Graph theorems for manifolds

Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds. Research supported by NSF Grants GP 8470 and GP 19221.     

Isoperimetric comparison theorems for manifolds with density

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r ^α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.     

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.     

De rham homology for networks of manifolds

This paper describes in terms of differential forms the real homology of a certain class of spaces, which we call networks. Networks include, besides smooth manifolds, singular sets of toral actions, classifying spaces of Lie groups, etc. A generalized Thom isomorphism theorem is also proved in this context.     

Rigidity and vanishing theorems for almost quaternionic manifolds

We prove the rigidity under (compatible) circle actions of several twisted Dirac operators on almost quaternionic manifolds, and the vanishing of the indices of some of them as a consequence.      

Peripheral acyclicity and homology manifolds

In this paper we study various concepts of peripheral acyclicity and local homotopical properties, generalizing and extending earlier work due to R. C. Lacher, D. R. McMillan jr. and the authors. Then we obtain some applications for cell-like decompositions of manifolds and construct exotic factors (i.e. homology manifolds) of certain Euclidean spaces.Work performed under the auspices of the G.N.S.A.G.A of Italian C.N.R. and during the permanence of the second author at the Dept. of Math. (Univ. of Modena) as a visiting professor.Supported in part by the Ministero della Università e della Ricerca Scientifica e Tecnologica within the project «Geometria Reale e Complessa».Supported in part by the Commission of the European Communities Go West Fellowship No. 792 and by the Ministry for Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-94.     

Volume and distance comparison theorems for sub-Riemannian manifolds

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in [3] and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of [3].     

Liouville-type theorems for subharmonic functions on complete manifolds

Applying the generalized maximum principle, a Liouvilletype theorem of subharmonic functions on complete Riemannian manifolds is shown and a Liouville-type differential inequality on properly immersed complete submanifolds is given. Project supported by the National Natural Science Foundation of China and K.C. Wong Educational Fund     

Chebotarev-type theorems in homology classes

We describe how closed geodesics lying in a prescribed homology class on a negatively curved manifold split when lifted to a finite cover. This generalizes a result of Zelditch in the case of compact hyperbolic surfaces.

     

Upper bound multigraphs for posets

The study of upper bound graphs of posets can be extended naturally to multigraphs. This paper characterizes such upper bound multigraphs, shows they determine the associated posets up to isomorphism, and extends results of D. Scott to characterize posets having chordal or interval upper bound multigraphs.Research partially supported by Office of Naval Research contract N00014-88-K-0163.     

Injectivity theorems on compact complex manifolds

We use analytic methods in this paper to prove some new Enoki type injectivity theorems on compact complex manifolds which generalize more or less the original Enoki injectivity theorem.     

Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt–Caffarelli–Friedman and Caffarelli–Jerison–Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace–Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.     

A note on bochner type theorems for complete manifolds

In this note we give an extension of Bochner's vanishing theorem to complete manifolds; this generalizes earlier results of J. Dodziuk, K.D. Elworthy and S. Rosenberg. This research was partially done under the E.E.C. Contract # SC 1-0105-C “GADGET” at the C.N.R.S. U.R.A. 188