Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with their induced metrics. In this paper, we provide a sufficient condition to determine whether a real analytic Kähler manifold is not a relative to a complex space form equipped with its canonical metric. As an application, we show that minimal domains, bounded homogeneous domains and some Hartogs domains equipped

This article shows that for generic choice of Riemannian metric on a compact oriented manifold M of dimension four, the tangent planes at any self-intersection \(p \in M\) of any prime closed parametrized minimal surface in M are not simultaneously complex for any orthogonal complex structure on M at p. This implies via geometric measure theory that \(H_2(M;{{\mathbb {Z}}})\) is generated by homology

Let Ω and Ω′ be open subsets of a flat (2,3,5)-distribution. We show that a C1-smooth contact mapping f: Ω → Ω′ is a \(C^\infty\)-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group. (The Tanaka prolongation of the Lie algebra is of finite type.) The conclusion is reached through a careful study of some differential identities satisfied by

We show that every 3-\((\alpha ,\delta )\)-Sasaki manifold of dimension \(4n + 3\) admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature \(16n(n+2)\alpha \delta\). In the non-degenerate case we describe all homogeneous 3-\((\alpha ,\delta )\)-Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If \(\alpha

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler–Einstein metric whose differential has a constant length. Then, we will construct

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally occurring, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane—as a subject—is new

We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded 1-manifolds diffeomorphic to \({\mathbb {R}}\), which approach the zero set

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504). We introduce

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

We exploit conformal transformations of gluing formulas to realize connections between zeta functions of Laplacians and associated Dirichlet-to-Neumann map zeta functions. Furthermore, the geometric content in gluing formulas is identified and explicit results are given for a three-dimensional manifold.

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain \(\Omega \), with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue

The authors gratefully acknowledge the support of the Grants GA19-06357S, GAUK 700217 and SVV-2017-260456.

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor \(\psi \) is a spinor that satisfies the equation \(\nabla _X\psi =AX\cdot \psi \) with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

In this paper, we present a proof of Schauder estimate on Euclidean space and use it to generalize Donaldson’s Schauder estimate on space with conical singularities in the following two directions. The first is that we allow the total cone angle to be larger than 2\(\pi \) and the second is that we discuss higher-order estimates.

We study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics \((g_a,\{J_\alpha \}_{\alpha =1}^3)\) on the domain Y of the standard quaternion space \({\mathbb {H}}^n\) one of which, say \((g_a,J_1)\) is a Bochner flat Kähler metric. To do so, we deform conformally the standard

Let \((T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times \cdots \times S_{r_k}^1, dt_1^2+dt_2^2+\cdots +dt_k^2)\) be flat tori, \(r_k\ge \cdots \ge r_2\ge r_1>0\) and \(({\mathbb {R}}^n,g_E)\) the Euclidean space with the flat metric. We compute the isoperimetric profile of \((T^2\times {\mathbb {R}}^n, h_2+g_E)\), \(2\le n\le 5\), for small and big values of the volume. These computations give explicit lower

We give a method to resolve four-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.

We begin the study of completeness of affine connections, especially those on statistical manifolds or on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.

We consider a shape optimization problem for the first mixed Steklov–Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton’s shell theorem.

We establish the following Hadamard–Stoker-type theorem: Let \(f:M^n\rightarrow \mathscr{H} ^{\,\,\, n}\times \mathbb{R}\) be a complete connected hypersurface with positive definite second fundamental form, where \(\mathscr{H} ^{\,\,\, n}\) is a Hadamard manifold. If the height function of f has a critical point, then it is an embedding and M is homeomorphic to \(\mathbb{S}^n\) or \(\mathbb{R}^n.\)

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