Bi-harmonic maps and bi-Yang-Mills fields

We study biharmonic homogeneous real hypersurfaces in complex projective space and quaternion projective space. We provide a classification of biharmonic homogeneous real hypersurfaces in quaternion projective space. We also classify pseudo-harmonic, subelliptic biharmonic, and Levi-harmonic homogeneous Hopf hypersurfaces in complex space forms.

One of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology. Since the universal covering space of a complete space of nonpositive curvature is contractible, this amounts to relations between the geometry of the space (or its universal covering space) and the algebraic structure of its fundamental group. A paradigm is the well-known result of Avez [Av] that the fundamental group of a compact Riemannian manifold M of nonpositive sectional curvature has exponential growth if and only if M is not flat. Gromov observed that the proof of Avez yields a somewhat stronger statement: the fundamental group of M is amenable if and only if M is flat, see [G1, p. 93]. Zimmer generalized this to the case where M is complete and of finite volume [Zi]. Further generalizations were obtained by Anderson [An] and Burger and Schroeder [BS]. The objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are Alexandrov spaces of nonpositive curvature. For the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature, it is convenient to study the isometric action of the fundamental group on the universal covering space. In many situations, it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature. This is the case in most of our arguments below. Let X be a Hadamard space, that is, a complete simply connected geodesic space of nonpositive curvature in the sense of Alexandrov. A flat in X is a

Minimal 2-spheres in a complex projective space

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of -designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal -designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

On stable James numbers of stunted complex or quaternionic projective spaces

In this note we study a non-existence result of bi-harmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $$\phi {:}\,(M,g)\rightarrow (N, h)$$ is a bi-harmonic map, where $$(M, g)$$ is a complete Riemannian manifold and $$(N,h)$$ a Riemannian manifold with non-positive sectional curvature. We prove: We also prove Liouville-type theorems on complete Riemannian manifolds and give applications of our non-existence result to bi-harmonic submersions.

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $${\phi : (M, g) \to (N, h)}$$ is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that $${\phi}$$ is a harmonic map if one of the following conditions holds: (i) $${|d\phi|}$$ is bounded in L q (M) and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 \leq q \leq \infty}$$ , $${1 < p < \infty}$$ ; or (ii) $${Vol(M) = \infty}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . In addition, if N has strictly negative sectional curvature, we assume that $${rank\phi(q) \geq 2}$$ for some $${q \in M}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).

The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $\mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases of biharmonic maps and biharmonic maps are the critical points of the bienergy functional. The condition of biharmonicity for non-degenerate hypersurfaces in $\mathbb{Q}^{2m+1}$ is investigated for both cases: either the characteristic vector field of $\mathbb{Q}^{2m+1}$ is the unit normal vector field to the hypersurface or it belongs to the tangent space of the hypersurface. Some relevant examples are also illustrated.

In this paper we study biharmonic hypersurfaces of semi-Euclidean spaces. We consider the Chen conjecture which states that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. In particular, we obtain some conditions for biharmonic hypersurfaces in semi-Euclidean spaces. Finally, we give some examples for different conditions.

In order to derive the best possible estimates of the total mean curvature of a compact submanifold in a Euclidean space in terms of spectral geometry, in the late 1970s, Bang-Yen Chen introduced the theory of finite-type submanifolds, which could be viewed as $$\lambda $$ -biharmonic submanifolds in the sense of $$\lambda $$ -biharmonic maps. Interestingly, Chen proposed in 1991 the following problem (Chen in Soochow J Math 17:2:169–188, 1991, Problem 12): “Determine all submanifolds of Euclidean spaces which are of null 2-type. In particular, classify null 2-type hypersurfaces in Euclidean spaces.” In this paper, we give further support evidence to the above problem. We are able to prove that a linear Weingarten null 2-type or $$\lambda $$ -biharmonic hypersurface $$M^n$$ of a Euclidean space $$\mathbb R^{n+1}$$ has constant mean curvature and constant scalar curvature provided $$n<7$$ .

We prove that in many cases the geometric dimension of the p-fold Whitney sum pHk of the Hopf bundle Hk over quaternionic projective space Qe is the smallest n such that for all / < k the reduction of the j'th symplectic Pontryagin class of pHk to coefficients t4i._1((R>/R>"_1)A bo) is zero, where bo is the spectrum for connective KO-ti\toiy localized at 2. We im-4&-H3 mediately obtain new immersions of real projective space RP in Euclidean space if the number of l's in the binary expansion of k is between 5 and 8.1. Introduction.In this paper we determine the geometric dimension (gd) of the stable class of many of the multiples pHk of the Hopf bundle Hk over quaternionic projective space QPk.This enables us immediately to obtain some new immersions of real projective spaces Rp4fc+3 in Euclidean space.Let v(2a(2b + 1)) = a, v(p, i) = v(p), and let a(n) equal the number of l's in the binary expansion of .Let P = Rf""/RF"_1 and Pn = F~.bo denotes the spectrum for real connective F-theory localized at 2 [3].it4_x(P" A bo) is a finite cyclic group ([3] or [7]).Theorem 1.1.(a) Ifgd(pHk) < , then v(p, i) > v(n4i_x(Pn A bo)) for all i < k.(b) The converse of (a) is true if n >2k and ir\_ x(Pn) - ^-^(F,, A bo) is infective for all i < k.Remark 1.2.gd(pHk) > 2k, because if the stable class of pHk contains a 2&-plane bundle, it has nonzero Euler class, since e2 = pk+ 0.By using the computations of [7] or [3] and [11], the observation that (p) is the coefficient of the symplectic Pontryagin class epHk), and an argument which is presented in 4 to eliminate one extraneous obstruction, Theorem 1.1 may be restated:

Biharmonic holomorphic maps into Kähler manifolds

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E (X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E (X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in [Formula: see text], for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space [Formula: see text] for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.

In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.