A note on the splitting theorem for the weighted measure

In this paper, we study complete manifolds equipped with smooth measures whose spectrum of the weighted Laplacian has an optimal positive lower bound and the m-dimensional Bakry–Émery Ricci curvature is bounded from below by some negative constant. In particular, we prove a splitting type theorem for complete smooth measure manifolds that have a finite-weighted volume end. This result is regarded as a study of the equality case of an author’s theorem (Wu, J Math Anal Appl 361:10–18, 2010).     

A splitting theorem on smooth metric measure spaces

We consider a smooth metric measure space (M, g, e ^?f dv). Let ?? _f be its weighted Laplacian. Assuming that ??₁(?? _f ) is positive and the m-dimensional Bakry-émery curvature is bounded below in terms of ??₁(?? _f ), we prove a splitting theorem for (M, g, e ^?f dv). This theorem generalizes previous results by Lam and Li-Wang (Trans Am Math Soc 362:5043?C5062, 2010; J Diff Geom 58:501?C534, 2001; see also J Diff Geom 62:143?C162, 2002).     

A splitting theorem for connected moravaK-theories

LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π_*(k(n)) ≅ ℤ _p [υ _n ] where |v _n | = 2(p ⁿ -1). We prove the following splitting theorem: Thek(n) ^*-torsion ofk(n) ^* (X) is already annihilated byv _n ^e (e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and ^r k(n) (0≤r≤e-1) where ^r k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for ^r k(n)ΛX.     

Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space

In this paper, let $(M^n,g,d\mu )$ be n-dimensional noncompact metric measure space which satisfies Poincaré inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation

${△}_{p,f}v+c{v}^{\sigma }=0,$

where $c\le 0,m>n\ge 1,1<p<\frac{m?1+\sqrt{(m?1)(m+3)}}{2},\sigma \le p?1$ are real constants.     

Interpolation theorem on Lorentz spaces over weighted measure spaces

In 1997 Ferreyra proved that it is impossible to extend the Stein-Weiss theorem in the context of Lorentz spaces. In this paper we obtain an interpolation theorem on Lorentz spaces over weighted measure spaces.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

     

A topological splitting theorem for sub-Riemannian manifolds

We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.     

A splitting theorem for holomorphic Banach bundles

This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided .     

A Liouville theorem for weighted p−Laplace operator on smooth metric measure spaces

We prove that on a smooth metric measure space with m ?Bakry–Émery curvature bounded from below by ?(m ? 1)K for some constant K ≥0 (i.e., Ric_{f ,m} ≥?(m ? 1)K ), the following degenerate elliptic equation (0.1) has no nonconstant positive solution when p > 1 and constant λ _{f ,p} satisfies Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng‐Yau's method, which was used by Wang‐Zhu in 2012 to derive a same Liouville theorem when 1 < p ≤2, Ric_{f ,m} ≥?(m ? 1)K and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.     

A radical splitting theorem for bernstein algebras

We introduce the notion of radical in Bernstein algebras and prove a splitting theorem, that is an analog of a well-known statement in classical varieties of algebras. Note that in this situation Bernstein algebras are more similar to solvable Lie and Malcev algebras (see [4], [6]) than to associative, Jordan or Binary Lie ones.

Throughout the paper all algebras and vector spaces are finite dimensional over an algebraically closed field k of characteristic 0.     

A splitting theorem for the Medvedev and Muchnik lattices

This is a contribution to the study of the Muchnik and Medvedev lattices of non‐empty Π⁰₁ subsets of 2^ω. In both these lattices, any non‐minimum element can be split, i. e. it is the non‐trivial join of two other elements. In fact, in the Medvedev case, ifP > _M Q, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ?‐theories.     

A large deviation theorem for weighted sums

Asymptotic representations are derived for large deviation probabilities of weighted sums of independent, identically distributed random variables. The main theorem generalizes a 1952 theorem of Chernoff which asserts that n ^–1 log P(S _n>cn)–log , where S _n is the partial sum of a sequence of independent, identically distributed random variables X ₁, X ₂, ... and is a constant depending on X ₁. The main result is similar in form to, but different in focus from, a particular case of Feller's (1969) theorem on large deviations for triangular arrays.This paper is based on work done for the author's doctoral dissertation written under Prof. Donald R. Truax of the University of Oregon, Eugene.     

A splitting theorem for open nonnegatively curved manifolds

We prove that an open nonnegatively curved manifold Mⁿ with soul S^k splits isometrically as S × ^2–k if (and only if) the holonomy group of the normal bundle NS is trivial.Supported by the Heinrich Hertz foundation     

A splitting theorem for groups acting on quasi-trees

It is well known that a group is free if and only if it acts freely without inversions on a tree. We prove a generalisation of this fact by defining a quasi-tree to be a graph with a bound on the size of its simple loops. It is shown that a finitely generated group acting freely on such a graph is isomorphic to a free product of free groups and finite groups.