In this paper, we study the eigenvalue of p-Laplacian on finite graphs. Under generalized curvature dimensional condition, we obtain a lower bound of the first nonzero eigenvalue of p-Laplacian. Moreover, a upper bound of the largest p-Laplacian eigenvalue is derived.

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σk-curvature in the interior and constant Hk-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type

In this paper, we provide a necessary and sufficient condition for a set in PSL(2,R) or in T1H2 to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane H2.

Let M be a real hypersurface in complex projective space. The almost contact metric structure on M allows us to consider, for any nonnull real number k, the corresponding k-th generalized Tanaka-Webster connection on M and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator Rξ

We classify and explicitly describe maximal antipodal sets of some compact classical symmetric spaces and those of their quotient spaces by making use of suitable embeddings of these symmetric spaces into compact classical Lie groups. We give the cardinalities of maximal antipodal sets and we determine the maximum of the cardinalities and maximal antipodal sets whose cardinalities attain the maximum

A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of SO(3) on R6, which can be used effectively to describe

We construct 1-parameter families of well known solutions to the Yamabe problem defined on maximal flag manifolds to determine bifurcation instants for these families looking at changes of the Morse index of these metrics when the parameter varies on the interval [0,1]. A bifurcation point for such families is an accumulation point of others solutions to the Yamabe problem and a local rigidity point

In this paper, we investigate a duality between Hermitian and almost Kähler structures on the tangent manifold TM induced by pairs of conjugate connections on its base, affine Riemannian manifold M. In the context of information geometry, the classical theory of statistical manifold (which we call S-geometry) prescribes a parametrized family of probability distributions with a Fisher-Rao metric g and

We prove that the structure Jacobi operator Rξ on a real hypersurface M in the complex projective space CP2 satisfies Rξ+κϕ2=0 with κ∈R⁎ (or equivalently, constant Reeb sectional curvature) if and only if either M is of type (A) or M is locally congruent to a non-homogeneous Hopf hypersurface with vanishing Hopf principal curvature.

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover,

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show S3 does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration

Let M13(c) be an 3-dimensional Lorentz space form and C(M13(c)) denote the conformal transformation group of M13(c). A spacelike surface x:M2→M13(c) is called a conformal homogeneous spacelike surface. If there exists a subgroup G⊂C(M13(c)) such that the orbit G(p)=x(M2),p∈x(M2). In this paper, we classify completely conformal homogeneous spacelike surfaces up to a conformal transformation of M13(c)

We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to S1 has infinitely many geometrically distinct geodesics joining any given pair of points p and q. In the special case in which β1(M;Z)≥1 and M is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.

Donaldson-Friedman constructed anti-self-dual classes on K3#3CP2‾ using twistor space. We show that some of these conformal classes have almost-Kähler representatives.

A submanifold Mn of a Euclidean space EN is called biharmonic if ΔH→=0, where H→ is the mean curvature vector of Mn. A well known conjecture of B.Y. Chen states that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. Ideal submanifolds were introduced by Chen as those which receive the least possible tension at each point. In this paper we prove that every δ(r)-ideal biharmonic

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature η–invariants for the family of Berger metrics on the odd dimensional spheres.

In this paper, we mainly investigate curvature properties and harmonicity of invariant vector fields on the four-dimensional Oscillator groups endowed with three left-invariant pseudo-Riemannian metrics of signature (2,2). We determine all harmonic vector fields, vector fields which define harmonic maps and the vector fields which are critical points for the energy functional restricted to vector fields

This note studies invariant generators for a certain class of invariant smooth generalized subbundles of exact Courant algebroids, which generalizes existing results for invariant vector subbundles of exact Courant algebroids. As a result, we provide a simple and geometric proof for Theorem 1 in [1].

We investigate the holomorphic differential operators on a Riemann surface M. This is done by endowing M with a projective structure. Let L be a theta characteristic on M. We explicitly describe the jet bundle Jk(E⊗L⊗n), where E is a holomorphic vector bundle over M equipped with a holomorphic connection, for all k and n. This provides a description of global holomorphic differential operators from

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥5. Denote Lg the Paneitz-Branson type operator. In this paper, we show that there exists a nodal solution (solution with changing sign) of the nonlinear Paneitz-Branson type equation Lgv=ϵ|v|N−2v where ϵ<0. At the end, we give a geometric application of the above equation.

In this paper we prove that, under certain conditions, in a quasi-Einstein semi-Riemannian warped product the fiber is necessarily an Einstein manifold. We provide all the quasi-Einstein manifolds when r-Bakry-Emery tensor is null, the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n−1)-dimensional translation group and the fiber is Ricci-flat. As an

We prove that a non-totally vicious n-dimensional compact spacetime (M,g) admitting a parallel lightlike vector field is foliated by compact totally geodesic null hypersurfaces. As a consequence, assuming non-negative Ricci curvature on the leaves then the first Betti number of M is bounded above by n with equality if and only if M is diffeomorphic to the torus.

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and

The spherical Fourier transform on a harmonic Hadamard manifold (Xn,g) of positive volume entropy is studied. If (Xn,g) is of hypergeometric type, namely spherical functions of X are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric

We discuss natural transformations in the context of Lie groupoids, and their infinitesimal counterpart. Our main result is an integration procedure that provides smooth natural transformations between Lie groupoid morphisms.

