can be expressed in the generated disparity map. In order to address these issues, we propose a novel training approach that incorporates diffusion models into the iterative optimization process. We designed a Time-based Gated Recurrent Unit (T-GRU) to correlate temporal and disparity outputs. Unlike standard recurrent units, we employ Agent Attention to generate more expressive features. We also designed an attention-based context network to capture a large amount of contextual information. Experiments on several public benchmarks show that we have achieved competitive stereo matching performance. Our model ranks first in the Scene Flow dataset, achieving over a 7% improvement compared to competing methods, and requires only 8 iterations to achieve state-of-the-art results.
In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^{n},g)$ with scalar curvature $R_{g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to $(\mathbb{S}^{3}\times \mathbb{T}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{T}^{n-3}})$, for $4\leq n\leq 7$, then $(M^{n},g)$ is locally isometric to $\mathbb{S}^{3}\times\mathbb{T}^{n-3}$. Similar results are established for noncompact cases as $(\mathbb{S}^{3}\times \mathbb{R}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{R}^{n-3}})$ being model spaces (see Theorem \ref{noncompactrigidity1}, Theorem \ref{noncompactrigidity2}, Theorem \ref{noncompactrigidity3}, Theorem \ref{noncompactrigidity4}). We observe that the results differ significantly when $n=4$ compared to $n\geq 5$. Our results imply that the $ε$-gap length extremality of the standard $\mathbb{S}^3$ is stable under the Riemannian product with $\mathbb{R}^m$, $1\leq m\leq 4$ (see $D_{3}$. Question in Gromov's paper \cite{Gromov2017}, p.153).
We show that the hyperbolic manifold $\mathbb{H}^n/\mathbb{Z}^{n-2}$ is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound $-n(n-1)$, and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results.
One of the key problems in 3D object detection is to reduce the accuracy gap between methods based on LiDAR sensors and those based on monocular cameras. A recently proposed framework for monocular 3D detection based on Pseudo-Stereo has received considerable attention in the community. However, so far these two problems are discovered in existing practices, including (1) monocular depth estimation and Pseudo-Stereo detector must be trained separately, (2) Difficult to be compatible with different stereo detectors and (3) the overall calculation is large, which affects the reasoning speed. In this work, we propose an end-to-end, efficient pseudo-stereo 3D detection framework by introducing a Single-View Diffusion Model (SVDM) that uses a few iterations to gradually deliver right informative pixels to the left image. SVDM allows the entire pseudo-stereo 3D detection pipeline to be trained end-to-end and can benefit from the training of stereo detectors. Afterwards, we further explore the application of SVDM in depth-free stereo 3D detection, and the final framework is compatible with most stereo detectors. Among multiple benchmarks on the KITTI dataset, we achieve new state-of-the-art performance.
By using Gromov's $μ$-bubble technique, we show that the $3$-dimensional spherical caps are rigid under perturbations that do not reduce the metric, the scalar curvature, and the mean curvature along its boundary. Several generalizations of this result will be discussed.
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001} and Chruściel-Herzlich \cite{ChruscielHerzlich} respectively. Near the future null-infinity ($s=0$), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on $\mathbb{S}^{2}$ up to three order and obtain a compatibility condition for fourth order derivatives near $s=0$. We also show that if the trace free part of the second fundamental forms $\mathring A$ of this spacelike CMC surface decay fast enough then the restriction of its associate function $P$ (for definition, see \eqref{defofp} ) on the null-infinity must be a first eigenfunction of the Laplace on $\mathbb{S}^2$ or constant. In particular in Minkowski spacetime, a uniqueness result and constructions of spacelike CMC surfaces near $s=0$ are proved. Also, we show that the inner boundary of certain spacelike CMC surfaces are totally geodesic.
In this paper, we prove for $n\leq 7$ that if a differentiable $n$-manifold contains a relatively incompressible essential hypersurface in some class $\mathcal C_{deg}$, then it admits no complete metric with positive scalar curvature. Based on this result, we show for $n\leq 7$ that surgeries between orientable $n$-manifolds and $n$-torus along incompressible sub-torus with codimension no less than $2$ still preserve the obstruction for complete metrics with positive scalar curvature. As an application, we establish positive mass theorem with incompressible conditions for asymptotically flat/conical manifolds with flat fiber $F$ (including ALF and ALG manifolds), which can be viewed as a generalization of the classical positive mass theorem from \cite{SY79PMT} and \cite{SY2017}. Finally, we investigate Gromov's fill-in problem and bound the total mean curvature for nonnegative scalar curvature fill-ins of flat $2$-toruses (an optimal bound is obtained for product $2$-toruses). This confirms the validity of Mantoulidis-Miao's definition of generalized Brown-York mass in \cite{MM2017} for flat $2$-toruses.
Recently, three-dimensional (3D) detection based on stereo images has progressed remarkably; however, most advanced methods adopt anchor-based two-dimensional (2D) detection or depth estimation to address this problem. Nevertheless, high computational cost inhibits these methods from achieving real-time performance. In this study, we propose a 3D object detection method, Stereo CenterNet (SC), using geometric information in stereo imagery. SC predicts the four semantic key points of the 3D bounding box of the object in space and utilizes 2D left and right boxes, 3D dimension, orientation, and key points to restore the bounding box of the object in the 3D space. Subsequently, we adopt an improved photometric alignment module to further optimize the position of the 3D bounding box. Experiments conducted on the KITTI dataset indicate that the proposed SC exhibits the best speed-accuracy trade-off among advanced methods without using extra data.
In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces $\mathbb R^{n-1}\times \mathbb S^1$ and $\mathbb R^{n-2}\times \mathbb T^2$ respectively in dimensions no greater than $7$ (Theorem \ref{ALFPMT0}). { Different from the compatibility condition for spin structure in \cite[Theorem 2]{minerbe2008a}, we show that some type of incompressible condition for $\mathbb S^1$ and $\mathbb T^2$ is enough to guarantee the nonnegativity of the mass.} As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-existence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to $n$-torus. { Finally, we investigate certain fill-in problems and obtain an optimal bound for total mean curvature of admissible fill-ins for flat product $2$-torus $\mathbb S^1(l_1)\times \mathbb S^1(l_2)$.}
By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected $C^2$-smooth surface as initial data in asymptotically ADS-Schwarzschild manifolds with positive mass is bigger than or equal to the total mass, which is completely different from the situation in asymptotically flat case.