Nonnegative Scalar Curvature Research Articles

The Obata–Vétois argument and its applications

Abstract 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.

Implications of Some Mass-Capacity Inequalities

Applying a family of mass-capacity related inequalities proved in Miao (Peking Math J 2023, https://doi.org/10.1007/s42543-023-00071-7), we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity.

Monotonicity of the p-Green Functions

Abstract On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang [ 43] in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

ADM mass for C 0 C^{0} metrics and distortion under Ricci–DeTurck flow

Abstract We show that there exists a quantity, depending only on C 0 C^{0} data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the C 0 C^{0} sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the C 0 C^{0} mass at infinity is independent of choice of C 0 C^{0} -asymptotically flat coordinate chart, and the C 0 C^{0} local mass has controlled distortion under Ricci–DeTurck flow when coupled with a suitably evolving test function.

A new proof of the Riemannian Penrose inequality

We give an overview of our recent new proof of the Riemannian Penrose inequality in the case of a single black hole. The proof is based on a new monotonicity formula, holding along the level sets of the p -capacitary potential of the connected boundary of an asymptotically flat 3 -manifold, with nonnegative scalar curvature.

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1 < p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.

An extreme limit with nonnegative scalar curvature

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the W1,p sense for p<2 to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov–Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.

Rigidity results for complete manifolds with nonnegative scalar curvature

Rigidity results for complete manifolds with nonnegative scalar curvature

Nonnegative scalar curvature on manifolds with at least two ends

AbstractLet be an orientable connected ‐dimensional manifold with and let be a two‐sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's ‐bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if does not admit a metric of psc and , then does not carry a complete metric of psc and does not carry a complete metric of uniformly psc, provided that and , respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

Mass, Capacitary Functions, and the Mass-to-Capacity Ratio

We study connections among the ADM mass, positive harmonic functions tending to zero at infinity, and the capacity of the boundary of asymptotically flat $3$-manifolds with nonnegative scalar curvature. First we give new formulae that detect the ADM mass via harmonic functions. Then we derive a family of monotone quantities and geometric inequalities if the underlying manifold has simple topology. As an immediate application, we observe several additional proofs of the $3$-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions that imply positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such sufficient conditions shows, if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on spatial Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by our findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio. We point out such manifolds satisfy improved inequalities, their mass has an upper bound depending only on the boundary data, there are no closed minimal surfaces enclosing the boundary, and these manifolds include static extensions in the context of the Bartnik quasi-local mass.

Sobolev inequalities and convergence for Riemannian metrics and distance functions

If one thinks of a Riemannian metric, $$g_1$$ , analogously as the gradient of the corresponding distance function, $$d_1$$ , with respect to a background Riemannian metric, $$g_0$$ , then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper, we study the sub-critical case $$p < \frac{m}{2}$$ where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an $$L^{\frac{p}{2}}$$ bound on a Riemannian metric implies an $$L^q$$ bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov’s conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.

Dp–convergence and 𝜖–regularity theorems for entropy and scalar curvature lower bounds

Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $\epsilon_i\to 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on. Instead, we introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $\epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,\mu_i \geq -\epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^\infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$.

Complete Self-similar Hypersurfaces to the Mean Curvature Flow with Nonnegative Constant Scalar Curvature

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. In this paper we give complete classifications of codimension one complete self-shrinkers with nonnegative constant scalar curvature. We also give alternative proofs of the classifications of codimension one translating solitons and self-expanders with nonnegative constant scalar curvature.

Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

In [52], Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be “far away” from being round. The resulting manifolds, called extensions, are geometrically not “close” to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality as discussed in [53,19]. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao [22]. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum [1] and by Cabrera Pacheco, Cederbaum, and McCormick [21], a similar construction was performed for 3-dimensional asymptotically Euclidean, electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively.This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n≥2. We study in detail the sub-extremality of these manifolds, consider the so far unstudied case of extremality in extensions with electric charge and allow more general conditions for the metric of our extensions.Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

Isometries on almost Ricci–Yamabe solitons

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere S^n(r) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with alpha ne 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space E^n or Euclidean sphere S^n. Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

First Boundary Dirac Eigenvalue and Boundary Capacity Potential

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface \(\Sigma \) bounding a noncompact domain in a spin asymptotically flat manifold \((M^n,g)\) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of \(\Sigma \) in \((M^n,g)\) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

Higher localised [formula omitted]-genera for proper actions and applications

For a finitely generated discrete group Γ acting properly on a spin manifold M, we formulate new topological obstructions to Γ-invariant metrics of positive scalar curvature on M that take into account the cohomology of the classifying space B_Γ for proper actions.In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher Aˆ-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of Γ-invariant positive scalar curvature on M. For classes arising from the subring of H⁎(B_Γ,R) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted L2-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of Wang-Wang [24] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of Mathai [17], which provided a partial answer to a conjecture of Gromov-Lawson on higher Aˆ-genera.If M is non-cocompact, we obtain obstructions to M being a partitioning hypersurface inside a non-cocompact Γ-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [12], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.

The Gauss–Bonnet inequality beyond aspherical conjecture

Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group \(H_k\) for all \(k\ge 3\), either it is flat or it has Gauss–Bonnet quantity (defined by (1.3)) no greater than \(8\pi \). In the second case, the equality for Gauss–Bonnet quantity yields that the universal covering splits as the Riemannian product of a 2-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical 2-systole estimate unifies the results from Bray et al. (Comm Anal Geom 18(4):821–830, 2010) and Zhu (Proc Am Math Soc 148(8):3479–3489, 2020).

On the Chern conjecture for isoparametric hypersurfaces

For a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, we show that if \({\rm{tr}}\left({{{\cal A}^k}} \right)\) are constants for k = 3, …, n − 1 and the shape operator \({\cal A}\) then M is isoparametric. The result generalizes the theorem of de Almeida and Brito (1990) for n = 3 to any dimension n, strongly supporting the Chern conjecture.

Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Abstract Let ( M , g T ⁢ M ) {(M,g^{TM})} be a noncompact complete Riemannian manifold of dimension n, and let F ⊆ T ⁢ M {F\subseteq TM} be an integrable subbundle of TM. Let g F = g T ⁢ M | F {g^{F}=g^{TM}|_{F}} be the restricted metric on F and let k F {k^{F}} be the associated leafwise scalar curvature. Let f : M → S n ⁢ ( 1 ) {f:M\to S^{n}(1)} be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if k F &gt; rk ⁢ ( F ) ⁢ ( rk ⁢ ( F ) - 1 ) {k^{F}&gt;{\rm rk}(F)({\rm rk}(F)-1)} on the support of d ⁢ f {{\rm d}f} , and either TM or F is spin, then inf ⁡ ( k F ) &lt; 0 {\inf(k^{F})&lt;0} . As a consequence, we prove Gromov’s sharp foliated ⊗ ε {\otimes_{\varepsilon}} -twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about Λ 2 {\Lambda^{2}} -enlargeable metrics (and/or manifolds) to the foliated case.