Paracontact Metric    κ , μ   -Manifold Satisfying the Miao-Tam Equation

Dehe Li,Jiabin Yin,Remi Léandre

doi:10.1155/2021/6687223

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

Dehe Li, Jiabin Yin + Show 1 more

Open Access

https://doi.org/10.1155/2021/6687223

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Abstract

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

Highlights

Inspired by the positive mass theorem and the variational characterization of Einstein metrics on a closed manifold, with an aim to find a proper concept of metrics that would sit between constant scalar curvature metrics and Einstein metrics, in [1], Miao and Tam studied the variational properties of the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric
Let Ω be a compact n -dimensional Riemannian manifold with smooth boundary Σ, γ be a given metric on Σ, and MKγ be the space of metrics on Ω which have constant scalar curvature K and have induced metric on Σ given by γ
Some explicit examples of Miao-Tam critical metrics can be found in [1, 2], including the standard metrics on geodesic balls in space forms but the spatial Schwarzschild metrics and AdS-Schwarzschild metrics restricted to certain domains containing their horizon and bounded by two spherically symmetric spheres

Summary

Introduction

Inspired by the positive mass theorem and the variational characterization of Einstein metrics on a closed manifold, with an aim to find a proper concept of metrics that would sit between constant scalar curvature metrics and Einstein metrics, in [1], Miao and Tam studied the variational properties of the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric They derived the following sufficient and necessary condition for a metric to be a critical point: Theorem 1 (Theorem 5 in [1]). In [2], the authors classified all Einstein and conformally flat MiaoTam critical metrics They proved that any connected, compact, Einstein manifold with smooth boundary satisfying Miao-Tam critical condition is isometric to a geodesic ball in a connected space form. ΛÞ is a nonconstant solution of the Miao-Tam equation, M2n+1 is locally flat in dimension 3, and in higher dimensions ðn > 1Þ, it is locally isometric to the product of a flat ðn + 1Þ -dimensional manifold and an n-dimensional manifold of negative constant curvature equal to −4

Preliminaries

The Proof of Theorem 2