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a

In this paper, we study the cross curvature soliton. We study the cross curvature soliton with a warped product structure. On the other hand, we show that the volume entropy is decreasing along the cross curvature flow.

In this paper, we classify n-dimensional (n≥4) gradient shrinking Ricci solitons with bounded Ricci curvature. Precisely, we obtain that such a soliton satisfying Ric+Hessf=λg with 0≤Ric≤λng is a finite quotient of the Gaussian shrinking soliton Rn. In the case of dimension 4, we prove that a radially flat gradient shrinking Ricci soliton with 0≤Ric≤λ2g is a finite quotient of R4 or R2×S2.

Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotient N˜11=(Sp(2)×S4)/S3, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotient N˜11/S3, we construct a transnormal system with transnormal hypersurfaces diffeomorphic

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.

The aim of this paper is to investigate the mean curvature flow soliton solutions on the Heisenberg group H when the initial data is a ruled surface by straight lines. We give a family of those solutions which are generated by Iso0(H) (the isometries of H for which the origin is a fix point). We conclude that the function which describe the motion of these surfaces under MCF, is always a linear affine

We review the derived algebraic geometry of derived zero loci of sections of vector bundles, with particular emphasis on derived critical loci. In particular we some of the structures carried by derived critical loci: the homotopy Batalin-Vilkovisky structure, the action of the 2-monoid of the self-intersection of the zero section, and the derived symplectic structure of degree −1. We also show how

Consider a closed non-degenerate 3-form ω with an infinitesimal action of a Lie algebra g. Motivated by the fact that the observables associated to ω form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra g. We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal

A contravariant pseudo-Hessian manifold is a manifold M endowed with a pair (∇,h) where ∇ is a flat connection and h is a symmetric bivector field satisfying a contravariant Codazzi equation. When h is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifolds and, in fact, to any contravariant pseudo-Hessian

We found that we need to revise some proofs in the paper “Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor”. Hence, we correct them in the present paper.

We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of dω and its Hodge dual ⁎dω, where ω is the fundamental 2-form of the nearly Kähler structure. The proof is based on a fundamental integrability condition for conformal Killing

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n

We define an operator ΔJ on any almost Hermitian manifold by first projecting the gradient of a function to the tangent space of its level set, and then taking the divergence. We call ΔJ the J-Laplacian operator. It is then shown that J-harmonic functions, which are functions H satisfying ΔJH=0, are first integrals of the vector field δω# where ω is the fundamental 2-form. This generalizes the notion

We establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n-sphere. This result is used to give conditions under which a minimizing geodesic flower degenerates into a simple closed geodesic.

We investigate the distribution of lengths of closed geodesics on complete and simply connected Sasakian space forms of dimension greater than 1. When its constant ϕ-sectional curvature k is irrational and satisfies k>−9/4 or k<−4, we can see that two closed geodesics on this manifold are congruent to each other in strong sense by an isometry of this manifold if and only if they have a common length

In this paper, our purpose is to obtain uniqueness results related to the mean curvature equation for entire Killing graphs constructed over the base Pn of a warped product of the type Pfn×ρR with warping function ρ and density f. For this, we establish a suitable f-parabolicity criterion and, under appropriate constraints on the Bakry-Émery-Ricci tensor and on the f-mean curvature, we prove some rigidity

Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ\G/H is by definition a quotient of G/H by a discrete uniform subgroup Γ≤G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, Γ\G/H is an S1-Seifert bundle over a locally homogeneous aspherical Kähler orbifold. We discuss the structure

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce the concepts of these submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.

In this note we provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.

The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow evolving on [0,T), whose first eigenvalue λ0 of the operator −△+(1−(n−1)ρ)24(1−2(n−1)ρ)R for the initial metric g(0) is positive, or T>0 is finite, an upper bound

By combining the ideas of Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, we develop an efficient method for solving a wide variety of equivalence problems. The key is a pseudo-group analog of the classic result that characterizes congruence of submanifolds in Lie groups in terms of equivalence of the Lie group's Maurer-Cartan forms. This result, when combined

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in S4 and derive a necessary condition for such (possibly branched) isotropic surfaces to descend to (possibly branched) maps from RP2 to S4. The Veronese sphere and several other examples

In this article we consider solvable hypersurfaces of the form Nexp⁡(RH) with induced metrics in the symmetric space M=SL(3,C)/SU(3), where H a suitable unit length vector in the subgroup A of the Iwasawa decomposition SL(3,C)=NAK. Since M is rank 2, A is 2-dimensional and we can parametrize these hypersurfaces via an angle α∈[−π/2,π/2] determining the direction of H. We show that one of the hypersurfaces

We consider the Laplacian coflow of a G2-structure on warped products of the form M7=M6×fS1 with M6 a compact 6-manifold endowed with an SU(3)-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the SU(3)-structure on M6 and the warping function f. Necessary and sufficient conditions for the existence of solution for this flow

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.

In this paper, we focus on almost regular Douglas (α,β)-spaces. We find that any almost regular Douglas (α,β)-metrics on a manifold M of dimension n>2 must be Berwald metric, if they are weak Landsberg metric.