We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length at most t; then $\lim \nolimits _{t \to \infty } P_t / ({e^{ht}}/{ht}) = 1$, where h is the entropy of the geodesic flow on the space $GX$ of parametrized unit-speed geodesics in X.

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

In this paper, from the property of Killing for structure Jacobi tensor $\mathbb {R}_{\xi }$, we introduce a new notion of cyclic parallelism of structure Jacobi operator $R_{\xi }$ on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of $R_{\xi }$ and Killing property of $\mathbb {R}_{\xi }$. Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, Minervini proved the currential “fundamental Morse equation” of Harvey–Lawson but without the restrictive tameness condition for Morse gradient flows. Here, we construct local resolutions for the flow of a section of a fiber bundle endowed with a vertical vector field which is of Morse gradient type in every fiber in order to remove the tameness hypothesis from the currential homotopy formula proved by the first author. We apply this to produce currential deformations of odd degree closed forms naturally associated to any hermitian vector bundle endowed with a unitary endomorphism and metric compatible connection. A transgression formula involving smooth forms on a classifying space for odd K-theory is also given.

We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $\mathbb {R}^{n}\times \mathbb {R}$, $n\ge 2$, of the form $(r,y(r))$ or $(r(y),y)$, where $r=|x|$, $x\in \mathbb {R}^{n}$, is the radially symmetric coordinate and $y\in \mathbb {R}$. More precisely, for any $\lambda>\frac {1}{n-1}$ and $\mu>0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$ in $\mathbb {R}$ which satisfies $r(0)=\mu $, $r^{\prime }(0)=0$ and $r(y)>yr^{\prime }(y)>0$ for any $y\in \mathbb {R}$. We will prove that $\lim _{y\to \infty }r(y)=\infty $ and $a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$ exists with $0\le a_{1}<\infty $. We will also give a new proof of the existence of a constant $y_{1}>0$ such that $r^{\prime \prime }(y_{1})=0$, $r^{\prime \prime }(y)>0$ for any $0<y<y_{1}$, and $r^{\prime \prime }(y)<0$ for any $y>y_{1}$.

In this paper we obtain some improved $L^p$-Hardy and $L^p$-Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$th mild $\partial \overline {\partial }$-lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$-Bott–Chern numbers.

For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$, we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$. Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$-symmetry.

We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim\vert z^{N/K} \mathrm{d}z \vert$, where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$, where $\mu$ is a regulated version of the L2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).

We characterise the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

Given a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$, where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\]) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\]. As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\].

We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\], we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation.

We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

We prove that any metric with curvature less than or equal to $-1$ (in the sense of A. D. Alexandrov) on a closed surface of genus greater than $1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$. The proof is by approximation, using a result about isometric immersion of smooth metrics by Labourie and Schlenker.

In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$.

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.