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Yuguang Shi

Publications and source records attributed to Yuguang Shi.

At least 37 records · Page 2Linked to original sources
arXiv2021

Nonexistence of NNSC-cobordism of Bartnik data

In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data $(Σ_1^{n-1}, γ_1, H_1)$ and $(Σ_2^{n-1}, γ_2, H_2)$. We prove that given two metrics $γ_1$ and $γ_2$ on $S^{n-1}$ ($3\le n\le 7$) with $H_1$ fixed, then $(S^{n-1}, γ_1, H_1)$ and $(S^{n-1}, γ_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{highdimnoncob0}). Moreover, we show that for $n=3$, a much weaker condition that the total mean curvature $\int_{S^2}H_2dμ_{γ_2}$ is large enough rules out NNSC cobordisms(Theorem \ref{2-d0}); if we require the Gaussian curvature of $γ_2$ to be positive, we get a criterion for non existence of trivial NNSC-cobordism by using Hawking mass and Brown-York mass(Theorem \ref{cobordism20}). For the general topology case, we prove that $(Σ_1^{n-1}, γ_1, 0)$ and $(Σ_2^{n-1}, γ_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{highdimnoncob10}).

math.DG↗
arXiv2020

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question \ref{extension1}). Then we introduce a fill-in invariant (see Definition \ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov's conjectures formulated in \cite{Gro19} (see Conjecture \ref{conj0} and Conjecture \ref{conj1} below)

math.DG↗
arXiv2020

NNSC-Cobordism of Bartnik Data in High Dimensions

In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are $(n-1)$-dimensional Bartnik data $\big(Σ_i ^{n-1}, γ_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(Ω^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial Ω=Σ_{1} \cup Σ_{2}$ such that $γ_i$ is the metric on $Σ_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $Σ_i$ in $\big(Ω^n, g\big)$). If $\big(\mathbb{S}^{n-1},γ_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(Σ_1 ^{n-1}, γ_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, γ_{\rm std}\big)$ denotes the standard round unit sphere then $\big(Σ_1 ^{n-1}, γ_1, H_1\big)$ admits an NNSC fill-in. Just as Gromov's conjecture is connected with positive mass theorem, our problems are connected with Penrose inequality, at least in the case of $n=3$. Our third problem is on $Λ\big(Σ^{n-1}, γ\big)$ defined below.

math.DG↗
arXiv2019

On the Fill-in of Nonnegative Scalar Curvature Metrics

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Σ,γ,H)$. We prove that given a metric $γ$ on $\mathbf{S}^{n-1}$ ($3\leq n\leq 7$), $(\mathbf{S}^{n-1},γ,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem \ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $γ$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $\mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $\int_{\mathbf S^{n-1}}H\,\mathrm dμ_γ$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.\,23 in \cite{Gromov4}). In the second part of this paper, we investigate the $θ$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

math.DG↗
arXiv2018

Characterization of large isoperimetric regions in asymptotically hyperbolic initial data

Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on work of A.~Neves and G.~Tian and of the first-named author, we show that the leaves of the canonical foliation of $(M, g)$ are the unique solutions of the isoperimetric problem for their area. The assumption $R \geq -6$ is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.

math.DG↗
arXiv2017

Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u) $ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

math.AP↗
arXiv2016

Scalar curvature and singular metrics

Let $M^n$, $n\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $σ(M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$, smooth outside a compact set $Σ$, and is in $W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $σ(M)$ outside $Σ$. We prove that $g_0$ is Einstein outside $Σ$ if the codimension of $Σ$ is at least $2$. If in addition, $g_0$ is Lipschitz then $g_0$ is smooth and Einstein after a change the smooth structure. If $Σ$ is a compact embedded hypersurface, and $g_0$ is smooth up to $Σ$ from two sides of $Σ$, and if the difference of the mean curvatures along $Σ$ at two sides of $Σ$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $Σ$. For manifolds with dimension between $3$ and $7$ without spin assumption, we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least $2$.

math.DG↗
arXiv2016

Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R\geq -6$

In this paper, aimed at exploring the fundamental properties of isoperimetric region in $3$-manifold $(M^3,g)$ which is asymptotic to Anti-de Sitter-Schwarzschild manifold with scalar curvature $R\geq -6$, we prove that connected isoperimetric region $\{D_i\}$ with $\mathcal{H}_g ^3(D_i)\geq δ_0>0$ cannot slide off to the infinity of $(M^3,g)$ provided that $(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region $\{D_i\}$ with topological sphere $\{\partial D_i\}$ as boundary is exhausting regions of $M$ if Hawking mass $m_H(\partial D_i)$ has uniform bound. In the case of exhausting isoperimetric region, we obtain a formula on expansion of isoperimetric profile in terms of renormalized volume.

math.DG↗
arXiv2014

Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant.

math.DG↗
arXiv2013

Gauss map of translating solitons of mean curvature flow

In this short note we study Bernstein's type theorem of translating solitons whose images of their Gauss maps are contained in compact subsets in an open hemisphere of the standard $\mathbf{S}^n$ (see Theorem 1.1). As a special case we get a classical Bernstein's type theorem in minimal submanifolds in $\mathbf{R}^{n+1}$ (see Corollary 1.2).

math.DG↗
arXiv2012

On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature

In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be asymptotically locally hyperbolic. One interesting application of this theorem is to show a rigidity result for the hyperbolic space under an integral condition for the curvature.

math.DG↗
arXiv2011

Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity

In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum $(M^n,g,V),$ we also compute the asymptotic expansions of $g$ and $V$ at conformal infinity.

math.DG↗
arXiv2011

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in \cite{GL} \cite{Lee} \cite{Bi} of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.

math.DG↗