On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g ₀) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g ₀) is not conformally equivalent to the unit sphere S ⁿ and that the set of all such metrics is compact. When (M, g ₀) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2. This work was supported by the Australian Research Council.     

Rigidity for closed manifolds with positive curvature

Let M be an n-dimensional complete connected Riemannian manifold with sectional curvature sec(M) ≥ 1 and radius rad(M) > π/2. In this article, we show that M is isometric to a round n-sphere if for any x ∈ M, the first conjugate locus of x is a single point and if M contains a geodesic loop of length 2 · rad(M). We also show that the same conclusion is true if the conjugate value function at any point of M is a constant function. This work was done while the author was visiting MPI for Mathematics in Leipzig, Germany. The author is very grateful to MPI for Mathematics in Leipzig for its hospitality and CAPES.     

Generalized connected sum construction for scalar flat metrics

In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M₁  \sharp_K  M₂{M = M_1 \, \sharp_K \, M_2} of two compact Riemannian scalar flat manifolds (M ₁, g ₁) and (M ₂, g ₂) along a common Riemannian submanifold (K, g _K ) whose codimension is ≥3. Here we present two constructions: the first one produces a family of “small” (in general nonzero) constant scalar curvature metrics on the generalized connected sum of M ₁ and M ₂. It yields an extension of Joyce’s result for point-wise connected sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1₊) class in the Kazdan–Warner classification, we refine the first construction in order to produce a family of scalar flat metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact time symmetric initial data sets, extending the Isenberg–Mazzeo–Pollack gluing construction.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p)?× U(n ? p)

We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

Flat Riemannian Manifolds as Torus Bundles

Vasquez showed that for any finite group G there exists a number n(G) such that for every flat Riemannian manifold M with holonomy group G there exists a fiber bundle T → M → N, where T is a flat torus and N is a flat manifold of dimension less than or equal to n(G). We show that n(H) ≤ n(G) if H Δ leftG or G = N ? H and use this result to describe groups with the Vasquez number equal to 2 or 3.     

Compactness for conformal metrics with constant Q curvature on locally conformally flat manifolds

In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n ≥ 5 and with Poincaré exponent less than , the set of conformal metrics of positive constant Q and positive scalar curvature is compact in the C∞ topology.     

The Jump of the Laplacian on a Submanifold

Assume that a submanifold M ? ?ⁿ of an arbitrary codimension k ? {1, …, n} is closed in some open set O→?ⁿ. With a given function u ? C²(O\M) we may associate its trivial extension u: O→? such that u|_O\M=u and u|m ≡ 0. The jump of the Laplacian of the function u on the submanifold M is defined by the distribution Δu — Δu. By applying some general version of the Fubini theorem to the nonlinear projection onto M we obtain the formula for the jump of the Laplacian (Theorem 2.2).     

Contact metric manifolds with <Emphasis Type="Italic">η</Emphasis>-parallel torsion tensor

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E ⁿ⁺¹ × S ⁿ (4) in higher dimensions.      

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δ _f , acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to Δ _f . As a consequence, we construct a large family of pairwise Δ _f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than . Supported by Conicet, Secyt-UNC.     

On non-negatively curved metrics on open five-dimensional manifolds

Let V ⁿ be an open manifold of non-negative sectional curvature with a soul Σ of co-dimension two. The universal cover of the unit normal bundle N of the soul in such a manifold is isometric to the direct product M ^n-2 × R. In the study of the metric structure of V ⁿ an important role plays the vector field X which belongs to the projection of the vertical planes distribution of the Riemannian submersion on the factor M in this metric splitting . The case n = 4 was considered in [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] where the authors prove that X is a Killing vector field while the manifold V ⁴ is isometric to the quotient of by the flow along the corresponding Killing field. Following an approach of [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] we consider the next case n = 5 and obtain the same result under the assumption that the set of zeros of X is not empty. Under this assumption we prove that both M ³ and Σ³ admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres, the vector field X is Killing, while the whole manifold V ⁵ is isometric to the quotient of by the flow along corresponding Killing field. Supported by the Faculty of Natural Sciences of the Hogskolan i Kalmar (Sweden).     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

Sparse Dynamic Programming for Longest Common Subsequence from Fragments

Sparse Dynamic Programming has emerged as an essential tool for the design of efficient algorithms for optimization problems coming from such diverse areas as computer science, computational biology, and speech recognition. We provide a new sparse dynamic programming technique that extends the Hunt–Szymanski paradigm for the computation of the longest common subsequence (LCS) and apply it to solve the LCS from Fragments problem: given a pair of strings X and Y (of length n and m, respectively) and a set M of matching substrings of X and Y, find the longest common subsequence based only on the symbol correspondences induced by the substrings. This problem arises in an application to analysis of software systems. Our algorithm solves the problem in O(|M| log |M|) time using balanced trees, or O(|M| log log min(|M|, nm/|M|)) time using Johnson's version of Flat Trees. These bounds apply for two cost measures. The algorithm can also be adapted to finding the usual LCS in O((m + n) log |Σ| + |M| log |M|) time using balanced trees or O((m + n) log |Σ| + |M| log log min (|M|, nm/|M|)) time using Johnson's version of Flat Trees, where M is the set of maximal matches between substrings of X and Y and Σ is the alphabet. These bounds improve on those of the original Hunt–Szymanski algorithm while retaining the overall approach.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.