Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds

We prove that if $$M^n(n\ge 4)$$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $$R_0>\sigma _{n}K_{\max }$$ , where $$\sigma _n\in (\frac{1}{4},1)$$ is an explicit positive constant depending only on $$n$$ , then $$M$$ must be isometric to a spherical space form. Moreover, we prove that if an $$n(\ge {\!\!4})$$ -dimensional compact Einstein manifold satisfies $$K_{\min }\ge \eta _n R_0,$$ where $$\eta _n\in (\frac{1}{4},1)$$ is an explicit positive constant, then $$M$$ is locally symmetric. It should be emphasized that the pinching constant $$\eta _n$$ is optimal when $$n$$ is even. We then obtain some rigidity theorems for Einstein manifolds under $$(n-2)$$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $$M$$ is an $$n(\ge {\!\!4})$$ -dimensional compact Einstein submanifold in the simply connected space form $$F^{N}(c)$$ with constant curvature $$c\ge 0$$ , and the normalized scalar curvature $$R_0$$ of $$M$$ satisfies $$R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$$ where $$A_n=n^3-5n^2+8n$$ , and $$H$$ is the mean curvature of $$M$$ , then $$M$$ is isometric to a standard $$n$$ -sphere.

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We

We prove that if a complete Riemannian [Formula: see text]-manifold with non-trivial codimension 1 homology with [Formula: see text]-coefficients or [Formula: see text]-coefficients has positive macroscopic scalar curvature large enough, then it contains a non-nullhomologous hypersurface of small Urysohn [Formula: see text]-width. This constitutes a macroscopic analogue of a theorem by Bray–Brendle–Neves on the area of non-contractible 2-spheres in a closed Riemannian 3-manifold with positive scalar curvature. Our proof is based on an adaptation of Guth’s macroscopic version of the Schoen–Yau descent argument.

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S n is a group.

Let (M, g) be an n-dimensional Riemannian manifold. We say M has k-positive Ricci curvature if at each point p ∈ M the sum of the k smallest eigenvalues of the Ricci curvature at p is positive. We say that the k-positive Ricci curvature is bounded below by α if the sum of the k smallest eigenvalues is greater than

We give a sufficient condition for a higher dimensional Kleinian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma subset-of upper I s o m left-parenthesis double-struck upper H Superscript n plus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>Isom</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \subset \operatorname {Isom} (\mathbb {H}^{n+1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be convex cocompact in terms of the critical exponent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As a consequence, we see that the fundamental group of a compact conformally flat manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally flat manifolds with positive scalar curvature.

Motivated by the work of Li and Mantoulidis [C. Li, C. Mantoulidis, Positive scalar curvature with skeleton singularities, Math. Ann. 374(1–2) (2019) 99–131], we study singular metrics which are uniformly Euclidean [Formula: see text] on a compact manifold [Formula: see text] ([Formula: see text]) with negative Yamabe invariant [Formula: see text]. It is well known that if [Formula: see text] is a smooth metric on [Formula: see text] with unit volume and with scalar curvature [Formula: see text], then [Formula: see text] is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles [Formula: see text] along codimension-2 submanifolds. We also show in three dimensions, if the Yamabe invariant of connected sum of two copies of [Formula: see text] attains its minimum, then the same is true for [Formula: see text] metrics with isolated point singularities.

In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.

Diameter estimate for closed manifolds with positive scalar curvature

We consider compact hypersurfaces in an (n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when n = 2 and is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ≥ 3 and is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product 𝕄 × ℝ1.

Positive scalar curvature on foliations: The noncompact case

As shown by Gromov–Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the non-vanishing of the [Formula: see text]-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov–Lawson surgery procedure for constructing positive scalar curvature metrics.

The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.

In this short note, we obtain an integral inequality for closed Riemannian manifolds with positive scalar curvature and give some rigidity characterization of the equality case, which generalizes the recent results of Catino which deal with the conformally flat case, and of Huang and Ma which deal with the harmonic curvature case. Moreover, we obtain an integral pinching condition with non-negative constant $$\sigma _2(A^{\tau })$$ , which can be seen as a complement to Bo and Sheng who considered conformally flat manifolds with constant quotient curvature of $$\sigma _k(A^{\tau })$$ .

In this paper we study manifolds, $X_{\Sigma}$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ is homotopy invariant under certain surgeries on $X_{\Sigma}$.