Singular metrics with non-negative scalar curvature and RCD

This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness. We show, in dimensions greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

In this note, we look at the difference, or rather the absence of a difference, between the space of metrics of positive scalar curvature and metrics of non-negative scalar curvature. The main tool to analyze the former on a spin manifold is the spectral theory of the Dirac operator and refinements thereof. This can be used, for example, to distinguish between path components in the space of positive scalar curvature metrics. Despite the fact that non-negative scalar curvature a priori does not have the same spectral implications as positive scalar curvature, we show that all invariants based on the Dirac operator extend over the bigger space. Under mild conditions we show that the inclusion of the space of metrics of positive scalar curvature into that of non-negative scalar curvature is a weak homotopy equivalence.

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.

We show that closed hypersurfaces in Euclidean space with non-negative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the $k$th mean curvature, for $k$ greater than 2, as we construct the counterexamples for all $k$ greater than 2. Our proof relies on a new geometric argument which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete noncompact asymptotically flat hypersurfaces with non-negative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed in our paper, Hypersurfaces with non-negative scalar curvature (J. Differential Geom., vol. 95 (2013), pp. 249–278). This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation.

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.

Let M be a noncompact manifold with nonnegative Ricci curvature outside a compact set and with nonnegative scalar curvature. First we generalize and extend the famous Kazdan-Warner and Bourguignon-Ezin condition, from compact manifolds to M. Second we show that the Yamabe problem of prescribing constant positive scalar curvature on those M with positive Yamabe invariant does not have a finite energy solution in most cases. We also prove qualitatively sharp two-sided large time estimates of Schrodinger heat kernels including those of the conformal Laplacian with long range scalar curvature.

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam’s result is known to be equivalent to the Riemannian positive mass theorem. In this paper, we provide a supplement to Shi–Tam’s result by including the boundary effect of minimal hypersurfaces. More precisely, given a compact manifold $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_H$ and $\Sigma_O$, where $\Sigma_H$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_O$ is assumed to be isometric to a suitable 2-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of mass $m$, we establish an inequality relating $m$, the area of $\Sigma_H$, and two weighted total mean curvatures of $\Sigma_O$ and $\Sigma$. In $3$-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of $\Sigma_O$ being greater than or equal to the Hawking mass of $\Sigma_H$. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in an asymptotically flat $3$-manifold $M$ into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of $M$. It follows that our result on the compact manifold $\Omega$ is equivalent to the Riemannian Penrose inequality.

In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.

Motivated by the torus stability problem, in this work, we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly.

Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and is a critical point for the L^2 norm of the full curvature R among compact perturbations of (N, g). (ii) (N, g) has non-negative scalar curvature, a free isometric S^1 action, and is a critical point of the L^2 norm of R among compact volume non-increasing perturbations of (N, g) with non-negative scalar curvature. In either case, (N, g) is flat. The Schwarzschild metric (on the space-like hypersurface) has an isometric S^1 action and satisfies the other assumptions in (ii), showing that this result is sharp.

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat3-manifold with non-negative scalar curvature which has zero total mass is isometricto (IR 3 ,δ ij ). In this paper, we quantify this statement using spinors and prove thatif a complete, asymptotically flat manifold with non-negative scalar curvature hassmall mass and bounded isoperimetric constant, then the manifold must be close to(IR 3 ,δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure. This curvatureestimate allows us to extend the case of equality of the Positive Mass Theorem toinclude non-smooth manifolds with generalized non-negative scalar curvature, whichwe define. 1 Introduction We introduce our problem in the context of General Relativity. Consider a 3 + 1 dimen-sional Lorentzian manifold N with metric g αβ of signature (− + ++). We denote theinduced Levi-Civita connection by ∇¯. Then the corresponding Ricci tensor R¯

We extend Vétois’ Obata-type argument and use it to identify a closed interval I n I_{n} , n ≥ 3 n\geq 3 , containing zero such that if a ∈ I n a\in I_{n} and ( M n , g ) (M^{n},g) is a compact conformally Einstein manifold with nonnegative scalar curvature and Q 4 + a ⁢ σ 2 Q_{4}+a\sigma_{2} constant, then it is Einstein. We also relax the scalar curvature assumption to the nonnegativity of the Yamabe constant under a more restrictive assumption on 𝑎. Our results allow us to compute many Yamabe-type constants and prove sharp Sobolev inequalities on compact Einstein manifolds with nonnegative scalar curvature. In particular, we show that compact locally symmetric Einstein four-manifolds with nonnegative scalar curvature extremize the functional determinant of the conformal Laplacian, partially answering a question of Branson and Ørsted.

An extreme limit with nonnegative scalar curvature

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature.