Einstein 4-manifolds and nonpositive isotropic curvature

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.     

Noncompact 4-manifolds with uniformly positive isotropic curvature

In this paper, we study Ricci flow on noncompact 4-manifolds with uniformly positive isotropic curvature and with no essential imcompressible space form. That means there is positive lower bound of isotropic curvature and bounded geometry. Then by Perelman's technique, we can analyze the structures of such manifolds.     

A note on scalar curvature type problems in dimension 4

Let (M,g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem Δgu+a(x)u=f(x)u³ always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem.     

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Let (M, g) be a 4-dimensional Einstein Riemannian manifold. At each point p of M, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p. In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O(4) acts as a transformation group between ST bases at T _p M and for the so-called 2-stein curvature tensors, the group Sp(1) ? SO(4) acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of SO(4) which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups SO(2), T ², Sp(1) or U(2) are obtained and the classes of curvature tensors whose transformation group into new ST bases is one of the mentioned groups are determined.     

Four-manifolds with positive isotropic curvature

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.     

Besse conjecture with positive isotropic curvature

Annals of Global Analysis and Geometry - The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of...     

The prescribed curvature problem in dimension four

Necessary and sufficient conditions for a g-valued differential 2-form on a 4-dimensional manifold to be, locally, a curvature form, are given. The dimension four is exceptional for the problem of prescribed curvature as, in this dimension, Bianchi's identities can be eliminated for a large class of Lie algebras, including semisimple algebras. Hence, the curvature forms are characterized as the solutions to a second-order partial differential system, which is proved to be formally integrable.     

A uniform estimate for scalar curvature equation on manifolds of dimension 4

We consider the solutions to the prescribed scalar curvature equation on a four-dimensional Riemannian manifold M. We prove an upper bound for the supremum of all the solutions on every compact subset K of M, provided that all the solutions on M are bounded from below by a positive number.     

Positive scalar curvature and minimal hypersurfaces

We show that the minimal hypersurface method of Schoen and Yau can be used for the ``quantitative' study of positive scalar curvature. More precisely, we show that if a manifold admits a metric with or , where is the scalar curvature of , any 2-tensor on and the Weyl tensor of , then any closed orientable stable minimal (totally geodesic in the second case) hypersurface also admits a metric with the corresponding positivity of scalar curvature. A corollary pertaining to the topology of such hypersurfaces is proved in a special situation.

     

On compact manifolds with positive isotropic curvature

In this paper we construct new Riemannian metrics with positive isotropic curvature on compact manifolds which fiber over the circle. We also study the relationship between the positivity of the isotropic curvature and the positivity of the -curvature.

     

Weyl curvature and the Euler characteristic in dimension four

We give lower bounds, in terms of the Euler characteristic, for the L²-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups which can act properly discontinuously, isometrically, and cocompactly on hyperbolic -space in terms of the combinatorics of the action of on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the -sphere.

     

Symmetry and self-duality in nonlinear programming

This paper is concerned with a pair of naturally symmetric problems related by duality. Self-duality has been investigated for the class of non-differentiable convex programs.     

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L ² harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L ² harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic manifold can not admit any almost K?hler structure of positive isotropic curvature.     

Positive curvature and eigenfunctions of the Laplacian

Examples are given of noncompact Riemannian manifolds having nonnegative Ricci cuvature and infinitely many square integrable eigenfunctions for the Laplace operator.

     

Positive curvature,symmetry, and topology

We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.     

Positive scalar curvature in dim⩾8

We announce a first series of new results and techniques extending the scope of applications of minimal hypersurfaces in scalar curvature geometry. For instance, the restriction to dimensions ?7 which arises from subtle analytic problems in higher dimensions is entirely removed. To cite this article: J. Lohkamp, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Local Poincaré inequalities in non-negative curvature and finite dimension

In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres.