We prove a unique continuation theorem for non-minimal biharmonic hypersurfaces of spheres, based on Aronszajn’s 1957 article. Under the right hypotheses, this result shows that, for these immersions, CMC on an open subset implies globally CMC. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

Recently, the blow-up formulae of cohomologies on complex manifolds have been extensively studied. The purpose of this paper is to give a proof for the blow-up formula of integral Bott–Chern cohomology on compact complex manifolds in terms of relative Dolbeault sheaves.

We study the Steklov problem on a subgraph with boundary \((\Omega ,B)\) of a polynomial growth Cayley graph \(\Gamma\). For \((\Omega _l, B_l)_{l=1}^\infty\) a sequence of subgraphs of \(\Gamma\) such that \(|\Omega _l| \longrightarrow \infty\), we prove that for each \(k \in {\mathbb {N}}\), the kth eigenvalue tends to 0 proportionally to \(1/|B|^{\frac{1}{d-1}}\), where d represents the growth rate

It is shown that on every closed oriented Riemannian 4-manifold (M, g) with positive scalar curvature, $$\begin{aligned} \int _M|W^+_g|^2d\mu _{g}\ge 2\pi ^2(2\chi (M)+3\tau (M))-\frac{8\pi ^2}{|\pi _1(M)|}, \end{aligned}$$ where \(W^+_g\), \(\chi (M)\) and \(\tau (M)\), respectively, denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M. This generalizes Gursky’s inequality

In this paper, we show that any convex affine domain with a nonempty limit set on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension ≤ 5.

For a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s

We show that the spherical equilateral triangle of diameter \(\frac{\pi }{2}\) is a strict local minimizer of the fundamental gap on the space of the spherical triangles with diameter \(\frac{\pi }{2}\), which partially extends Lu-Rowlett’s result–(Commun Math Phys 319(1): 111–145, 2013) from the plane to the sphere.

In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle \(\mathbb {S}^1\) that intuitively represents the length of the portion of curve which tangent vector points in a certain direction

In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin \(j+1/2\) over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power \(j+1\) and understand how the spinor fields with spin \(j+1/2\) are related to the spinors with lower

In this paper, we introduce the notion of taut contact hyperbola on three-manifolds. It is the hyperbolic analogue of the taut contact circle notion introduced by Geiges and Gonzalo (Invent. Math., 121: 147–209, 1995), (J. Differ. Geom., 46: 236–286, 1997). Then, we characterize and study this notion, exhibiting several examples, and emphasizing differences and analogies between taut contact hyperbolas

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left-invariant deformations \(\{M_t\}_{t\in B}\)

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and

We study the dynamics of magnetic flows on Heisenberg groups, investigating the extent to which properties of the underlying Riemannian geometry are reflected in the magnetic flow. Much of the analysis, including a calculation of the Mañé critical value, is carried out for \((2n+1)\)-dimensional Heisenberg groups endowed with any left invariant metric and any exact, left-invariant magnetic field. In

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge–Wheeler and Zerilli equations obtained recently by Anderson et al. (Ann. Henri Poincaré 21:61–813, 2020) are sufficient to obtain the conformal

We give a new construction of the holonomy groupoid of a regular foliation in terms of a partial connection on a diffeological principal bundle of germs of transverse parametrisations, which may be viewed as a systematisation of Winkelnkemper’s original construction using ideas from gauge theory. We extend these ideas to construct a novel holonomy groupoid for any foliated bundle, which we prove sits

We investigate the maximal open domain \({\mathscr {E}}(M)\) on which the orthogonal projection map p onto a subset \(M\subseteq {{\mathbb {R}}}^d\) can be defined and study essential properties of p. We prove that if M is a \(C^1\) submanifold of \({{\mathbb {R}}}^d\) satisfying a Lipschitz condition on the tangent spaces, then \({\mathscr {E}}(M)\) can be described by a lower semi-continuous function

Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized B-opers, where the successive quotients of the oper filtration are allowed to have

We study the Anomaly flow on 2-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. In the case of flat holomorphic bundle, the general solution to the Anomaly flow is given for any initial invariant Hermitian metric. The solutions depend on two constants \(K_1\) and \(K_2\), and we study the qualitative behaviour of the Anomaly flow in terms of their signs, as well as

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on \(\mathbb {R}^2\times S^2\) and

The fluid ball conjecture states that a static perfect fluid space-time is spherically symmetric. In this paper we construct a Robinson’s divergence formula for the static perfect fluid space-time. Inspired by this conjecture, a rigidity result for the spatial factor of a static perfect fluid space-time satisfying some boundary conditions is proved, provided that an equation of state holds.

We classify all rotational symmetric solutions of the singular minimal surface equation in both cases \(\alpha <0\) and \(\alpha >0\). In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger–Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem associated with various energy functionals in the setting of mappings between singular metric spaces. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, (1) we extend the existence result

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable

Let X be a compact Kähler manifold. Let \(T_1, \ldots , T_m\) be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of \(T_1, \ldots , T_m\). We recover the well-known non-pluripolar product of \(T_1, \ldots , T_m\) when T is the current of integration along X. Our main results are a monotonicity property

We construct, for \(p>n\), a concrete example of a complete non-compact n-dimensional Riemannian manifold of positive sectional curvature which does not support any \(L^p\)-Calderón–Zygmund inequality: $$\begin{aligned}\Vert {{\,\mathrm{Hess}\,}}\varphi \Vert _{L^p}\le C(\Vert \varphi \Vert _{L^p}+\Vert \Delta \varphi \Vert _{L^p}), \qquad \forall \,\varphi \in C^{\infty }_c(M). \end{aligned}$$ The

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below \(8\pi \). In particular, every constrained Willmore torus with Willmore energy below \(8\pi \) and non-rectangular conformal class is non-degenerated.

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric

We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram–Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated with the Gauss hyper-sphere. By using sequences of inscribed polygonals with

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 Kahler 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 Kahler 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 Kahler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer true for non-Kahler 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 $$180^\circ$$ 180 ∘ -rotation with respect to L . However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces