Given a homogeneous pseudo-Riemannian space \((G/H,\langle \ , \ \rangle),\) a geodesic \(\gamma :I\rightarrow G/H\) is said to be two-step homogeneous if it admits a parametrization \(t=\phi (s)\) (s affine parameter) and vectors X, Y in the Lie algebra \({\mathfrak{g}}\), such that \(\gamma (t)=\exp (tX)\exp (tY)\cdot o\), for all \(t\in \phi (I)\). As such, two-step homogeneous geodesics are a natural

We give a lower bound on the boundary injectivity radius of the Margulis tubes with smooth boundary constructed by Buser, Colbois, and Dodziuk. This estimate depends on the dimension and a curvature bound only.

In this note, we prove that a special family of Killing potentials on certain Hirzebruch complex surfaces, found by Futaki and Ono [18], gives rise to new conformally Kähler, Einstein–Maxwell metrics. The correspondent Kähler metrics are ambitoric [7, 9] but they are not given by the Calabi ansatz [31]. This answers in positive questions raised in [18, 19].

We study toric nearly Kähler manifolds, extending the work of Moroianu and Nagy. We give a description of the global geometry using multi-moment maps. We then investigate polynomial and radial solutions to the toric nearly Kähler equation.

We find two different families of \(\mathbf{Sp}(4,\mathbb{R})\) symmetric \(G_2\) structures in seven dimensions. These are \(G_2\) structures with \(G_2\) being the split real form of the simple exceptional complex Lie group \(G_2\). The first family has \(\tau _2\equiv 0\), while the second family has \(\tau _1\equiv \tau _2\equiv 0\), where \(\tau _1\), \(\tau _2\) are the celebrated \(G_2\)-invariant

Shortly after the introduction of Seiberg–Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact Kähler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun’s Theorem is no longer true for non-Kähler surfaces. In particular, we show that the Yamabe invariants of Inoue surfaces and their blowups are all zero. We also take this opportunity to record

This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally 2-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces

As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment L, the surface is invariant under the \(180^\circ\)-rotation with respect to L. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong

The aim of this paper is to extend classic results of the theory of constant mean curvature surfaces in the product space \({\mathbb {H}}^2\times {\mathbb {R}}\) to the class of immersed surfaces whose mean curvature is given as a \(C^1\) function depending on their angle function. We cover topics such as the existence of a priori curvature and height estimates for graphs and a structure-type result

We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those \(\text{G}_2\)-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold. In the process, we

In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action

We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of \(L^p\) solutions with \(0< p< 1\) and improves the \(L^1\) uniqueness result of Li (J Differ Geom 20:447–457, 1984) by weakening the curvature assumption.

In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products \({{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n\) and \({{\mathbb {R}}}^m \ltimes \mathrm

We construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a gluing construction for Ricci-positive manifolds with corners that gives a sufficient condition

Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds.

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the pointwise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand-new

This paper is devoted to the role played by the Higgs algebra \(H_3\) in the generalisation of classical harmonic analysis from the sphere \(S^{m-1}\) to the (oriented) Grassmann manifold \({{\text {Gr}}}_o(m,2)\) of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group \({\text {SO}}(m)\) acting on functions on the Grassmannian. This is then

We present a novel approach to the study of Yang–Mills instantons on quaternionic Kähler manifolds, based on an extension of the harmonic space method of constructing instantons on hyperkähler manifolds. Our results establish a bijection between local equivalence classes of instantons on quaternionic Kähler manifolds M and equivalence classes of certain holomorphic maps on an appropriate \({\mathrm

We give a shorter proof of the well-posedness of the Laplacian flow in \({\rm G}_2\)-geometry. This is based on the observation that the DeTurck–Laplacian flow of \({\mathrm{G}}_2\)-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of (not necessarily closed) \({\mathrm{G}}_2\)-structures, which fits in the general framework introduced by Hamilton

In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the given minimisation problem. Further we generalise Arnold’s results concerning the problem of finding the minimum magnetic energy in an orbit of the group of volume-preserving

In the original article the formula chapter 4, just above Remark 4.2 in the paragraph starting with Case (2) the formula should read.

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest

We point out that Ecker’s local integral quantity agrees with Huisken’s global integral quantity at infinity for ancient mean curvature flows if Huisken’s one is finite on each time-slice. In particular, this means that the finiteness of Ecker’s integral quantity at infinity implies the finiteness of the entropy at infinity.

Let X be a connected smooth complex projective variety of dimension \(n \ge 1\). Let D be a simple normal crossing divisor on X. Let G be a connected complex Lie group, and \(E_G\) a holomorphic principal G-bundle on X. In this article, we give criterion for existence of a logarithmic connection on \(E_G\) singular along D.

In this paper, we classify three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function.

We prove that every conformal submersion from a round sphere onto an Einstein manifold with fibers being geodesics is—up to an isometry—the Hopf fibration composed with a conformal diffeomorphism of the complex projective space of appropriate dimension. We also show that there are no conformal submersions with minimal fibers between manifolds satisfying certain curvature assumptions.

In this paper, we study the prescribing Webster scalar curvature problem on strictly pseudoconvex CR manifolds of real dimension \(2n+1\). First, we study the CR Nirenberg problem and prove some existence results. Second, we provide the existence and a positive lower bound for a solution of the CR Yamabe problem with zero Webster scalar curvature on noncompact complete manifolds.

We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing

In this paper, we continue to consider Willmore Legendrian surfaces and csL Willmore surfaces in \({\mathbb {S}}^5\), notions introduced by Luo (Calc Var Partial Differ Equ 56, Art. 86, 19, 2017. https://doi.org/10.1007/s00526-017-1183-z). We will prove that every complete Willmore Legendrian surface in \({\mathbb {S}}^5\) is minimal and find nontrivial examples of csL Willmore surfaces in \({\mathbb

In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical n-dimensional minimal submanifolds with index of relative nullity \(n-2\), fully described by Dajczer and Florit (Ill J Math 45:735–755, 2001) in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal

We consider the Sampson Laplacian acting on covariant symmetric tensors on a Riemannian manifold. This operator is an example of the Lichnerowicz-type Laplacian. It is of fundamental importance in mathematical physics and appears in many problems in Riemannian geometry including the theories of infinitesimal Einstein deformations, the stability of Einstein manifolds and the Ricci flow. We study the

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable s in some right-half plane of \({\mathbb {C}}\). In Spilioti (Ann Glob Anal Geom 53(2):151–203, 2018), it was proved that they admit a meromorphic continuation to the whole complex plane.

We prove some existence and nonexistence results for complete area minimizing surfaces in the homogeneous space \({\mathbb {E}}(-1,\tau )\). As one of our main results, we present sufficient conditions for a curve \(\Gamma\) in \(\partial _{\infty } {\mathbb {E}}(-1,\tau )\) to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in \({\mathbb

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: Can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? By demonstrating that the Poisson relation is an equality for a generic bi-invariant metric on a compact Lie group, we establish that the length spectrum of a generic bi-invariant metric on a compact

If a Spin(7)-manifold \(N^8\) admits a free \(S^1\) action preserving the fundamental 4-form, then the quotient space \(M^7\) is naturally endowed with a \(G_2\)-structure. We derive equations relating the intrinsic torsion of the Spin(7)-structure to that of the \(G_2\)-structure together with the additional data of a Higgs field and the curvature of the \(S^1\)-bundle; this can be interpreted as

We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we generalize Leray–Hirsch, Künneth and Poincaré–Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen and Yang (Ann Global Anal Geom 56:277–290, 2019).

We present a diamond shaped diagram for even Clifford manifolds similar to the quaternion-Kähler diamond studied by Ch. Boyer and K. Galicki. We define two spaces \({\mathcal {S}}\) and \({\mathcal {U}}\) which fiber over an even Clifford manifold which, together with the twistor space \({\mathcal {Z}}\) defined by G. Arizmendi and Ch. Hadfield, form a diamond shaped diagram of fibrations. Moreover

Let \(\Gamma\) be a nonelementary discrete subgroup of \({\mathrm {Sp}}(n,1)\). We show that if the trace skew-field of \(\Gamma\) is commutative, then \(\Gamma\) stabilizes a copy of complex hyperbolic subspace of \({\mathbf {H}}^n_{{\mathbb {H}}}\).

Given a smooth closed oriented manifold M of dimension n embedded in \({\mathbb {R}}^{n+2}\), we study properties of the ‘solid angle’ function \(\varPhi :{\mathbb {R}}^{n+2}{{\setminus }} M\rightarrow S^1\). It turns out that a non-critical level set of \(\varPhi\) is an explicit Seifert hypersurface for M. This gives an explicit analytic construction of a Seifert surface in higher dimensions.

Let \(\pi : {\mathbb {S}}^{2n+1}\rightarrow {\mathbb {C}}P^n\) be the Hopf map and let \(\phi\) be a totally real immersion of a \(k(\ge 3)\)-dimensional simply connected manifold \(\Sigma\) into \({\mathbb {C}}P^n\). It is well known that there exists an isotropic lift \({\overline{\phi }}\) into \({\mathbb {S}}^{2n+1}\) preserving the second fundamental form. Using this isotropic lift, we obtain

In this paper, we prove higher-order Minkowski-type formulas for closed real hypersurfaces in complex space forms. As applications, we obtain several integral inequalities.

We give a proof of the classical Schwarz reflection principle for Jenkins–Serrin type minimal surfaces in the homogeneous three manifolds \(\mathbb E(\kappa ,\tau )\) for \(\kappa \leqslant 0\) and \(\tau \geqslant 0\). In our previous paper, we proved a reflection principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.

For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete.

The wrong funding number has been given in the acknowledgements section. It should be read: Nos. ZZ201818 and LZD014.

We study the shape of an elastic rod subject to both bending and twisting, when the rod’s resistance to bending depends on the direction of the deformation. In this sense, we develop the theory of anisotropic rods in the plane and in space.

Almost para-quaternionic structures on smooth manifolds of dimension 2n are equivalent to almost Grassmannian structures of type (2, n). We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves

We study the geometry of a stable drop of incompressible liquid constrained to lie in a spherical container. The energy functional is comprised of the surface tension, the wetting energy and the line tension. It is shown that the only stable equilibrium drops having the topology of a disc are flat discs and spherical caps. Sharp conditions for the stability of equilibrium spherical caps and flat discs

The Kreck–Stolz \(s\) invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the \(s\) invariant for metrics on \(S^n\) bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely

We construct an infinite-dimensional Lie rackoid Y which hosts an integration of the standard Courant algebroid. As a set, \(Y={{\mathcal {C}}}^{\infty }([0,1],T^*M)\) for a compact manifold M. The rackoid product is by automorphisms of the Dorfman bracket. The first part of the article is a study of the Lie rackoid Y and its tangent Leibniz algebroid, a quotient of which is the standard Courant algebroid

In this paper, we will first build an \(L^2\) Dolbeault lemma by analytic methods and Hörmander \(L^2\) estimates. Then as applications, we will prove some log Nadel type vanishing theorems on compact Kähler manifolds and some log Kawamata–Viehweg type vanishing theorems on projective manifolds. Some log Nakano–Demailly type vanishing theorems for vector bundles will be also discussed by the same methods;

We introduce a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups \(\mathbf{SO }(n)\), \(\mathbf{SU }(n)\), \(\mathbf{Sp }(n)\), \(\mathbf{SL }_{n}({\mathbb {R}})\), \(\mathbf{Sp }(n,{\mathbb {R}})\), \(\mathbf{SU }(p,q)\), \(\mathbf{SO }(p,q)\), \(\mathbf{Sp }(p,q)\), \(\mathbf{SO }^*(2n)\) and \(\mathbf{SU

Conditions for the existence of Kähler–Einstein metrics and central Kähler metrics (Maschler in Trans Am Math Soc 355:2161–2182, 2003) along with examples, both old and new, are given on classes of Lorentzian 4-manifolds with two distinguished vector fields. The results utilize the general construction (Aazami and Maschler in Kähler metrics via Lorentzian geometry in dimension four, Complex Manifolds

Thomas Friedrich, who founded the journal ‘Annals of Global Analysis and Geometry’ together with Rolf Sulanke in 1983 and acted as editor in chief for more than 3 decades, died in Marburg (Germany) on February 27, 2018, at the age of sixty-eight of COPD and lung cancer. Besides sketching Thomas’s biography and scientific work, it is our goal in this obituary to tell the founding story of ‘his’ journal

Podestà and Spiro (Osaka J Math 36(4):805–833, 1999) introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kähler structure (g, J) (“standard G-manifolds”) and studied invariant Kähler and Kähler–Einstein metrics on M. In the first part of this paper, we gave a combinatoric description of the standard non-compact G-manifolds

[This corrects the article DOI: 10.1007/s10455-016-9514-4.].

Let \((M_i, g_i)_{i \in \mathbb {N}}\) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov–Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator \(\mathcal {D}^B\) on B. We give an explicit description