A simple proof on the non-existence of shrinking breathers for the Ricci flow

Suppose M is a compact n-dimensional manifold, n≥ 2, with a metric g _ij (x, t) that evolves by the Ricci flow ∂ _t g _ij = −2R _ij in M× (0, T). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. Mathematics Subject Classification (1991) Primary 58J35, 53C44 Secondary 58C99     

A variational proof of the Gauss-Bonnet formula

The classical Gauss-Bonnet formula has the form I(g_ij)=2, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation I of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric _ij we have I(_ij)=2. From this the formula I(g_ij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].     

Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

On solutions of the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space (R ⁿ , g), with n ≥ 3 and g _ij = δ _ij ε _i , ε _i = ±1, where at least one ε _i = 1 and nondiagonal tensors of the form T = Σ _ij f _ij dx _i dx _j such that, for i ≠ j, f _ij (x _i , x _j ) depends on x _i and x _j . We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R ⁿ , on the n-dimensional torus T ⁿ and on cylinders T ^k ×R ^n-k , that solve the Ricci equation or the Einstein equation. Partially supported by CAPES/PROCAD. Partially Supported By Cnpq, Capes/Procad.     

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

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.      

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Lorentz Ricci Solitons on 3-dimensional Lie groups

The three-dimensional Heisenberg group H ₃ has three left-invariant Lorentzian metrics g ₁, g ₂, and g ₃ as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g ₁ as a Lorentz Ricci Soliton. This Ricci Soliton g ₁ is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.     

Local audibility of a hyperbolic metric

A Riemannian metric g on a compact boundaryless manifold is said to be locally audible if the following statement is true for every metric g′ sufficiently close to g: if g and g′ are isospectral then they are isometric. The local audibility is proved of a metric of constant negative sectional curvature.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

Differential Harnack inequalities for heat equations with potentials under geometric flows

In the paper we consider a closed Riemannian manifold M with a time-dependent Riemannian metric g _ij (t) evolving by ? _t g _ij  = ?2S _ij , where S _ij is a symmetric two-tensor on (M,g(t)). We prove some differential Harnack inequalities for positive solutions of heat equations with potentials on (M,g(t)). Some applications of these inequalities will be obtained.     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

Structure of the Projective Group in A Pseudo-Riemannian Space

We study n-dimensional pseudo-Riemannian spaces Vⁿ(g_ij) with an arbitrary signature that admit projective motions, i.e., groups of continuous transformations preserving geodesics. In particular, we find the metric of a pseudo-Riemannian space of special type and establish important projective-group properties of this space.     

Patterns of exponential decay for solutions to second order elliptic equations in a sector ofR 2

We consider solutions ψ to equations of the form in a sector Ω ofR ². The basic assumptions are that the limitsa _ij(x)→δ_ij,b _i(x)→0,c _i→E at infinity are achieved at certain rates and thatg decays faster than ψ. We then discuss the possible patterns of exponential decay for ψ in Ω. NSERC University Research Fellow. Research partially supported by USNEF grant MCS-83-01159.     

On Isometric Immersion of Three-Dimensional Geometries $$\widetilde{SL}_2$$ , Nil,and Sol into a Four-Dimensional Space of Constant Curvature

We prove the nonexistence of an isometric immersion of the geometries Nil ³ and into a four-dimensional space M _c ⁴ of constant curvature c. We establish that the geometry Sol ³ cannot be immersed into M _c ⁴ for c ≠ −1 and find the analytic immersion of this geometry into the hyperbolic space H ⁴ (−1). __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 421–426, March, 2005.     

Recovery of Immersions from Their Metric Tensors and Nonlinear Korn Inequalities: A Brief Survey

The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω  R~n→ E~n with prescribed metric tensor field(g ij) : Ω→ S~n, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.     

Separation properties forMW-fractals

Corresponding to the irreducible 0–1 matrix (a _ij ) _n×n , take similitude contraction mappingsϕ _ij for eacha _ij =1, ina _ij =1, in R ^d with ratio 0<r _ij <1. There are unique nonempty compact setsF ₁,…,F _n satisfying for each1≤i≤n, F _i. We prove that open set condition holds if and only ifF _i is ans-set for some1≤i≤n, wheres is such that the spectral radius of matrix (r _ij ³ ) _{n x n} is 1. Partly supported by Natural Science Foundation of China, and partly by Natural Science Foundation of Hubei Province