A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e ^h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \triangle_m{\triangle_{\mu}} has a positive lower bound λ₁(M) > 0 and the m(m > n)-dimensional Bakry-émery curvature is bounded from below by -\fracm-1m-2l₁(M){-\frac{m-1}{m-2}\lambda_1(M)}, then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

Comparing the relative volume with a revolution manifold as a model

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair where ℳ is a revolution manifold and a “parallel” of ℳ. Work partially supported by a DGICYT Grant No. PB91-0324.     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.     

Geodesics On The Symplectomorphism Group

Let M be a compact manifold with a symplectic form ω and consider the group D_w{\mathcal{D}_\omega} consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L ²-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on D_w{\mathcal{D}_\omega} . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of D_w{\mathcal{D}_\omega} . Then, estimating the growth of such geodesics, we show that they extend globally.     

Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

A compact set K ì \mathbbC^N{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that

Inverse Scattering at Fixed Energy in de Sitter�CReissner�CNordstr?m Black Holes

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter–Reissner–Nordstr?m black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ ≠ 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ ≠ 0 is the fixed energy and n ? \mathbbN^*{n \in \mathbb{N}^{*}} denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M, the square of the charge Q ² and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T(λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ? L{n \in {\mathcal{L}}} where L{\mathcal{L}} is a subset of \mathbbN^*{\mathbb{N}^{*}} that satisfies the Müntz condition ?_{n ? L}\frac1n = +￥{\sum_{n \in{\mathcal{L}}}\frac{1}{n} = +\infty} . Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the “unphysical” scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities \frac1T(l,z){\frac{1}{T(\lambda,z)}}, \fracR(l,z)T(l,z){\frac{R(\lambda,z)}{T(\lambda,z)}} and \fracL(l,z)T(l,z){\frac{L(\lambda,z)}{T(\lambda,z)}} belong to the Nevanlinna class in the region ${\{z \in \mathbb{C}, Re(z) > 0 \}}${\{z \in \mathbb{C}, Re(z) > 0 \}} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.     

Quasiregularly elliptic link complements

We extend the theorem of B. Daniel about the existence and uniqueness of immersions into \mathbbSⁿ × \mathbbR or \mathbbHⁿ × \mathbbR{\mathbb{S}^{n}\,\times\,\mathbb{R}\, {\rm or}\, \mathbb{H}^{n}\,\times\,\mathbb{R}} to the Riemannian product of two space forms. More precisely, we prove the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two space forms.     

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.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

The classification of ϕ-symmetric Sasakian manifolds

A -symmetric spaceM is a complete connected regular Sasakian manifold, that fibers over an Hermitian symmetric spaceN, so that the geodesic involutions ofN lift to define global (involutive) automorphisms of the Sasakian structure onM. In the present paper the complete classification of -symmetric spaces is obtained. The groups of automorphisms of the Sasakian structures and the groups of isometries of the underlying Riemannian metrics are determined. As a corollary, the Sasakian space forms are also determined.     

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S ² ×  M ² for any closed Riemann surface M. Explicitly we show that Y (S ² ×  M ²) >  (2/3)Y(S ⁴).     

Blow-up examples for the Yamabe problem

It has been conjectured that if solutions to the Yamabe PDE on a smooth Riemannian manifold (M ⁿ , g) blow-up at a point p ? M{p \in M} , then all derivatives of the Weyl tensor W _g of g, of order less than or equal to [\fracn-62]{[\frac{n-6}{2}]} , vanish at p ? M{p \in M} . In this paper, we will construct smooth counterexamples to the Weyl Vanishing Conjecture for any n ≥ 25.     

Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation: x úy = x úzx \lor y = x \lor z and x ùy = x ùzx \land y = x \land z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M ₃ or N ₅ as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ù\land and ú\lor no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M ₃ or N ₅ that ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x úz = y úzx \lor z = y \lor z and x ùz = y ùzx \land z = y \land z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

Isometric Equivalence of Integration Operators

We are interested in the isometric equivalence problem for the Cesàro operator C(f) (z) = \frac1z ò₀^zf(x) \frac11-xd x{C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi} and an operator T_g(f)(z)=\frac1zò₀^zf(x) g^￠(x) d x{T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence problem of two operators T_g1{T_{g_{1}}} and T_g2{T_{g_{2}}} on the Hardy space and Bergman space. We show that the operators T_g1{T_{g_{1}}} and T_g2{T_{g_{2}}} satisfy T_g1U₁=U₂T_g2{T_{g_{1}}U_{1}=U_{2}T_{g_{2}}} on H ^p , 1 ≤ p < ∞, p ≠ 2 if and only if g₂(z) = lg₁(e^iqz){g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }, where λ is a modulus one constant and U _i , i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence of composition operators on the Hardy space.     

Uniform approximation of the heat kernel on a manifold

We approximate the heat kernel h(x, y, t) on a compact connected Riemannian manifold M without boundary uniformly in \((x,y,t)\in M\times M\times [a,b]\), \(a>0\), by n-fold integrals over \(M^n\) of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate corollary, we get a uniform approximation of solutions of the Cauchy problem for the heat equation on M.     

Classification and Liouville type theorems for <Emphasis Type="Italic">p</Emphasis>-harmonic morphisms

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the L ^q -energy is finite, then is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).