arXiv subjects

Hongyi Sheng

Publications and source records attributed to Hongyi Sheng.

8 recordsLinked to original sources

arXiv2026

Rigidity and Classification of Manifolds with Maximal Boundary Static Kernels

Boundary static potentials arise from the adjoint equations associated with simultaneous variations of scalar curvature and boundary mean curvature. On a connected manifold of dimension $n\ge2$ with nonempty smooth boundary, the space of boundary static potentials has dimension at most $n$. We classify all complete manifolds attaining this bound. For $n\ge3$, every such manifold is isometric to a domain in a simply connected space form and has at most two boundary components. In dimension two, the classification also includes coverings of annuli and exteriors, as well as spherical Möbius bands. Finally, for $n\ge3$, the dimension hypothesis is sharp for constant-curvature rigidity, and for $n\ge2$, it is sharp for the bound on the number of boundary components.

math.DG↗

arXiv2026

Supercritical dual k-Minkowski flows with general prescribed data: a priori estimates and asymptotic convergence

Let k be an integer from 1 to n, let f be a positive smooth function on the unit sphere, and consider the normalized flow X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X, beta = C_n^k. In the supercritical range alpha > k+1, we prove the flow exists for all time and converges exponentially in C^{infty}, from every smooth strictly convex initial hypersurface enclosing the origin, to the unique smooth strictly convex solution of f(x)r^{alpha} sigma_k({kappa}) = beta u. The convergence is driven by the relative residual p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u, whose L^{infty} norm is nonincreasing and decays with the explicit exponent beta(alpha-k-1); the residual estimates are proved before, and independently of, the curvature estimates. Existence of a strictly convex solution of the stationary equation in this range, for arbitrary positive angular data, was previously obtained by Bryan-Ivaki-Scheuer through an expanding-type flow started from a barrier; the contribution here is the convergence of the normalized contracting flow from arbitrary initial data, with an explicit exponential rate, together with uniqueness of the limit.

math.DG↗

arXiv2026

Non-homogeneous curvature flows in a hemisphere

Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

math.DG↗

arXiv2026

Non-homogeneous curvature flows in hyperbolic space

Let H^{n+1} be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^{beta}r; we treat both regimes: beta>1+k{alpha} and beta=1+k{alpha}. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_{infty} respectively in two different cases.

math.DG↗

arXiv2026

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↗

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↗

arXiv2025

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↗

arXiv2025

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↗