arXiv subjects
Publications and source records attributed to Pengzi Miao.
We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is also given.
Given a complete Riemannian metric of nonnegative scalar curvature on $\Sigma \times (-\infty, 0 ] $, where $\Sigma$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the boundary.
Applying a family of mass-capacity related inequalities proved in \cite{M22}, we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity.
We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $ p \to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing.
Given a metric $γ$ of nonnegative Gauss curvature and a positive function $H$ on a $2$-sphere $Σ$, we estimate the Bartnik quasi-local mass of $(Σ, γ, H)$ in terms of the area, the total mean curvature, and a quantity depending only on $γ$, measuring the roundness of the metric. If $γ$ has positive Gauss curvature, the roundness of $γ$ in the estimate is controlled by the ratio $κ$ between the maximum and the minimum of the Gauss curvature. As $κ\to 1$, the estimate approaches a sharp estimate for round spheres with arbitrary, positive mean curvature functions. Enroute we observe an estimate of the supremum of the total mean curvature among nonnegative scalar curvature fill-ins of a closed manifold with positive scalar curvature.
We study connections among the ADM mass, positive harmonic functions tending to zero at infinity, and the capacity of the boundary of asymptotically flat $3$-manifolds with nonnegative scalar curvature. First we give new formulae that detect the ADM mass via harmonic functions. Then we derive a family of monotone quantities and geometric inequalities if the underlying manifold has simple topology. As an immediate application, we observe several additional proofs of the $3$-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions that imply positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such sufficient conditions shows, if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on spatial Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by our findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio. We point out such manifolds satisfy improved inequalities, their mass has an upper bound depending only on the boundary data, there are no closed minimal surfaces enclosing the boundary, and these manifolds include static extensions in the context of the Bartnik quasi-local mass.
We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has scalar curvature lower bound -n(n-1) and is isometric to hyperbolic space outside a coordinate horosphere, then the manifold is isometric to hyperbolic space. In addition, we apply our formula to investigate regions near infinity that do not contribute to the mass quantity, which leads to improved rigidity results of hyperbolic space.
We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $\infty$ which include the scaling of any fixed coordinate-convex surface.
We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and asymptotically hyperbolic manifolds.
We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the $n$-dimensional mass as a suitable integral of geometric quantities that determine the $(n-1)$-dimensional mass.
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature.
In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.
We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.
We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.
Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat $3$-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss-Bonnet theorem.
We construct asymptotically flat, scalar flat extensions of Bartnik data $(Σ, γ, H)$, where $γ$ is a metric of positive Gauss curvature on a two-sphere $Σ$, and $H$ is a function that is either positive or identically zero on $Σ$, such that the mass of the extension can be made arbitrarily close to the half area radius of $(Σ, γ)$. In the case of $H \equiv 0$, the result gives an analogue of a theorem of Mantoulidis and Schoen, but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon $(Σ, γ)$, for any metric $γ$ with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi-Tam type metric construction from \cite{ShiTam02} and a refined Shi-Tam monotonicity, found by the first named author in \cite{Miao09}.
It is conjectured that the full (spacetime) Bartnik mass of a surface $Σ$ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by $Σ$. Assuming this holds true for a 1-parameter family of surfaces $Σ_t$ evolving in an initial data set {with the dominant energy condition}, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of $Σ_t$ is monotone non-decreasing whenever $Σ_t$ flows outward. It is our pleasure to dedicate this paper to Robert Bartnik on the occasion of his $60$th birthday.
Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.