arXiv subjects

Hongyi Sheng

Publications and source records attributed to Hongyi Sheng.

4 recordsLinked to original sources

arXiv2026

Obata-Type Rigidity on Static Manifolds with Boundary

We investigate static metrics on simple manifolds with compact boundary and establish an Obata-type rigidity theorem. We identify new sufficient geometric conditions under which the combined curvature map $g\mapsto (R_g, H_g)$ is a local surjection. Consequently, we demonstrate that in contrast to manifolds without boundary, where staticity obstructs deformability, the scalar curvature map can be locally surjective at static metrics on manifolds with boundary.

math.DG↗

arXiv2024

Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature

Let $(M, g)$ be a compact Riemannian manifold with boundary $\partial M$. Given a function $f$ on $\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$ on $\partial M$. Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions.

math.DG↗

arXiv2024

Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere.

math.DG↗

arXiv2024

Localized Deformation of the Scalar Curvature and the Mean Curvature

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. This result is a generalization of Corvino's result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the linearized problem is non-variational. We also discuss generic conditions that guarantee localized deformations, and related geometric properties.

math.DG↗