arXiv subjects

Haobin Yu

Publications and source records attributed to Haobin Yu.

8 recordsLinked to original sources

arXiv2026

Foliation of area-minimizing hypersurfaces in asymptotically flat manifolds of higher dimension

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})

math.DG↗

arXiv2026

Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature

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.

math.DG↗

arXiv2025

Positive mass theorems on singular spaces and some applications

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.

math.DG↗

arXiv2024

Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture

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.

math.DG↗

arXiv2020

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

Let $(M^3, g)$ be an asymptotically flat 3-manifold with positive ADM mass. In this paper, we show that each leaf of the canonical foliation is the unique isoperimetric surface for the volume it encloses. Our proof is based on the "fill-in" argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with nonnegative scalar curvature.

math.DG↗

arXiv2017

On the Rigidity of Riemannian-Penrose Inequality for Asymptotically Flat 3-manifolds with Corners

In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.

math.DG↗

arXiv2016

Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds

Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature and positive mass. We show that each leaf of the canonical foliation through stable constant mean curvature surfaces of the end of $(M, g)$ is uniquely isoperimetric for the volume it encloses.

math.DG↗

arXiv2014

Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the background manifold is sufficiently pinched and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

math.DG↗