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Yuchen Bi

Publications and source records attributed to Yuchen Bi.

17 recordsLinked to original sources
arXiv2026

Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends

We characterize equality in the mass-capacity inequality for asymptotically flat manifolds with arbitrary ends in every dimension \(n\ge 3\). If \(u\) is the capacity minimizer, then the equality forces \((M,u^{4/(n-2)}g)\) to be isometric to \((\mathbb R^n\setminus S, g_{\mathrm{Euc}})\) for a compact set \(S\) satisfying \(\operatorname{dim}_{\mathcal H}S\le (n-2)/2\). A key ingredient is that conformal changes by positive harmonic functions transform nonnegative Ricci curvature into nonnegative Bakry--\'Emery Ricci curvature of effective dimension \(4-n\).

math.DG↗
arXiv2026

Positive Scalar Curvature Obstructions via Singular Dimension Descent

In light of recent advances in conformal blow-up methods for the positive mass theorem, including He--Shi--Yu, Bi--Hao--He--Shi--Zhu, and Brendle--Wang, we develop a Schoen--Yau type singular dimension descent method for positive scalar curvature obstructions in arbitrary dimensions. We prove obstructions to positive scalar curvature on enlargeable manifolds and establish the corresponding cubical width inequalities and two-systole estimates. The method also applies to enlargeable AM--PI spaces, giving a positive scalar curvature obstruction when the singular set has Assouad codimension greater than \(3-2/n\).

math.DG↗
arXiv2026

Curvature-free effects from volume growth and ends-counting and their applications

In this paper, we investigate two curvature-free effects from volume growth and ends-counting, respectively. Motivated by generalizing classical results from Ricci curvature to other common curvatures, we establish two main theorems. First, any complete non-compact manifold with lower sublinear volume growth admits a smooth bounded mean-concave exhaustion. Second, any complete manifold with infinitely many ends contains escaping geodesic lines outside every compact subset. As applications, we provide new proofs of the Calabi--Yau minimal volume growth theorem and the Cai--Li--Tam finite-ends theorem for nonnegative Ricci curvature, without relying on the Bishop--Gromov volume comparison theorem or analytic tools specific to Ricci curvature. We further extend these results to Riemannian manifolds with nonnegative scalar curvature and K\"ahler manifolds with positive holomorphic sectional curvature.

math.DG↗
arXiv2026

Riemannian Penrose inequality in all dimensions

We prove the Riemannian Penrose inequality in arbitrary dimension for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary, where the boundary is allowed to have a singular set of Hausdorff dimension at most \(n-8\). Moreover, the equality holds exactly when the manifold is isometric to the Riemannian Schwarzschild exteriors. Our proof extends Bray's conformal-flow method to higher dimensions, where the outer-minimizing enclosures along the flow may be singular.

math.DG↗
arXiv2026

A proof for the Riemannian positive mass theorem up to dimension 19

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 $12$.

math.DG↗
arXiv2026

Linear Quantitative Rigidity for Almost-CMC Surfaces

We prove a quantitative rigidity result for almost constant mean curvature spheres in $\mathbb{R}^3$. Under a sub--two--sphere Willmore bound and a small $L^2$--CMC defect, we show that an almost--CMC surface is close to the round sphere, with linear control of the $W^{2,2}$--distance of the parametrization and the $L^\infty$--norm of the conformal factor. An analogous statement holds under an a priori area bound below that of two spheres.The proof relies on a linearized analysis around the sphere. A previously established qualitative rigidity result provides the initial closeness required to enter the perturbative regime. The estimate further extends to integral $2$--varifolds of unit density using known regularity and density results.

math.DG↗
arXiv2025

Quantitative Stability of the Clifford Torus as a Willmore Minimizer

For an integral $2$-varifold $V\subset \mathbb{S}^3$ with square-integrable mean curvature, unit density, and support of genus at least $1$, assume that its Willmore energy satisfies \[ \mathcal{W}(V)\le 2\pi^2+\delta^2,\qquad \delta<\delta_0\ll1. \] We show that the support $\Sigma=\operatorname{spt}V$ is, after applying a suitable conformal transformation of $\mathbb{S}^3$, quantitatively close to the Clifford torus. More precisely, under an appropriate conformal normalization, the surface $\Sigma$ admits a $W^{2,2}$ conformal parametrization by the flat torus whose conformal factor and metric coefficients differ from those of the Clifford torus by at most $C\delta$.

math.DG↗
arXiv2025

Mass-capacity inequality modeled on conformally flat manifolds

In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.

math.DG↗
arXiv2025

Deblending Overlapping Galaxies in DECaLS Using Transformer-Based Algorithm: A Method Combining Multiple Bands and Data Types

