اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the Fill-in of Nonnegative Scalar Curvature Metrics
96
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 $(Sigma,gamma,H)$. We prove that given a metric $gamma$ on $mathbf{S}^{n-1}$ ($3leq nleq 7$), $(mathbf{S}^{n-1},gamma,H)$ 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 $gamma$ 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 $int_{mathbf S^{n-1}}H,mathrm dmu_gamma$ 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 $theta$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
قيم البحث
اقرأ أيضاً
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 r
ef{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 Gromovs conjectures formulated in cite{Gro19} (see Conjecture ref{conj0} and Conjecture ref{conj1} below)
In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m e -1$) of scalar flag curvature must be locally Minkowskian in
dimension $nge 3$. We characterize by some PDEs a Kropina metric ($m=-1$) which is respectively of scalar flag curvature and locally projectively flat in dimension $nge 3$, and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.
Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ w
ith unit volume and with scalar curvature $R(g)ge sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $leq 2pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^infty$ metrics with isolated point singularities.
Extending Aubins construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, tinmathbb{R}$. In particul
ar, there are no topological obstructions for metrics with $varepsilon$-pinched Weyl curvature and negative scalar curvature.
In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulat
ions of $mathbb{S}^3$; namely, the boundary complexes of cyclic polytopes. We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
