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Pengzi Miao

Publications and source records attributed to Pengzi Miao.

At least 55 records · Page 3Linked to original sources
arXiv2013

Deformation of Scalar Curvature and Volume

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

math.DG↗
arXiv2013

On second variation of Wang-Yau quasi-local energy

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We also give examples of functions H, which can be made arbitrarily close to the constant 2, on the standard sphere such that the boundary data consisting of the standard spherical metric and H has positive Brown-York mass while the associated functional is negative somewhere.

math.DG↗
arXiv2012

A New Monotone Quantity along the Inverse Mean Curvature Flow in $\mathbb R^n$

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypersurface.

math.DG↗
arXiv2011

Critical points of Wang-Yau quasi-local energy

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $Σ$ has positive Gaussian curvature and all boundary components of $Ω$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $Σ$ in $Ω$ and $H_0$ is the mean curvature of $Σ$ when isometrically embedded in $R^3$. If $Ω$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $Σ$ in $Ω$ is a strict local minimum of the Wang-Yau quasi-local energy of $Σ$, 2. on a small perturbation $\tildeΣ$ of $Σ$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tildeΣ$.

math.DG↗
arXiv2010

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(Ω, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component $Σ_i$ has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${\hat Σ}_i \subset \R^n$, then \int_{Σ_i} H d σ\le \int_{{\hat Σ}_i} \hat{H} d {\hat σ} where $H$ is the mean curvature of $Σ_i$ in $(Ω, g)$, $\hat{H}$ is the Euclidean mean curvature of ${\hat Σ}_i$ in $\R^n$, and where $d σ$ and $d {\hat σ}$ denote the respective volume forms. Moreover, equality in (\ref{eqn: main theorem}) holds for some boundary component $Σ_i$ if, and only if, $(Ω, g)$ is isometric to a domain in $\R^n$. In the proof, we make use of a foliation of the exterior of the $\hat Σ_i$'s in $\R^n$ by the $\frac{H}{R}$-flow studied by Gerhardt \cite{Gerhardt90} and Urbas \cite{Urbas90}. We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in \cite{ShiTam02}

math.DG↗
arXiv2009

Some estimates of Wang-Yau quasilocal energy

Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0 $ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $ E(Σ, X, T_0) $ itself depends on the choice of $X$, however the infimum of $ E(Σ, X, T_0)$ over $ T_0 $ does not. In particular, when $Σ$ lies in a time symmetric 3-slice in $N$ and has nonnegative Brown-York quasilocal mass $\mby(Σ)$, our estimates show that $\inf\limits_{T_0}E(Σ, X, T_0)$ equals $ \mby (Σ)$. We also study the spatial limit of $ \inf\limits_{T_0}E(S_r,X_r,T_0)$, where $S_r$ is a large coordinate sphere in a fixed end of an asymptotically flat initial data set $(M, g, p)$ and $X_r$ is an isometric embeddings of $S_r$ into $\mathbb{R}^3 \subset \mathbb{R}^{3,1}$. We show that if $(M, g, p)$ has future timelike ADM energy-momentum, then $\lim\limits_{r\to\infty}\inf\limits_{T_0}E(S_r,X_r,T_0)$ equals the ADM mass of $(M, g, p)$.

math.DG↗
arXiv2009

On geometric problems related to Brown-York and Liu-Yau quasilocal mass

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensional manifold. As an interesting by-product, we are able to write the ADM mass \cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere $S_r$ and an integral of the scalar curvature plus a geometrically constructed function $Φ(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $Σ$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $Σ$ bounds a compact spacelike hypersurface in $\R^{3,1}$, then the Liu-Yau mass of $Σ$ is strictly positive unless $Σ$ lies on a hyperplane. We also show that the examples given by Ó Murchadha, Szabados and Tod \cite{MST} are special cases of this result.

math.DG↗
arXiv2009

On a Localized Riemannian Penrose Inequality

Consider a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary is the disjoint union of two pieces: the horizon boundary and the outer boundary, where the horizon boundary consists of the unique closed minimal surfaces in the manifold and the outer boundary is metrically a round sphere. We obtain an inequality relating the area of the horizon boundary to the area and the total mean curvature of the outer boundary. Such a manifold may be thought as a region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. The inequality we establish has close ties with the Riemannian Penrose Inequality, proved by Huisken and Ilmanen, and by Bray.

math.DG↗
arXiv2009

Einstein and conformally flat critical metrics of the volume functional

Let $R$ be a constant. Let $\mathcal{M}^R_γ$ be the space of smooth metrics $g$ on a given compact manifold $Ω^n$ ($n\ge 3$) with smooth boundary $Σ$ such that $g$ has constant scalar curvature $R$ and $g|_Σ$ is a fixed metric $γ$ on $Σ$. Let $V(g)$ be the volume of $g\in\mathcal{M}^R_γ$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V(\cdot)$ in $\mathcal{M}^R_γ$.

math.DG↗
arXiv2008

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and ''small'' hyperbolic and spherical balls in dimensions 3 to 5, the standard space form metrics are indeed saddle points for the volume functional.

math.DG↗
arXiv2007

On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

math.DG↗
arXiv2006

A Note on Existence and Non-existence of Minimal Surfaces in Some Asymptotically Flat 3-manifolds

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let $g$ be a metric on the three-sphere $S^3$ satisfying $Ric(g) \geq 2 g$. If the volume of $(S^3, g)$ is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in the asymptotically flat manifold $(S^3 \setminus \{P \}, G^4 g)$. Here $G$ is the Green's function of the conformal Laplacian of $(S^3, g)$ at an arbitrary point $P$. We also give an example of $(S^3, g)$ with $Ric(g) > 0$ where $(S^3 \setminus \{P \}, G^4 g)$ does have closed minimal surfaces.

math.DG↗
arXiv2005

A Remark on Boundary Effects in Static Vacuum Initial Data sets

Let (M, g) be an asymptotically flat static vacuum initial data set with non-empty compact boundary. We prove that (M, g) is isometric to a spacelike slice of a Schwarzschild spacetime under the mere assumption that the boundary of (M, g) has zero mean curvature, hence generalizing a classic result of Bunting and Masood-ul-Alam. In the case that the boundary has constant positive mean curvature and satisfies a stability condition, we derive an upper bound of the ADM mass of (M, g) in terms of the area and mean curvature of the boundary. Our discussion is motivated by Bartnik's quasi-local mass definition.

gr-qc↗
arXiv2003

Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface

We study a class of non-smooth asymptotically flat manifolds on which metrics fails to be $C^1$ across a hypersurface $Σ$. We first give an approximation scheme to mollify the metric, then we prove that the Positive Mass Theorem still holds on these manifolds if a geometric boundary condition is satisfied by metrics separated by $Σ$.

math-ph↗
arXiv2003

Ricci Curvature Rigidity for Weakly Asymptotically Hyperbolic Manifolds

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.

math.DG↗
arXiv2003

Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity

We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the extension is dominated by but not identically equal to the one determined by the given domain, we can decrease its ADM mass while raising its boundary mean curvature. Thus our analysis implies that, for a domain with quasi-convex boundary, the geometric boundary condition holds in Bartnik's minimal mass extension conjecture \cite{Bartnik_energy}.

math-ph↗