Manifolds Of Positive Scalar Curvature Research Articles

Contractible 3-manifolds and positive scalar curvature (II)

In this article, we are interested in the question whether any complete contractible 3- manifold of positive scalar curvature is homeomorphic to \mathbb{R}^3 . We study the fundamental group at infinity, \pi_1^\infty , and its relationship to the existence of complete metrics of positive scalar curvature. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial \pi_1^\infty 1 is homeomorphic to \mathbb{R}^3 .

Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold

Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

Positive scalar curvature and minimal hypersurface singularities

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

De Sitter space from dilatino condensates in massive IIA supergravity

We use the superspace formulation of (massive) IIA supergravity to obtain the explicit form of the dilatino terms, and we find that the quartic-dilatino term is positive. The theory admits a ten-dimensional de Sitter solution, obtained by assuming a nonvanishing quartic-dilatino condensate which generates a positive cosmological constant. Moreover, in the presence of dilatino condensates, the theory admits formal four-dimensional de Sitter solutions of the form $d{S}_{4}\ifmmode\times\else\texttimes\fi{}{M}_{6}$, where ${M}_{6}$ is a six-dimensional K\"ahler-Einstein manifold of positive scalar curvature.

Minimal hypersurfaces and bordism of positive scalar curvature metrics

Let (Y, g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen–Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume $$(Y_0,g_0)$$ and $$(Y_1,g_1)$$ are closed psc-manifolds equipped with stable minimal hypersurfaces $$X_0 \subset Y_0$$ and $$X_1\subset Y_1$$ . Under natural topological conditions, we show that a psc-bordism $$(Z,{\bar{g}}) : (Y_0,g_0)\rightsquigarrow (Y_1,g_1)$$ gives rise to a psc-bordism between $$X_0$$ and $$X_1$$ equipped with the psc-metrics given by the Schoen–Yau construction.

Remarks on GJMS operator of order six

We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n \geq 10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $\tilde g$ of $g$ such that its $Q$-curvature $Q_{\tilde g}^6$ equals $f$.

Eigenvalue Estimates of the spincDirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero.

Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-K\"ahler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.

Eigenvalue estimates for the Dirac operator on Kähler–Einstein manifolds of even complex dimension

In the case of a Kähler–Einstein manifold of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.

A Level Set Analysis of the Witten Spinor with Applications to Curvature Estimates curves of even genus

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set of small surface area. These estimates involve either a volume bound or a spectral bound for the Dirac operator on a conformal compactification, but they are independent of the isoperimetric constant.

Manifolds of Positive Scalar Curvature: A Progress Report

Manifolds of Positive Scalar Curvature: A Progress Report

An upper bound for a Hilbert polynomial on quaternionic Kähler manifolds

In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kähler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover, the article contains some details on differential equations of finite type. Part of this article is used in the proof of the main theorem.

Manifolds of positive scalar curvature and conformal cobordism theory

For a supergoup \(\), we study closed \(\)-manifolds with positive conformal classes. We use the relative Yamabe invariant from [2] to define the conformal cobordism relation on the category of such manifolds. We prove that the corresponding conformal cobordism groups \(\) are isomorphic to the cobordism groups \(\) defined by Stolz in [19]. As a corollary, we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics. Our main technical tools come from analysis and conformal geometry.

Curvature estimates in asymptotically flat manifolds of positive scalar curvature

AbstractWe consider an asymptotically flat Riemannian spin manifold of positive scalarcurvature. An inequality is derived which bounds the Riemann tensor in terms of thetotal mass and quantifies in which sense curvature must become small when the totalmass tends to zero. 1 Introduction Suppose that (M n ,g) is an asymptotically flat Riemannian spin manifold of positive scalarcurvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifoldis always positive, and is zero if and only if the manifold is flat. This result suggests thatthere should be an inequality which bounds the Riemann tensor in terms of the total massand implies that curvature must become small when the total mass tends to zero. In [4]such curvature estimates were derived in the context of General Relativity for 3-manifoldsbeing hypersurfaces in a Lorentzian manifold. In the present paper, we study the problemmore generally on a Riemannian manifold of dimension n≥ 3. Our curvature estimatesthen give a quantitative relation between the local geometry and global properties of themanifold.The main difficulty in higher dimensions is to bound the Weyl tensor (which for n= 3vanishes identically). Our basic strategy for controlling the Weyl tensor can be understoodfrom the following simple consideration. The existence of a parallel spinor in an open setU⊂ Mimplies that the manifold is Ricci flat in U. Thus it is reasonable that by gettingsuitable estimates for the derivatives of a spinor, one can bound all components of theRicci tensor. This method is used in [4], where a solution of the Dirac equation is analyzedusing the Weitzenbo¨ck formula. But the local existence of a parallel spinor does not implythat the Weyl tensor vanishes. This is the underlying reason why in dimension n>3,our estimates cannot be obtained by looking at one spinor, but we must consider a family(ψ

FANO MANIFOLDS WITH GEOMETRIC STRUCTURES MODELED AFTER HOMOGENEOUS CONTACT MANIFOLDS

In this paper we present a study on geometric structures modeled after homogeneous contact manifolds and show that on Fano manifolds these geometric structures are locally isomorphic to the standard geometric structures on the model spaces. This conclusion is analogous to those of [13, 7]. We expect that this work will help prove the conjecture that a compact quaternionic Kähler manifold of positive scalar curvature is a quaternionic symmetric space [21].

Holomorphic Spinors and the Dirac Equation

In the first part of this paper, the closed spin Kahler manifolds of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator, are characterized by holomorphic spinors. In the second part, the space of holomorphic spinors on a Kahler–Einstein manifold is described by eigenspinors of the square of the Dirac operator and vanishing theorems for holomorphic spinors are proved.

Arithmetic manifolds of positive scalar curvature

Arithmetic manifolds of positive scalar curvature

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smalles possible eigenvalue is attained are also listed. Moreover, a complete classification of the compact odd-dimensional manifolds whose universal covering space is S n−1 × R is given in the limiting case. All such manifolds are diffeomorphic but not necessarily isometric to S n−1 × S 1.

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let π : M → N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kahler structure andM a Sasakian structure. (b) Let \( \pi:{M^{2n + 1} \to N^{2n}} \) be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature \( \geqslant 4n(n + 1)\). If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold \(\left( {M^{2m} ,g} \right)\) of even complex dimension satisfies the inequality \(\left( {M^{2m} ,g} \right)\), where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T 2×N, where T 2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.