In large-scale galaxy surveys, particularly deep ground-based photometric studies, galaxy blending is inevitable and poses a potential primary systematic uncertainty for upcoming surveys. Current deblenders predominantly rely on analytical modeling of galaxy profiles, facing limitations due to inflexible and imprecise models. We present a novel approach using a U-net structured transformer-based network for deblending astronomical images, which we term the CAT-deblender. It was trained using both RGB and grz-band images, spanning two distinct data formats from the Dark Energy Camera Legacy Survey (DECaLS) database, including galaxies with diverse morphologies. Our method requires only the approximate central coordinates of each target galaxy, bypassing assumptions on neighboring source counts. Post-deblending, our RGB images retain a high signal-to-noise peak, showing superior structural similarity to ground truth. For multi-band images, the ellipticity of central galaxies and median reconstruction error for the r-band consistently lie within +/-0.025 to +/-0.25, revealing minimal pixel residuals. In our comparison focused on flux recovery, our model shows a mere 1 percent error in magnitude recovery for quadruply blended galaxies, significantly outperforming SExtractor's higher error rate of 4.8 percent. By cross-matching with publicly accessible overlapping galaxy catalogs from the DECaLS database, we successfully deblended 433 overlapping galaxies. Furthermore, we demonstrated effective deblending of 63,733 blended galaxy images randomly selected from the DECaLS database.

astro-ph.GA↗
arXiv2024

Anomalous diffusion in quantum system driven by heavy-tailed stochastic processes

In this paper, we study a stochastically driven non-equilibrium quantum system where the driving protocols consist of hopping and waiting processes. The waiting times between two hopping processes satisfy a heavy-tailed distribution. By calculating the squared width of the wavepackets, our findings demonstrate the emergence of various anomalous transport phenomena when the system remains unchanged within the heavy-tailed regime, including superdiffusive, subdiffusive, and standard diffusive motion. Only subdiffusion occurs when the system has evolved during the waiting process. All these transport behaviors are accompanied by a breakdown of ergodicity, highlighting the complex dynamics induced by the stochastic driving mechanism.

cond-mat.str-el↗
arXiv2024

Optimal rigidity estimates for varifolds almost minimizing the Willmore energy

For an integral $2$-varifold $V=\underline{v}(\Sigma,\theta_{\ge 1})$ in $\mathbb{R}^n$ with generalized mean curvature $H\in L^2$ such that $\mu(\mathbb{R}^n)=4\pi$ and $\int_{\Sigma}|H|^2d\mu\le 16\pi(1+\delta^2)$ , we show that $\Sigma$ is $W^{2,2}$ close to the standard embedding of the round sphere in a quantitative way when $\delta< \delta_0\ll 1$. For $n=3$, we prove that the sharp constant is $\delta_0^2=2\pi$.

math.DG↗
arXiv2022

Bi-Lipschitz rigidity for $L^2$-almost CMC surfaces

For smooth surfaces properly immersed in the unit ball of $\RR^n$ with density close to one and small Willmore energy, the optimal a priori estimate(bi-Lipschitz and $W^{2,2}$ parametrization)is provided. We also discuss the quantitative rigidity for $L^2$-almost CMC surfaces.

math.DG↗
arXiv2022

Bi-Lipschitz Regularity of 2-Varifolds with the Critical Allard Condition

For an intergral $2$-varifold $V=\underline{v}(\Sigma,\theta_{\ge 1})$ in the unit ball $B_1$ passing through the original point, assuming the critical Allard condition holds, that is, the area $\mu_V(B_1)$ is close to the area of a unit disk and the generalized mean curvature has sufficient small $L^2$ norm, we prove $\Sigma$ is bi-Lipschitz homeomorphic to a flat disk in $\mathbb{R}^2$ locally.

math.DG↗
arXiv2022

The $C^0$-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estimate of Li \cite{Li}. In this paper, we will prove a $C^0$-convergence result in this boundary blow-up process. Our method is different from \cite{Zhang,Gluck}.

math.AP↗
arXiv2021

The Prescribed $Q$-Curvature Flow for Arbitrary Even Dimension in a Critical Case

In this paper, we study the prescribed $Q$-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold $(M,g)$, which was introduced by S. Brendle in \cite{B2003}, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and $\int_M Qd\mu < (n-1)!\Vol\left( S^n \right) $. In this paper we study the critical case that $\int_M Qd\mu = (n-1)!\Vol\left( S^n \right)$, we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in \cite{LLL2012} in dimensiona 4 and extend the work of Li-Zhu \cite{LZ2019} in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs.

math.AP↗