On asymptotics of complete Ricci-flat Kähler metrics on open manifolds

Tian and Yau constructed in J. Am. Math. Soc., 3(3):579–609, 1990, a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. For the explicitly constructed referential metric ω of Tian and Yau (J. Am. Math. Soc., 3(3):579–609, 1990) we prove a property that ${\|\partial\overline\partial u\|_\omega}$ has the same decay rate as Δ _ω u provided u satisfies some decay conditions on higher Laplacians. As an application we describe the behaviour of this metric towards the boundary divisor and prove the best possible decay rate of the difference to ω.     

K?hler–Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain ${M\subset X}$ in a complex manifold carries a complete K?hler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ${\partial M}$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over ${\partial M}$ . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K?hler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of K?hler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c ₁?<?0 admits a complete K?hler–Einstein metric.     

A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations

Let ${f:\Omega \rightarrow \mathbb{R}}$ be a smooth function on a domain   ${\Omega \subset \mathbb{C}^n}$ with its Hessian matrix ${\left( \frac{\partial^2 f}{\partial z^i \partial\bar{z}^j}\right)}$ positive Hermitian. In this paper, we investigate a class of partial differential equations $$\Delta \ln \det (f_{i\bar{j}}) = \beta \;\| \text{grad} \ln \det (f_{i\bar{j}}) \|^2, $$ where Δ and ${\| \cdot \|}$ are the Laplacian and tensor norm, respectively, with respect to the metric ${G = \sum f_{i\bar{j}} \,dz^i \otimes d\bar{z}^j}$ , and β > 1 is some real constant depending on the dimension n. We prove that the above PDEs have a Bernstein property when the metric G is complete, provided that ${\det (f_{i\bar{j}})}$ and the Ricci curvature are bounded.     

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products ${\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}$ .     

Concordance and Isotopy of Metrics with Positive Scalar Curvature

Two positive scalar curvature metrics g ₀, g ₁ on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g ₀, g ₁ of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., there exists a metric ${\bar{g}}$ of positive scalar curvature on the cylinder ${M \times I}$ which extends the metrics g ₀ on ${M \times \{0\}}$ and g ₁ on ${M \times \{1\}}$ and is a product metric near the boundary. The main result of the paper is that if psc-metrics g ₀, g ₁ on M are psc-concordant, then there exists a diffeomorphism ${\Phi : M \times I \rightarrow M \times I}$ with ${\Phi|_{M \times \{0\}} = Id}$ (a pseudo-isotopy) such that the metrics g ₀ and ${(\Phi|_{M \times \{1\}})^{*}g_{1}}$ are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥  5, psc-metrics g ₀, g ₁ are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to the Gromov–Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.     

Complete gradient shrinking Ricci solitons with pinched curvature

We prove that any n-dimensional complete gradient shrinking Ricci soliton with pinched Weyl curvature is a finite quotient of ${\mathbb{R}^{n}, \mathbb{R}\times \mathbb{S}^{n-1}}$ or ${\mathbb{S}^{n}}$ . In particular, we do not need to assume the metric to be locally conformally flat.     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

Modular Curvature and Morita Equivalence

The curvature of the noncommutative torus \({T^2_\theta}\) (\({\theta \in \mathbb{R}{\setminus}\mathbb{Q}}\)) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper (Connes and Moscovici in J Am Math Soc 27:639–684, 2014) we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the C ^*-algebra \({A_\theta = C(T^2_\theta)}\) with other toric algebras \({A_{\theta'}=C(T^2_{\theta'})}\). In the enlarged context the conformal metric on \({T^2_\theta}\) is exchanged with an arbitrary Hermitian metric on the Heisenberg \({(A_\theta, A_{\theta'})}\)-bimodule E′ for which \({{\rm End}_{A_{\theta'}}(E') = A_\theta }\). We prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on E′, attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule E′ such that \({A_\theta = {\rm End}_{A_{\theta'}}(E')}\), and in this sense it is Morita invariant. To prove the above results we extend Connes’ pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.     

Extension theorems, non-vanishing and the existence of good minimal models

We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension ${\kappa (K_X+S+B)\ge 0}$ . The main new ingredient is a refinement of the Ohsawa–Takegoshi L ² extension theorem involving singular Hermitian metrics.     

Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3}

Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ .     

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ , where $B=\sum_{i=1}^{k} (\frac{\partial^{2}}{\partial x_{k}^{2}} + \frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} )$ . The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω ₁ are given so that B sublinear operators satisfies the condition (1.2) are bounded from $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ to $L_{p,\omega_{1},\gamma}(\mathbb{R}_{k,+}^{n})$ .     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure ${(\phi, \xi, \eta, g)}$ in a complex projective space ${P_{n}\mathbb{C}}$ . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of ${P_{n}\mathbb{C}}$ whose Ricci tensors S satisfy ${\nabla_{\phi\nabla_{\xi}\xi}S = 0}$ . Under some further conditions we characterize Hopf hypersurfaces of ${P_{n}\mathbb{C}}$ .     

On an Important Non-Riemannian Quantity in Finsler Geometry

In this paper, we study a non-Riemannian quantity ${\bar{{\bf E}}}$ -curvature. We prove that if F is a projectively flat Finsler metric of nonzero flag curvature, then it is Riemannian if and only if ${{\bar{\bf E}}}$ -curvature vanishes. Further, we characterize the Einstein-Douglas metrics with vanishing ${{\bar{\bf E}}}$ -curvature.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

Given an f-structure ${\varphi}$ on a manifold M, together with a compatible metric g and connection ${\nabla}$ on M, we construct an odd firstorder differential operator D whose principal symbol is of the type considered in [13]. In the special case of a CR-integrable almost ${\mathcal {S}}$ -structure, we show that when ${\nabla}$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator D is given by D = ${{\sqrt {2} (\overline {\partial}_b + \overline{\partial}_{b}^{\ast})}}$ , where ${\overline {\partial}_b}$ is the tangential Cauchy-Riemann operator. We then describe two types of “quantization” of manifolds with f-structure that reduce to familiar methods in symplectic geometry in the case that ${\varphi}$ is a compatible almost complex structure, and to the contact quantizations defined in [16] when ${\varphi}$ comes from a contact metric structure.