We construct biharmonic non-harmonic maps between Riemannian manifolds $(M,g)$ and $(N,h)$ by first making the ansatz that $\varphi :(M,g)\to (N,h)$ be a harmonic map and then deforming the metric on $N$ by $${\tilde{h}}_{\alpha }=\alpha h+(1-\alpha )df\otimes df$$ to render $\varphi$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N,h)$ and $\alpha \in (0,1)$. We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

We study local reflections ${\varphi }_{\sigma }$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if ${\varphi }_{\sigma }$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ${\varphi }_{\sigma }$ is harmonic for all geodesies $\sigma$.

We study local reflections ${\varphi }_{\sigma }$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if ${\varphi }_{\sigma }$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ${\varphi }_{\sigma }$ is harmonic for all geodesies $\sigma$.

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*}M,\overline{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces...

We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g,\pm \omega )$ with constant scalar curvature is either Einstein, or the dual field of $\omega$ is Killing. Next, let $(M^n,g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g,\pm \omega )$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

We generalize the concept of warped manifold to Riemannian submersions π: M → B between two compact Riemannian manifolds $(M,g_M)$ and $(B,g_B)$ in the following way. If f: B → (0,∞) is a smooth function on B which is extended to a function f̂ = f ∘ π constant along the fibres of π then we define a new metric $g_f$ on M by $g_f{|}_{\times }\equiv g_M{|}_{\times },g_f{{|}_{\times TM̂}\equiv f̂²g_M|}_{\times TM̂}$, where and denote the bundles of horizontal and vertical vectors. The manifold $(M,g_f)$ obtained that way is called a warped submersion. The function f is called a warping function. We show...

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{*}M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ and the $r$-th order cotangent $T^{r*}M=J^r{(M,R)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $\omega$-$\alpha$-Bloch space and characterize it in terms of $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|$$ and $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{\left|x\right|y-x^{\text{'}}}\right|$$ where $\omega$ is a majorant. Similar results are extended to harmonic little $\omega$-$\alpha$-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic...

The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f\left(e^{it}\right)=e^{i\phi \left(t\right)}$, $0\le t\le 2\pi$ where $\phi$ is a continuously non-decreasing function that satisfies $\phi \left(2\pi \right)-\phi \left(0\right)=2N\pi$, assume every value finitely many times in the disc?

We recall some classical results relating normality and some natural weakenings of normality in $\Psi$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda$-sets and $\sigma$-sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being ${\aleph }_0$-separated. This new class fits between $\lambda$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being...

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

For a metric continuum X, let C(X) (resp., $2^X$) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^f:2^X\to 2^Y$ be the induced functions given by $C\left(f\right)\left(A\right)=c{l}_Y\left(f\left(A\right)\right)$ and $2^f\left(A\right)=c{l}_Y\left(f\left(A\right)\right)$. In this paper, we prove that: • If $2^f$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1]...

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $𝒫$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom $D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone...

We study finite $G$-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If $M$ is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then $M$ has isospectral non-isometric covers.

A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of ${\mathbb{N}}^{*}$. One especially important application, due to Veličković, was to the existence of nontrivial involutions on ${\mathbb{N}}^{*}$. A tie-point of ${\mathbb{N}}^{*}$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost...

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then...

For odd-dimensional Poincaré–Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<n/2$) which are $C^m$ (with $m\in \mathbb{N}$) on the conformal compactification $\overline{X}$ of $X$. This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\overline{X},\partial \overline{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal infinity $(\partial \overline{X},\left[h_0\right])$. We also relate the set of $C^{n-2k+1}\left({\Lambda }^k\left(\overline{X}\right)\right)$ forms in the kernel of $d+{\delta }_g$...